JP3845029B2 - Nonlinear optimal solution search system - Google Patents

Description

【００５０】
ここで、関数ｆは、αがα1''とα2''との間で、ｆ0の値をとる場合があり、そのときのαをα1'''とすると、α1''、α2''をとおり、傾きｍを持つ直線Ｋの方程式をｇ(α)とし、(式１)を活用すると次式で表せる。
【００５１】
【数２】

【００５２】
αがα1''のときにｇ(α1'')＝ｆ1''の条件を用いると、α1'''は次式で表せる。
【００５３】
【数３】

【００５４】
また、このときｆ1''＜ｆ0＜ｆ2''の関係にあることも、図２から明らかである。
【００５５】
ステップＳ４１１においては、補正前のステップ幅α1(図２の場合はα1'')を(式３)によって求めたステップ幅α1'''に補正する。このようにして、ステップ幅αの補正処理が終了すると、最後の増減係数Ｍによるループが終了したか否かが判定される(ステップＳ４１２)。増減係数Ｍを１０→２→１.１と変化させる場合は、１.１が最後の増減係数Ｍである。
【００５６】
２回目の囲い込み処理においては、ステップＳ４０２において、増減係数Ｍが１回目よりも小さな値(例えば２)に設定される。そして、２回目の囲い込み処理においては、この減少させた増減係数Ｍを用いて前述した囲い込み処理が実行される。ただし、２回目の囲い込み処理のスタート点は、ステップＳ４１１で補正されたステップ幅α1に対応する点(図２の場合はα1''')となる。そして、このスタート点から、減少させた増減係数Ｍによる１回目よりも小さな探索点間隔での、探索方向ベクトルｄの方向(１回目と同じ)への囲い込み処理が行われる。
【００５７】
このような囲い込み処理が、最後の増減係数Ｍによるループが終了するまで繰り返される。Ｍの複数回のループが終了すると、最終的に求めるべきαが、α1に設定される(ステップＳ４１３ｃ)。この処理によって、基準点ｘ0からｆ(ｘ)が小さくなる方向にあり、ｆ(ｘ0)≒ｆ(ｘ0＋α・ｄ)の関係を有する最小点を含んだ区間［ｘ0，ｘ0＋α・ｄ］を求めることができる。
【００５８】
ところで、増減係数Ｍが大きくなると、少ない反復回数でｆ2''＞ｆ0となるｆ2''を求めることができる。ただしα2''はα1''のＭ倍なので、最悪ｆ1''とｆ0が近い値(最小点を挟んで互いに反対側に位置)であれば、ｆ2''はｆ0に比べて極端に大きな値となる可能性がある。このとき、目的関数が完全な２次式でなければ、近似式の精度は悪くなると推測される。
【００５９】
一方、増減係数Ｍの値を小さくすればｆ2'を求めるための反復回数は多くなるが、ｆ2''とｆ0の値が近くなり、近似式の精度が向上する。したがって、上述したように、初めにＭを大きくして粗くα1を求め、つぎにＭを小さくして細かくα1を求めることで、少ない反復回数で精度よい近似曲線を得ることができる。
【００６０】
このように、この改良された囲い込み法においては、ステップ幅αの増減係数Ｍをある程度大きくして、概略的にｆ1''＜ｆ0＜ｆ2''となるα1''、α2''を素早く探索し、その後徐々に増減係数Ｍを小さくしていくことにより、最終的にｆ2''≒ｆ0となるαを(式３)から求めることができる。このとき、最小点を含んだ区間［ｘ0，ｘ0＋α・ｄ］において、ｆ(ｘ0)≒ｆ(ｘ0＋α・ｄ)であり、区間の両端の関数値がほとんど等しいので、後に詳述する２次近似式最小解探索法において、区間の両端の数値が異なることによって生ずる、解の精度の劣化が、起こらないという利点も有している。また、Ｍの１つのループにおいて、増減係数Ｍを指数関数的に大きくすることで探索点間隔を徐々に大きくしているので、概略的にｆ1''＜ｆ0＜ｆ2''となるα1''、α2''を、能率良く短時間に探索することができる。
【００６１】
(２)最小解探索法の説明
図３は、最小２乗法による２次近似式または放物線近似式を用いた最小解探索法の処理の概念図である。同図では、関数ｆ(ｘ)＝ｆ(ｘ0＋α・ｄ)をαの関数と考え、関数ｆ(α)の最小点を挟む区間［０，α］において、放物線近似あるいは最小２乗法を用いて、区間［０，α］内の４点から２次近似式を得ている。この２次近似式から、下記の(式４)の導出手順に示すように、２次式の１階微分が０となる点から最小点を求めることができる。
【００６２】
【数４】

【００６３】
なお、上述した２次近似式は、区間［０，α］内の任意の３点以上を使用して導くことができる。また、２次近似式を求める際に、用いる点が少ないと、計算時間が短縮され、逆に用いる点が多くなれば、計算精度を向上させることができる。
【００６４】
図４は、２次近似式による最小解探索法の処理手順を示したフローチャートである。この手法は、図２に示した黄金分割法を用いずに、図３に示した最小２乗法による２次近似式または放物線近似式を用いて、最小解を探索する手法である。同図を用いて、処理の流れを説明する。
【００６５】
まず、初期化手順を実施する。初期化手順は、ｆ0＝ｆ(ｘ0)、ｆ1＝ｆ(ｘ0＋α・ｄ)とし、分割点数Ｎを３以上とし、配列変数をａａ[１]＝０、ｙｙ[１]＝ｆ0、ａａ[Ｎ]＝α、ｙｙ[Ｎ]＝ｆ1とし、さらに、分割区間の幅Δａを、Δａ＝α／(Ｎ−１)とする(ステップＳ７０１)。つぎに、ループ変数Ｉにおける区間［０，α］を(Ｎ−１)に分割する各標本点と、その各標本点における、関数値をそれぞれ、ａａ[Ｉ]＝Δａ×(Ｉ−１)およびｙｙ[Ｉ]＝ｆ(ｘ0＋α・ｄ)の式で求める(ステップＳ７０２〜Ｓ７０４)。さらに上記で求めたＮ個の標本点ａａ[１]、ａａ[２]、・・・ａａ[Ｎ]と、この各標本点に対応する関数値ｙｙ[１]、ｙｙ[２]、・・・ｙｙ[Ｎ]から、２次式ｙ＝ｋ0＋ｋ1×α2＋ｋ2×αの係数ｋ0、ｋ1、ｋ2を最小２乗法による２次近似式または放物線近似式から求める。最後に、この得られたｋ0、ｋ1、ｋ2を用い、上記２次式ｙを最小にするαminを、(式４)の計算式を用いて求める(ステップＳ７０５)。
【００６６】
(３)直線探索法の説明
図５は、図１に示した改良された囲い込み法と、図４に示した２次近似式による最小解探索法とを組み合わせた、２次近似式による最小解／最大解を探索する直線探索法のフローチャートである。図５を用いて、その処理の流れについて簡単に説明する。
【００６７】
まず、上述した図１に示す改良された囲い込み法により、ある基準点ｘ0から関数ｆ(ｘ)が小さくなる方向にあって、最小点を含み、ｆ(ｘ0)≒ｆ(ｘ0＋α・ｄ)となる区間［ｘ0，ｘ0＋α・ｄ］を求める(ステップＳ８０１)。つぎに、α＝０か否かについて判定する(ステップＳ８０２)。α＝０の場合は、基準点ｘ0が最小解に一致するときであり、この処理を抜ける。α≠０の場合には、α＝−１か否かについて判定する(ステップＳ８０３)。α＝−１の場合は、上述した例外処理の場合である。この場合は、他の基準点を選び直すためにα＝１に設定し直し、この処理から抜ける。α＝０、α＝−１以外の場合には、上述した図４に示す２次近似式により、区間［ｘ0，ｘ0＋α・ｄ］を３点以上で分割し、分割点における関数値を近似した２次近似式から最小解を求める(ステップＳ８０５)。
【００６８】
（４）非線形最適解探索問題の説明
一般的な非線形最適化問題の定義を(式５)に示す。
【００６９】
【数５】

【００７０】
(式５)のｘは変数ベクトル、ｆ(ｘ)は目的関数、ｇ(ｘ)≦０は不等式制約条件、ｈ(ｘ)＝０は等式制約条件である。この場合の非線形最適化問題とは、制約条件ｇ(ｘ)≦０、ｈ(ｘ)＝０を満足し、目的関数ｆ(ｘ)が最小となる変数ベクトルｘを求める問題である。一方(式６)に、外点ペナルティ法の定義を示す。
【００７１】
【数６】

【００７２】
（式６）において、γはペナルティ係数、ｐはペナルティの大きさを制御する１以上の実数であり、逐次的に２次近似を用いる最適化手法では２が用いられる。外点ペナルティ法は、制約条件が存在する場合の最適化問題によく使用されるが、最適解探索の初期にγ＝０としてＵ(ｘ,γ)の最小解を探索し、探索の最中にγを徐々に大きくすることで、Ｕ(x,γ)が最小となる変数ｘを求めていく手法である。
【００７３】
図６および図７は、勾配法と図５に示した直線探索法を組み合わせた、最適解探索法の処理を示すフローチャートである。図６は、制約条件が無い場合の最適解探索手法であり、ある最小点を持つ目的関数ｆ(ｘ)が最小となる点ｘを探索する場合に用いる。図７は、制約条件がある場合の最適解探索手法であり、目的関数ｆ(ｘ)の変わりにペナルティを付加した目的関数Ｕ(ｘ,γ)を用い、図６に示した処理手順を活用している。まず、制約条件がない場合の処理手順について、図６を用いて説明する。
【００７４】
図６は、制約条件がない場合の非線形最適解探索手法を用いた処理手順を示すフローチャートである。まず、必要な初期化手順を実施する(ステップＳ９０１)。つぎに、勾配法を用い、ｘ0近傍において、ｆ(ｘ0)より小さくなる方向ｄを求める(ステップＳ９０２)。勾配法としては、ニュートン法、準ニュートン法、最急降下法等を用いればよい。特に、準ニュートン法、最急降下法を用いれば、前述したように、最適解が求められない場合や、探索方向ベクトルｄがｆ(ｘ)の小さくなる方向に向いていない場合等を回避することが可能である。
【００７５】
つぎに、ステップＳ９０２の処理で得られたｄについて、その値の大小について判定する(ステップＳ９０３)。ここで、ｄが十分小さい場合には、更新された基準点ｘ0が、所望の精度を有する最小解の場合であり、処理を終了する。ｄが十分小さくない場合は、図５のフローチャートで示した直線探索法の処理をおこない、ｆ(ｘ0＋α・ｄ)が最小となるαを求める(ステップＳ９０４)。そして、ステップＳ９０４で得られたαについて、その大小を判定する(ステップＳ９０５)。αが十分小さい場合とは、ステップＳ９０３におけるｄの判定のときと同様に、更新された基準点ｘ0が所望の精度を有する最小解の場合である。また、αが十分小さくない場合には、基準点ｘ0をｘ0＝ｘ0＋α・ｄで更新し(ステップＳ９０６)、ステップＳ９０２に戻って処理を繰り返す。
【００７６】
つぎに、制約条件がある場合の処理について説明する。図７は、この発明にかかる非線形最適解探索手法を用いた処理手順を示すフローチャートである。まず、ペナルティ係数γを設定し(ステップＳ１００１)、つぎに、図６に示した制約条件がない場合の非線形最適解探索手法において、ｆ(ｘ)を(式６)で示したＵ(ｘ，γ)で置き換え、図６のステップＳ９０３〜Ｓ９０７の処理を繰り返す(ステップＳ１００２)。そして、ペナルティ係数γを徐々に大きくする処理(ステップＳ１００１およびステップＳ１００３)を繰り返すことにより、Ｕ(ｘ，γ)が最小となる変数ｘを真の値に近づけていく。
【００７７】
（５）設計最適化装置の説明
図８は、熱流体・構造・電磁界などについての電機機器等に自動的に設計最適化を実行する機能を備えた設計最適化装置のブロック図である。この設計最適化装置は、上述した反復的な近似的直線探索法を用いている。同図において、１１は制御計算装置（コンピュータ）を、１２はＣＰＵ(Central Processing Unit)を、１３は入力装置を、１４は出力装置を、１５はメモリを、１６ａは制御プログラムを、１６ｂは制御データを、１７はＩ／Ｆ装置を、１８は計測装置を、１９は制御対象を、それぞれ示している。
【００７８】
この制御計算装置１１は、メモリ１５内に、上述した反復的な近似的直線探索法を用いた非線形最適解探索手法を備えた制御プログラム１６ａを格納している。また、制御計算装置１１は、計測装置１８から制御対象１９の計測結果を受信し、且つ制御プログラム１６ａの制御対象変更指示信号を、Ｉ／Ｆ装置１７を介して計測装置１８へ伝達する。計測装置１８は、制御対象１９の計測データを制御計算装置１１へ伝達する手段と、制御計算装置１１から受信した制御対象１９への変更指示信号にしたがって、制御対象１９の計測条件や制御条件などを変更する機能を有している。
【００７９】
この、近似的直線探索法を用いた設計最適化装置によれば、まず、制御計算装置１１の入力装置で、(式６)に基づく目的関数、設計変数、制約条件を入力し、メモリ１５へ格納する。つぎに、制御計算装置１１のメモリ１５に格納された制御プログラム１６ａを実行する。同図の制御計算装置１１のメモリ１５内の制御プログラム１６ａは、メモリ１５内の設計変数、Ｉ／Ｆ装置１７から受信した計測結果を用い、目的関数が最小となる設計変数の候補解を算出し、Ｉ／Ｆ装置１７へ、制御対象変更指示信号を送信する。その信号をＩ／Ｆ装置１７を介して受信した計測装置１８は、その内容に基づき制御対象１９の制御を更新し、再度計測を実施する。この動作を目的関数が最小となるまで自動的に繰り返すことができる。
【００８０】
このように、この実施の形態１によれば、制御プログラムが、図６または図７に示す近似的直線探索法を反復的に用いた非線形最適解探索手法を備えたことにより、直線探索１回あたりの反復計算回数を大幅に減らすことにより総反復計算回数を削減できるとともに、増減係数Ｍを効果的に漸減することにより反復計算回数のばらつきを少なくすることができるので、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００８１】
実施の形態２．
図９は、実施の形態２における熱流体・構造・電磁界などについての電機機器等に自動的に設計最適化を実行する機能を備えた設計最適化装置のブロック図であり、この場合は設計最適化の対象を計測対象モデルとしている。この設計最適化装置は、上述した実施の形態１の近似的直線探索法を反復的に用いている。図８に示した装置と同一符号のものは、基本的に同じ装置を表している。図９において、２１は制御用計算装置を、１２はＣＰＵ(Central Processing Unit)を、１３は入力装置を、１４は出力装置を、１５はメモリを、１６ａは制御プログラムを、１６ｂは制御データを、２２ａは計測プログラムを、２２ｂは計測データを、２３ａは解析プログラムを、２３ｂは解析データを、２４は制御対象モデルデータを、それぞれ示している。
【００８２】
制御計算装置２１は、メモリ１５内に、近似的直線探索法を反復的に用いた非線形最適解探索手法を備えた制御プログラム１６ａを格納している。また、メモリ１５は、制御プログラム１６ａのほかに、計測プログラム２２ａおよび計測データ２２ｂ、解析プログラム２３ａおよび解析データ２３ｂならびに制御対象モデルデータ２４が格納されている。ここで、解析プログラム２３ａは、制御対象モデルデータ２４に基づいた熱流体・構造・電磁界などの数値解析を実行する機能を持っている。また、計測プログラム２２ａは、解析プログラム２３ａの結果を取得し、制御対象モデルデータ２４の形状データ、計算条件などを決定するデータの変更機能を有している。
【００８３】
図９の設計最適化装置を使用するには、まず制御計算装置２１の入力装置１３で、(式５)に基づく目的関数、設計変数、制約条件を入力し、メモリ１５へ格納する。つぎに、制御計算装置２１のメモリ１５に格納された制御プログラム１６ａを実行する。制御計算装置２１のメモリ１５内の制御プログラム１６ａは、メモリ１５内の設計変数、計測プログラム２２ａから受信した計測結果を用い、目的関数が最小となる設計変数の候補解を算出し、計測プログラム２２ａへ制御対象変更指示信号を送信する。その信号を受信した計測プログラム２２ａは、その信号に基づき制御対象モデルデータ２４を更新し、解析プログラム２３ａに実行を指示し、解析プログラム２３ａの解析結果を取得する。そして、この動作を目的関数が最小となるまで自動的に繰り返すことができる。
【００８４】
このように、この実施の形態２によれば、制御対象モデルについての設計最適化をおこなうための、制御プログラムが、図７に示す近似的直線探索法を反復的に用いた非線形最適解探索手法を備えたことにより、直線探索１回あたりの反復計算回数を大幅に減らすことにより総反復計算回数を削減できるとともに、増減係数Ｍを効果的に漸減することにより反復計算回数のばらつきを少なくすることができるので、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００８５】
【発明の効果】
以上説明したように、この発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、区間の一方の端点を基準点ａとし、探索方向ベクトルｄ、基準点ａからの探索方向へのステップ幅αおよびステップ幅αを増減させるための１より大きい複数個の増減係数Ｍを記憶手段に保持し、基準点ａにステップ幅αを加えた点から記憶手段に保持された複数個の増減係数Ｍのうちの一つを用いてステップ幅αを増加させることにより算出した関数値が、基準点ａの関数値ｆ(ａ)を初めて上回ったときに、上回った関数値および上回った関数値を出力させる探索点を記憶手段に保持し、一つの増減係数Ｍについての当該探索点を求める処理を１回の囲い込み処理とするときに、１回の囲い込み処理が終了する度にステップ幅αを探索点の直前の探索点に復帰させ、かつ、記憶手段に保持された複数個の増減係数Ｍをその値が漸減する方向に順次読み出しつつ、当該増減係数Ｍごとに囲い込み処理を繰り返し実行し、基準点ａにおける関数値ｆ ( ａ ) と最後に読み出された増減係数Ｍについて求められた探索点における関数値との偏差が所定値以下となったときの探索点を他方の端点ｂとして求めることができ、最小解探索手段は、得られた区間［ａ，ｂ］内に存在する最小値を探索することができるので、直線探索１回あたりの反復計算回数を大幅に減らすことにより総反復計算回数を削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００８６】
つぎの発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、各１回の囲い込み処理中に、探索点の間隔が徐々に大きくなるステップ幅となるように増減係数Ｍを用いてステップ幅を増加させることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００８７】
つぎの発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、基準点ａの関数値ｆ(ａ)と同じ値を最初に超える探索点ｘ1と、この探索点ｘ1の直前の探索点ｘ2との間に、ｆ(ｘ1)＜ｆ(ａ)＜ｆ(ｘ2)の関係が得られたとき、ｘ1、ｘ2、ｆ(ｘ1)、ｆ(ｘ2)から１次式ｇ(ｘ)を作成し、ｇ(ｘ3)＝ｆ(ａ)となる点ｘ3を前記次回探索の開始点とするとともに、最終的に得られた探索点ｘ3を他方の端点ｂとすることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００８８】
この発明にかかる非線形最適解探索装置によれば、最小解探索手段は、区間［ａ，ｂ］内の任意の３点から放物線近似による２次近似式を作成し、該２次近似式の１階微分が０となる近似最小点を求めることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００８９】
この発明にかかる非線形最適解探索装置によれば、最小解探索手段は、区間［ａ，ｂ］内の任意の３点以上から最小２乗法による２次近似式を作成し、該２次近似式の１階微分が０となる近似最小点を求めることができるので直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９０】
この発明にかかる非線形最適解探索装置によれば、探索方向ベクトルｄを勾配法を用いて求めることができるので、基準点が最小近傍点に存在する場合や、α1を小さくしてもｆ1＜ｆ0の条件が成立しない場合等の、解が求められない状態になるのを避けることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９１】
この発明にかかる非線形最適解探索装置によれば、関数ｆ(ｘ)の代わりにペナルティ係数γを付加した目的関数Ｕ(ｘ，γ)を用い、探索の際にペナルティ係数γを段階的に大きくすることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９２】
つぎの発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、区間の一方の端点を基準点ａとし、探索方向ベクトルｄ、基準点ａからの探索方向へのステップ幅αおよびステップ幅αを増減させるための１より大きい複数個の増減係数Ｍを記憶手段に保持し、基準点ａにステップ幅αを加えた点から記憶手段に保持された複数個の増減係数Ｍのうちの一つを用いて該ステップ幅αを増加させることにより算出した関数値が、基準点ａの関数値ｆ(ａ)を初めて下回ったときに、下回った関数値および下回った関数値を出力させる探索点を記憶手段に保持し、一つの増減係数Ｍについての当該探索点を求める処理を１回の囲い込み処理とするときに、１回の囲い込み処理が終了する度にステップ幅αを探索点の直前の探索点に復帰させ、かつ、記憶手段に保持された複数個の増減係数Ｍをその値が漸減する方向に順次読み出しつつ、当該増減係数Ｍごとに囲い込み処理を繰り返し実行し、基準点ａにおける関数値ｆ ( ａ ) と最後に読み出された増減係数Ｍについて求められた探索点における関数値との偏差が所定値以下となったときの該探索点を他方の端点ｂとして求めることができ、また、最大解探索手段は、得られた区間［ａ，ｂ］内に存在する最大値を探索することができるので、直線探索１回あたりの反復計算回数を大幅に減らすことにより総反復計算回数を削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００９３】
つぎの発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、各１回の囲い込み処理中に、探索点の間隔が徐々に大きくなるステップ幅となるように増減係数Ｍを用いてステップ幅を増加させることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００９４】
つぎの発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、基準点ａの関数値ｆ(ａ)と同じ値を最初に小さくなる探索点ｘ1と、この探索点ｘ1の直前の探索点ｘ2との間に、ｆ(ｘ1)＜ｆ(ａ)＜ｆ(ｘ2)の関係が得られたとき、ｘ1、ｘ2、ｆ(ｘ1)、ｆ(ｘ2)から１次式ｇ(ｘ)を作成し、ｇ(ｘ3)＝ｆ(ａ)となる点ｘ3を前記次回探索の開始点とするとともに、最終的に得られた探索点ｘ3を他方の端点ｂとすることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減でき、従来の設計手法では得られなかった近似的最良設計解を短時間に求めることができる。
【００９５】
この発明にかかる非線形最適解探索装置によれば、最大解探索手段は、区間［ａ，ｂ］内の任意の３点から放物線近似による２次近似式を作成し、該２次近似式の１階微分が０となる近似最大点を求めることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９６】
この発明にかかる非線形最適解探索装置によれば、最大解探索手段は、区間［ａ，ｂ］内の任意の３点以上から最小２乗法による２次近似式を作成し、該２次近似式の１階微分が０となる近似最大点を求めることができるので直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９７】
この発明にかかる非線形最適解探索装置によれば、囲い込み処理手段は、探索方向ベクトルｄを勾配法を用いて求めることができるので、基準点が最小近傍点に存在する場合や、α1を小さくしてもｆ1＜ｆ0の条件が成立しない場合等の、解が求められない状態になるのを避けることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【００９８】
この発明にかかる非線形最適解探索装置によれば、関数ｆ(ｘ)の代わりにペナルティ係数γを付加した目的関数Ｕ(ｘ，γ)を用い、探索の際にペナルティ係数γを段階的に大きくすることができるので、直線探索１回あたりの反復計算回数を減らすことにより総反復計算回数を大幅に削減できるとともに、反復計算回数のばらつきを少なくすることができ、従来の設計手法では得られなかった近似的最良設計解を短時間、かつ、安定的に求めることができる。
【図面の簡単な説明】
【図１】 実施の形態１の囲い込み法の処理手順を示すフローチャートである。
【図２】 α1の補正処理を示す概念図である。
【図３】 最小２乗法による２次近似式または放物線近似式を用いた最小解探索法の処理の概念図である。
【図４】 ２次近似式による最小解探索法の処理手順を示したフローチャートである。
【図５】 図１に示した改良された囲い込み法と、図４に示した２次近似式による最小解探索法とを組み合わせた、２次近似式による最小解／最大解を探索する直線探索法のフローチャートである。
【図６】 制約条件が無い場合の、勾配法と図５に示した直線探索法を組み合わせた最適解探索法の処理を示すフローチャートである。
【図７】 制約条件がある場合の、勾配法と図５に示した直線探索法を組み合わせた最適解探索法の処理を示すフローチャートである。
【図８】 実施の形態１における、熱流体・構造・電磁界など電機機器等に自動的に設計最適化を実行する機能を備えた設計最適化装置のブロック図である。
【図９】 実施の形態２における、熱流体・構造・電磁界など電機機器等に自動的に設計最適化を実行する機能を備えた設計最適化装置のブロック図である。
【図１０】（ａ）は従来の囲い込み法の処理のフローチャートであり、（ｂ）は従来の囲い込み法の処理概念の説明図である。
【図１１】（ａ）は黄金分割法の処理のフローチャートであり、（ｂ）は黄金分割法の処理概念の説明図である。
【図１２】 従来の直線探索法のフローチャートを示したものである。
【符号の説明】
１１ 制御用計算装置、１２ ＣＰＵ、１３ 入力装置、１４ 出力装置、１５ メモリ、１６ａ 制御プログラム、１６ｂ 制御データ、１７ Ｉ／Ｆ装置、１８ 計測装置、１９ 制御対象、２１ 制御用計算装置、２２ａ 計測プログラム、２２ｂ 計測データ、２３ａ 解析プログラム、２３ｂ 解析データ、２４ 制御対象モデル。[0001]
BACKGROUND OF THE INVENTION
The present invention has a control program based on a straight line search method that obtains a section including the minimum / maximum points of a function by an enclosing process and obtains a minimum / maximum value in the section obtained by the minimum / maximum solution search process. The present invention relates to a nonlinear optimum solution search apparatus that searches for a nonlinear optimum solution by operating a computer using a program.
[0002]
[Prior art]
FIG. 10A is a flowchart of the process of the conventional enclosure method, and FIG. 10B is an explanatory diagram of the process concept of the conventional enclosure method. This conventional enclosing method has a function f (x) having a certain minimum point, and when the direction vector d where the function value f (x0) at the reference point x0 and the reference point x0 becomes small is known, x0 to x0. This is a method for obtaining a section including the minimum value of f (x) with x0 as a reference point when the value of is varied in the direction d in which the function value f (x0) decreases.
[0003]
Here, with reference to the flowchart shown in FIG. 10B, a method for obtaining a section including the minimum point using the conventional enclosing method will be briefly described. First, an initialization procedure is performed. In the initialization procedure, the reference point is x0, the function value at that time is f0 (= f (x0)), the step width α is a positive real number, and the step width increase coefficient M is a real number larger than one. Also, when the value of f (X) increases when x increases from x0, d = −1. Conversely, when the value of f (X) decreases when x increases from x0. , D = 1. Next, a processing procedure for changing x is performed. In this processing procedure, first, x is changed by α in the direction in which the value of f (x) decreases with reference to x0 (x1 = x0 + α × d), and the function value at x1 is changed to f1 (= f (x1)). ). At this time, with respect to f1 and f0, if f1 <f0, α is multiplied by M (α = α × M), and the value of x is changed by α · d using α multiplied by M, and the processing is continued. If f1> f0, the process is terminated. The interval [x0, x0 + α] including the minimum point is obtained from α finally obtained when the processing is completed. Note that the number of iterations of the enclosing method depends on the initial step width α and the increase coefficient M. The maximum point can be obtained in the same manner except that the sign of d is reversed and the end condition is f1 <f0.
[0004]
FIG. 11A is a flowchart of processing of the golden section method, and FIG. 11B is an explanatory diagram of processing concept of the golden section method. The golden section method has a function f (x) having a certain minimum point, and when the interval [a, b] including the minimum point is known, f (x) existing in the interval [a, b] is the minimum. This is a method for obtaining a point x.
[0005]
Next, a method for obtaining the minimum point in the interval [x1, x2] using the golden section method will be briefly described with reference to the flowchart shown in FIG. First, an initialization procedure is performed. As an initialization procedure, first, an interval [x1, x2] including the minimum point is set. For example, the golden division ratio F is set to F = (5^0.5-1) / (5^0.5Set to +1). It is assumed that the interval [x1, x2] including the minimum point is known in advance by another method (for example, the above-described enclosure method). Next, a processing procedure for changing x is performed. In this processing procedure, x1, x2, and F are used, and the width of the section [x1, x2] is sequentially narrowed. Specifically, x1 ′ and x2 ′ are obtained by the following equations.
x1 ′ = x1 + F × (x2−x1)
x2 ′ = x2−F × (x2−x1)
x1 ′ is the interval [x1, x2] (5^0.5-1): The point is internally divided into 2, and x2 'is the interval [x1, x2] 2: (5^0.5The point is to divide into -1).
[0006]
Next, when the function values f (x1 ′) and (x2 ′) at x1 ′ and x2 ′ are respectively f1 ′ and f2 ′, if f1 ′> f2 ′, then x1 = x1 ′ and f1 ′ ≦ f2 When ', x2 = x2' and the width of the section is narrowed one by one. Finally, | x1-x2 | is used for convergence determination, and the target minimum value is obtained. The maximum point can also be obtained by reversing the direction of the sign of the comparison condition, as in the enclosing method. The number of iterations of the golden section method depends on the difference between the absolute values of the function values f (x1) and f (x2) at both ends of the interval [x1, x2] and the convergence determination error.
[0007]
FIG. 12 shows a flowchart of a conventional line search method. In the conventional line search method, first, the section [a, b] including the minimum / maximum points is obtained by the conventional enclosing method shown in FIG. 10, and then the section [a, b] is obtained by the golden section method shown in FIG. b] The minimum / maximum points existing in the area are obtained.
[0008]
[Problems to be solved by the invention]
However, the enclosing method and the golden section method are both iterative methods, and in order to obtain a minimum / maximum solution with sufficient accuracy in a function that is the target of a certain minimum / maximization problem, usually about a dozen to several hundred times. It is said that iterative calculation is required.
[0009]
In a design optimization device equipped with a function to automatically perform design optimization for electrical equipment such as thermal fluid, structure, and electromagnetic field, to set the control target state optimally from the control target measurement data However, in the conventional technology, the iterative calculation time per search is large, and the iterative calculation time varies and the control is not stable. was there.
[0010]
The present invention has been made in view of the above, and an object of the present invention is to obtain a nonlinear optimum solution search apparatus that can shorten and stabilize a search time in a nonlinear optimization problem.
[0011]
[Means for Solving the Problems]
  In order to achieve the above object, a nonlinear optimum solution search apparatus according to the present invention is provided., SekiFind the interval containing the minimum number fBy the enclosure processing means and the enclosure processing meansI was askedSaidFind the minimum value of the function f in the intervalMinimum solution search means, and functions as at least the enclosure processing means and the minimum solution search meansControl programStorage means for holding the storage means, and the storage meansControl means for operating a control program held in a computerAnd comprising the aboveBased on the control of the control meansIn the function fIn the nonlinear optimum solution search device for searching for a nonlinear optimum solution,Enclosure processing meansIs a search direction vector with one end point of the section as a reference point a.Le d,TheStep in the search direction from the reference point aWidth α andGreater than 1 for increasing or decreasing the step width αMultipleIncrease / decrease coefficient MIs stored in the storage means.,SaidAt reference point aSaidFrom the point with step width α addedUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α exceeds the function value f (a) of the reference point a for the first time,The above function value andThe search point for outputting the above function value isWhen the processing for obtaining the search point for one increase / decrease coefficient M is held in the storage means as one enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andWhile sequentially reading out the plurality of increase / decrease coefficients M held in the storage means in the direction in which the values gradually decrease,Repeated enclosing processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.As the other end point bThe minimum solution searching means includes:Said seekingIsSearch for the minimum value existing in the interval [a, b]RukoIt is characterized by.
[0012]
  According to this invention,Enclosure processing meansIs the vector of the search direction with one end point of the section as the reference point aLe d, Step in the search direction from the reference point aWidth α andGreater than 1 to increase or decrease the step width αMultipleIncrease / decrease coefficient MIn the storage meansFrom the point of adding the step width α to the reference point aUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α exceeds the function value f (a) of the reference point a for the first time,Function value above andThe search point that outputs the function value exceedingWhen the processing for obtaining the search point for one increase / decrease coefficient M is stored in the storage means and is a single enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andFor each increase / decrease coefficient M, the plurality of increase / decrease coefficients M held in the storage means are sequentially read in the direction in which the value gradually decreases.Repeated enclosing processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.The other end point b can be obtained. Further, the minimum solution search means can search for the minimum value existing in the obtained section [a, b].
[0013]
In the non-linear optimum solution searching apparatus according to the next invention, in the above invention, the enclosing processing means may increase or decrease the coefficient M so that the interval between the search points becomes gradually larger during each enclosing process. Is used to increase the step width.
[0014]
According to the present invention, the enclosing processing means can increase the step width using the increase / decrease coefficient M so that the search point interval becomes gradually larger during each enclosing process.
[0015]
  In the non-linear optimum solution search apparatus according to the next invention, in the above invention, the enclosing processing means first includes a search point x1 that exceeds the same value as the function value f (a) of the reference point a, and the search point x1 The previous search point x2 andBetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g (x3) = f (a)searchPoint x3 was used as the starting point for the next search, and finally obtainedsearchThe point x3 is the other end point b
[0016]
  According to the present invention, the enclosing processing means includes the search point x1 that first exceeds the same value as the function value f (a) of the reference point a, the search point x2 immediately before the search point x1,BetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g The point x3 where (x3) = f (a) is set as the starting point of the next search and finally obtained.searchThe point x3 can be the other end point b.
[0017]
In the nonlinear optimum solution search apparatus according to the next invention, in the above invention, the minimum solution search means creates a quadratic approximation formula by parabolic approximation from arbitrary three points in the interval [a, b]. An approximate minimum point at which the first derivative of the next approximate expression is 0 is obtained.
[0018]
According to the present invention, the minimum solution search means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and the approximation in which the first derivative of the quadratic approximate expression is zero. The minimum point can be determined.
[0019]
In the nonlinear optimal solution search device according to the next invention, in the above invention, the minimum solution search means creates a quadratic approximate expression by a least square method from any three or more points in the interval [a, b], An approximate minimum point at which the first-order derivative of the quadratic approximate expression is 0 is obtained.
[0020]
According to this invention, the minimum solution search means creates a quadratic approximate expression by the least square method from any three or more points in the interval [a, b], and the first order differential of the quadratic approximate expression is 0. An approximate minimum point can be obtained.
[0021]
In the non-linear optimum solution search apparatus according to the next invention, the enclosing processing means obtains the search direction vector d using a gradient method.
[0022]
According to the present invention, the enclosure processing means can obtain the search direction vector d using the gradient method.
[0023]
The non-linear optimal solution search apparatus according to the next invention uses an objective function U (x, γ) to which a penalty coefficient γ is added instead of the function f (x) in the above invention, and the penalty coefficient γ is used for the search. It is characterized by increasing step by step.
[0024]
According to the present invention, the objective function U (x, γ) to which the penalty coefficient γ is added instead of the function f (x) can be used to increase the penalty coefficient γ step by step.
[0025]
  The nonlinear optimal solution search apparatus according to the next invention is, SekiFind the interval containing the maximum number of fBy the enclosure processing means and the enclosure processing meansI was askedSaidFind the maximum value of the function f in the intervalFunctions as maximum solution search means, at least the enclosure processing means and maximum solution search meansControl programStorage means for holding the storage means, and the storage meansControl means for operating the control program held in the computerAnd comprising the aboveBased on the control of the control meansIn the function fIn the nonlinear optimum solution search device for searching for a nonlinear optimum solution,Enclosure processing meansIs a search direction vector with one end point of the section as a reference point a.Le d,TheStep in the search direction from the reference point aWidth α andGreater than 1 for increasing or decreasing the step width αMultipleIncrease / decrease coefficient MIs stored in the storage means.,SaidAt reference point aSaidFrom the point with step width α addedUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α falls below the function value f (a) of the reference point a for the first time,The lower function value andThe search point for outputting the lower function value isWhen the processing for obtaining the search point for one increase / decrease coefficient M is held in the storage means as one enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andFor each increase / decrease coefficient M, the plurality of increase / decrease coefficients M held in the storage means are sequentially read in the direction in which the value gradually decreases.Repeat the enclosure processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.As the other end point bThe maximum solution searching means is:Said seekingIsSearch for the maximum value existing in the interval [a, b]RukoIt is characterized by.
[0026]
  According to this invention,Enclosure processing meansIs the vector of the search direction with one end point of the section as the reference point aLe d, Step in the search direction from the reference point aWidth α andGreater than 1 to increase or decrease the step width αMultipleIncrease / decrease coefficient MIn the storage meansFrom the point of adding the step width α to the reference point aUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α falls below the function value f (a) of the reference point a for the first time,Lower function value andThe search point that outputs the lower function valueWhen the processing for obtaining the search point for one increase / decrease coefficient M is stored in the storage means and is a single enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andFor each increase / decrease coefficient M, the plurality of increase / decrease coefficients M held in the storage means are sequentially read in the direction in which the value gradually decreases.Repeated enclosing processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.The other end point b can be obtained. Further, the maximum solution searching means can search for the maximum value existing in the obtained section [a, b].
[0027]
In the non-linear optimum solution searching apparatus according to the next invention, in the above invention, the enclosing processing means sets the increase / decrease coefficient M so that the interval between the search points becomes gradually larger during each enclosing process. Use to increase the step width.
[0028]
According to the present invention, the enclosing processing means can increase the step width using the increase / decrease coefficient M so that the search point interval becomes gradually larger during each enclosing process.
[0029]
  In the non-linear optimum solution searching apparatus according to the next invention, in the above invention, the enclosing processing means first has a search point x1 that is the same value as the function value f (a) of the reference point a, and the search point x1. Search point x2 immediately beforeBetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g (x3) = f (a)searchPoint x3 was used as the starting point for the next search, and finally obtainedsearchThe point x3 is the other end point b
[0030]
  According to the present invention, the enclosing processing means includes the search point x1 that first decreases the same value as the function value f (a) of the reference point a, and the search point x2 immediately before the search point x1.BetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g The point x3 where (x3) = f (a) is set as the starting point of the next search and finally obtained.searchThe point x3 can be the other end point b.
[0031]
In the non-linear optimum solution search apparatus according to the next invention, in the above invention, the maximum solution search means creates a quadratic approximation expression by parabolic approximation from arbitrary three points in the interval [a, b]. An approximate maximum point where the first-order derivative of the next approximate expression is 0 is obtained.
[0032]
According to this invention, the maximum solution search means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and the approximation in which the first derivative of the quadratic approximate expression is zero. The maximum point can be obtained.
[0033]
In the nonlinear optimal solution search device according to the next invention, in the above invention, the maximum solution search means creates a quadratic approximate expression by a least square method from any three or more points in the interval [a, b], An approximate maximum point where the first derivative of the quadratic approximate expression is 0 is obtained.
[0034]
According to this invention, the maximum solution searching means creates a quadratic approximate expression by the least square method from any three or more points in the interval [a, b], and the first-order derivative of the quadratic approximate expression is 0. An approximate maximum point can be obtained.
[0035]
In the non-linear optimum solution search apparatus according to the next invention, the enclosing processing means obtains the search direction vector d using a gradient method.
[0036]
According to the present invention, the enclosure processing means can obtain the search direction vector d using the gradient method.
[0037]
The non-linear optimal solution search apparatus according to the next invention uses an objective function U (x, γ) to which a penalty coefficient γ is added instead of the function f (x) in the above invention, and the penalty coefficient γ is used for the search. It is characterized by increasing step by step.
[0038]
According to the present invention, the objective function U (x, γ) to which the penalty coefficient γ is added instead of the function f (x) can be used to increase the penalty coefficient γ step by step.
[0039]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, preferred embodiments of the nonlinear optimum solution searching apparatus according to the present invention will be described in detail. Hereinafter, the problem of obtaining the minimum / maximum solution will be described by limiting to the processing procedure for obtaining the minimum solution. As described above, the method for obtaining the maximum solution is the same in concept and processing procedure except that the search direction and the determination criterion are reversed.
[0040]
Embodiment 1 FIG.
(1) Explanation of enclosure method
FIG. 1 is a flowchart showing a processing procedure of the enclosing method according to the first embodiment, which is an improvement of the conventional enclosing method shown in FIG. In FIG. 2, the horizontal axis is the direction vector d, and in an objective function f (x), the initial value is x0 (α = 0), and x0 is α1 · d, α1 ′ · d, α1 ″ · The range of values that the objective function f (x) can take when d and α2 ″ · d are increased is shown on the curve. α1 ′ = α1 × M, α1 ″ = α1 ′ × M, and α2 ″ = α1 ″ × M. In the conventional enclosing method, the case where there is one variable has been described. However, the contents described in Embodiment 1 deal with a plurality of variables. In a general nonlinear optimization problem, the objective function is usually a multivariate function, and therefore x0 and d shown below mean a multidimensional vector.
[0041]
Next, the processing procedure of the enclosing method of the first embodiment will be described in detail with reference to FIGS. First, an initialization procedure is performed (step S401). In this initialization procedure, the search direction vector is d, the reference point is x0, the function value at x0 is f0 (= f (x0)), and the step width α (positive real number) is initially set to α1. Is done. Here, it is assumed that the search vector d is obtained by the gradient method. The gradient method is a method for obtaining a direction vector d in which f (x) is small when obtaining a minimum solution of a multivariable function f (x) including one variable, and uses a quasi-Newton method, a steepest descent method, or the like. Can be obtained. Further, the step width α of the present invention is a width from the reference point x0 to each search point.
[0042]
Next, the value of the increase / decrease coefficient M is set (step S402). The value of the increase / decrease coefficient M is initially set to a large value, and is gradually changed (gradually decreased) to a value close to 1 each time one loop from step S402 to step S412 ends. The initial value of the increase / decrease coefficient M may be set to about M = 10, for example. In this case, the increase / decrease coefficient M is changed, for example, from 10 → 2 → 1.1. Then, a flag FLG for determining the branch of the process in step S405 or the like is set to 0 (step S403). The flag FLG is set so that it can take either 0 or 1. Next, the reference point x0 is varied by α1 · d in the direction of the direction vector d, and a function value f1 (= f (x0 + α1 · d)) at that time is obtained (step S404).
[0043]
Next, branching is determined based on the value of FLG (step S405). Since FLG = 0 is set in step S403, the first determination in step S405 is Yes, and the procedure proceeds to step S406a.
[0044]
In step S406a, the magnitudes of f1 and f0 are compared. When f1> f0, FLG is set to 1 (step S407), and the procedure proceeds to the process of step S408. When f1 ≦ f0, a process of multiplying the value of the step width α1 by M times (α1 = α1 · M) using the increase / decrease coefficient M is performed (step S409a, corresponding to α1 ′ in FIG. 2). It is determined whether α1 is sufficiently large (whether it is larger than a predetermined upper limit value) (step S410a). If the determination in step S410a is No, the process returns to step S404, and the function value f1 (= f (x0 + α1 · d)) at that time is obtained using the step width α1 (= α1 · M) multiplied by M. The function value f1 at this time corresponds to f1 'in FIG. Next, after passing through step S405, in step S406a, the magnitudes of f1 (= f1 ') and f0 are compared again. In the case of f1 ≦ f0, the value of the step width α1 is further increased M times using the increase / decrease coefficient M (M²(Step S409a, corresponding to α1 ″ in FIG. 2). In this way, in the comparison of step S406a, the procedure of steps S404 to S410a is repeatedly executed until f1> f0 is satisfied, thereby executing the enclosing search for multiplying the step width α1 by M using the increase / decrease coefficient M. .
[0045]
If it is determined in step S410a that the step width α1 is sufficiently large, the step width α is set to −1 (step S413a). In the process of α = −1 in step S413a, when the vector d in the search direction obtained by the gradient method is very small, the condition of f1> f0 is not satisfied even if the step width α1 is increased, and a solution is obtained. It is a flow of exception processing inserted assuming that there is no. Theoretically, it can be assumed that this occurs in the initial stage of search or when the reference point is located at the minimum neighborhood point, but it may be considered that there is almost no actual point. If exception processing actually occurs, re-searching may be performed by reselecting the reference point.
[0046]
As described above, when f1> f0 is satisfied in step S406a, FLG is changed to 1 (step S407). In the case of FIG. 2, the function value f1 (= f2 ″) when the step width is α2 ″ satisfies f1> f0. In the next step S408, the final processing result of the current enclosing process, that is, the final step width α1 (α2 ″ in the case of FIG. 2) and the function value f1 (= f2 ″) at that time are obtained. (Α2 = α1, f2 = f1). Next, by setting the step width α1 to 1 / M, the search point is returned to the previous point from the first point where f1> f0 is established (step S409b). This previous point corresponds to α1 ″ in FIG. Next, it is determined whether α1 is sufficiently small (whether it is smaller than a predetermined lower limit value) (step S410b). When it is determined that α1 is sufficiently small, the step width α is set to zero. (Step S413b). Step S413b is an exception process inserted on the assumption that the condition of f1 <f0 is not satisfied even if α1 is reduced and the solution cannot be obtained when the reference point exists at the minimum neighborhood point. In this case, the solution is considered to have converged. Theoretically, there are cases where the reference point is located at the minimum neighborhood point, or the search direction vector d is not oriented in the direction in which f (x) decreases, but the quasi-Newton method or the steepest descent method is used. If it is used, it is a phenomenon that can hardly occur, and if these methods are used in combination, there is no problem.
[0047]
If the determination in step S410b is No, the procedure proceeds to step S404, and the function value f1 (= f (= f (1) in the previous point (α1 ″ in FIG. 2) from the first point where f1> f0 is satisfied. x0 + α1 · d)) is obtained. The function value f1 at this time corresponds to f1 '' in FIG. Next, it is determined in step S405 whether FLG = 0. At this time, since FLG = 1, the procedure proceeds to step S406b. Then, in step S406b, the magnitude of f1 (f1 '' in the case of FIG. 2) and the magnitude of f0 are compared again. In this comparison process, since f1 <f0 is normally established, the procedure proceeds to step S411, and a process of correcting the step width α1 is executed.
[0048]
As described above, the step width α1 before correction corresponds to the previous point (α1 ″ in FIG. 2) from the first point where f1> f0 is satisfied. As described above, when α changes from α1 ″ to α2 ″, the relationship of f1 <f0 changes to the relationship of f1> f0. At this time, if the slope of the straight line K connecting f1 ″ at α1 ″ and f2 ″ at α2 ″ is m, m is expressed by the following equation.
[0049]
[Expression 1]

[0050]
Here, the function f may take the value of f0 between α1 ″ and α2 ″. If α is α1 ′ ″, α1 ″ and α2 ″ are As can be seen, the equation of the straight line K having the slope m is g (α), and when using (Equation 1), it can be expressed by the following equation.
[0051]
[Expression 2]

[0052]
When the condition of g (α1 ″) = f1 ″ is used when α is α1 ″, α1 ′ ″ can be expressed by the following equation.
[0053]
[Equation 3]

[0054]
It is also clear from FIG. 2 that the relationship of f1 ″ <f0 <f2 ″ is satisfied at this time.
[0055]
In step S411, the step width α1 before correction (α1 ″ in the case of FIG. 2) is corrected to the step width α1 ′ ″ obtained by (Equation 3). In this way, when the correction processing of the step width α is completed, it is determined whether or not the loop with the last increase / decrease coefficient M has been completed (step S412). When the increase / decrease coefficient M is changed from 10 → 2 → 1.1, 1.1 is the last increase / decrease coefficient M.
[0056]
In the second enclosing process, in step S402, the increase / decrease coefficient M is set to a smaller value (for example, 2) than the first time. In the second enclosing process, the enclosing process described above is executed using the decreased increase / decrease coefficient M. However, the start point of the second enclosing process is a point (α1 ′ ″ in the case of FIG. 2) corresponding to the step width α1 corrected in step S411. Then, the enclosing process in the direction of the search direction vector d (same as the first time) is performed from the start point at a search point interval smaller than the first time by the decreased increase / decrease coefficient M.
[0057]
Such an enclosing process is repeated until the loop with the last increase / decrease coefficient M is completed. When a plurality of M loops are completed, α to be finally obtained is set to α1 (step S413c). By this processing, an interval [x0, x0 + α · d] including a minimum point having a relationship of f (x0) ≈f (x0 + α · d) in a direction in which f (x) is reduced from the reference point x0 is obtained. Can do.
[0058]
By the way, when the increase / decrease coefficient M is increased, f2 ″ satisfying f2 ″> f0 can be obtained with a small number of iterations. However, α2 ″ is M times α1 ″, so if the worst f1 ″ and f0 are close to each other (positioned on opposite sides of the minimum point), f2 ″ is an extremely large value compared to f0. There is a possibility. At this time, if the objective function is not a perfect quadratic expression, it is presumed that the accuracy of the approximate expression is deteriorated.
[0059]
On the other hand, if the value of the increase / decrease coefficient M is reduced, the number of iterations for obtaining f2 ′ increases, but the values of f2 ″ and f0 become closer, and the accuracy of the approximate expression is improved. Therefore, as described above, an approximate curve with high accuracy can be obtained with a small number of iterations by first increasing M to roughly obtain α1, and then reducing M and finely obtaining α1.
[0060]
Thus, in this improved enclosing method, the increase / decrease coefficient M of the step width α is increased to some extent, and α1 ″ and α2 ″ that roughly satisfy f1 ″ <f0 <f2 ″ are quickly searched. Then, by gradually decreasing the increase / decrease coefficient M, α finally satisfying f2 ″ ≈f0 can be obtained from (Equation 3). At this time, in the interval [x0, x0 + α · d] including the minimum point, f (x0) ≈f (x0 + α · d) and the function values at both ends of the interval are almost equal. In the formula minimum solution search method, there is also an advantage that the degradation of the accuracy of the solution caused by the difference in the numerical values at both ends of the section does not occur. Further, in one loop of M, the increase / decrease coefficient M is increased exponentially, so that the search point interval is gradually increased. Therefore, α1 ″ which roughly satisfies f1 ″ <f0 <f2 ″. , Α2 ″ can be searched efficiently and in a short time.
[0061]
(2) Explanation of minimum solution search method
FIG. 3 is a conceptual diagram of processing of a minimum solution search method using a quadratic approximate expression or a parabolic approximate expression by the least square method. In this figure, the function f (x) = f (x0 + α · d) is considered as a function of α, and a parabolic approximation or a least square method is used in the interval [0, α] sandwiching the minimum point of the function f (α). The quadratic approximate expression is obtained from four points in the interval [0, α]. From this quadratic approximate expression, the minimum point can be obtained from the point where the first derivative of the quadratic expression becomes 0, as shown in the following derivation procedure of (Expression 4).
[0062]
[Expression 4]

[0063]
Note that the above-described quadratic approximate expression can be derived using any three or more points in the interval [0, α]. Further, when obtaining a quadratic approximate expression, if there are few points to be used, the calculation time is shortened. Conversely, if more points are used, the calculation accuracy can be improved.
[0064]
FIG. 4 is a flowchart showing the processing procedure of the minimum solution search method using a quadratic approximation. This method is a method of searching for the minimum solution using the quadratic approximation formula or the parabolic approximation formula by the least square method shown in FIG. 3 without using the golden section method shown in FIG. The process flow will be described with reference to FIG.
[0065]
First, an initialization procedure is performed. The initialization procedure is f0 = f (x0), f1 = f (x0 + α · d), the number of division points N is 3 or more, the array variables are aa [1] = 0, yy [1] = f0, aa [N ] = Α, yy [N] = f1, and the width Δa of the divided section is Δa = α / (N−1) (step S701). Next, each sample point that divides the interval [0, α] in the loop variable I into (N−1) and the function value at each sample point are aa [I] = Δa × (I−1), respectively. And yy [I] = f (x0 + α · d) (steps S702 to S704). Further, the N sample points aa [1], aa [2],... Aa [N] obtained above and the function values yy [1], yy [2],. From yy [N], the coefficients k0, k1, and k2 of the quadratic expression y = k0 + k1 × α2 + k2 × α are obtained from a quadratic approximate expression or a parabolic approximate expression by the least square method. Finally, using the obtained k0, k1, and k2, αmin that minimizes the quadratic expression y is obtained using the calculation formula (Formula 4) (Step S705).
[0066]
(3) Explanation of line search method
FIG. 5 shows a straight line search for searching for a minimum / maximum solution using a quadratic approximation formula, which combines the improved enclosing method shown in FIG. 1 and the minimum solution search method using a quadratic approximation formula shown in FIG. It is a flowchart of a method. The processing flow will be briefly described with reference to FIG.
[0067]
First, by the improved enclosing method shown in FIG. 1 described above, the function f (x) is reduced from a certain reference point x0 and includes the minimum point, and f (x0) ≈f (x0 + α · d) An interval [x0, x0 + α · d] is obtained (step S801). Next, it is determined whether α = 0 (step S802). When α = 0, it is when the reference point x0 coincides with the minimum solution, and this process is exited. If α ≠ 0, it is determined whether α = −1 (step S803). The case of α = −1 is the case of the exception processing described above. In this case, α = 1 is set again to select another reference point, and the process is exited. In cases other than α = 0 and α = −1, the section [x 0, x 0 + α · d] is divided into three or more points by the quadratic approximation shown in FIG. 4, and the function values at the division points are approximated. A minimum solution is obtained from the quadratic approximate expression (step S805).
[0068]
(4) Explanation of nonlinear optimal solution search problem
The definition of a general nonlinear optimization problem is shown in (Formula 5).
[0069]
[Equation 5]

[0070]
In Equation 5, x is a variable vector, f (x) is an objective function, g (x) ≦ 0 is an inequality constraint, and h (x) = 0 is an equation constraint. The nonlinear optimization problem in this case is a problem of obtaining a variable vector x that satisfies the constraint conditions g (x) ≦ 0 and h (x) = 0 and minimizes the objective function f (x). On the other hand, (Expression 6) shows the definition of the outer point penalty method.
[0071]
[Formula 6]

[0072]
In (Equation 6), γ is a penalty coefficient, p is one or more real numbers that control the magnitude of the penalty, and 2 is used in an optimization method that sequentially uses quadratic approximation. The outer point penalty method is often used for an optimization problem in the case where there are constraints. However, the minimum solution of U (x, γ) is searched at the initial stage of searching for an optimal solution with γ = 0. In this method, the variable x that minimizes U (x, γ) is obtained by gradually increasing γ.
[0073]
6 and 7 are flowcharts showing processing of the optimum solution search method in which the gradient method and the straight line search method shown in FIG. 5 are combined. FIG. 6 shows an optimum solution search method when there is no constraint condition, and is used when searching for a point x at which the objective function f (x) having a certain minimum point is minimum. FIG. 7 shows an optimal solution search method in the case where there is a constraint condition. The objective function U (x, γ) with a penalty is used instead of the objective function f (x), and the processing procedure shown in FIG. 6 is used. is doing. First, a processing procedure when there is no constraint condition will be described with reference to FIG.
[0074]
FIG. 6 is a flowchart showing a processing procedure using the nonlinear optimal solution search method when there is no constraint condition. First, a necessary initialization procedure is performed (step S901). Next, using the gradient method, a direction d smaller than f (x0) is obtained in the vicinity of x0 (step S902). As the gradient method, a Newton method, a quasi-Newton method, a steepest descent method, or the like may be used. In particular, if the quasi-Newton method or the steepest descent method is used, as described above, the case where the optimum solution cannot be obtained, or the case where the search direction vector d is not suitable for the direction in which f (x) decreases is avoided. Is possible.
[0075]
Next, with respect to d obtained by the process of step S902, the magnitude of the value is determined (step S903). Here, when d is sufficiently small, the updated reference point x0 is the minimum solution having the desired accuracy, and the process is terminated. If d is not sufficiently small, the process of the straight line search method shown in the flowchart of FIG. 5 is performed to obtain α that minimizes f (x0 + α · d) (step S904). Then, the magnitude of α obtained in step S904 is determined (step S905). The case where α is sufficiently small is a case where the updated reference point x0 is a minimum solution having a desired accuracy, as in the case of the determination of d in step S903. If α is not sufficiently small, the reference point x0 is updated with x0 = x0 + α · d (step S906), and the process returns to step S902 to repeat the process.
[0076]
Next, processing when there is a constraint condition will be described. FIG. 7 is a flowchart showing a processing procedure using the nonlinear optimum solution searching method according to the present invention. First, a penalty coefficient γ is set (step S1001). Next, in the nonlinear optimal solution search method when there is no constraint condition shown in FIG. 6, f (x) is expressed as U (x, γ), and the processing of steps S903 to S907 in FIG. 6 is repeated (step S1002). Then, by repeating the process of gradually increasing the penalty coefficient γ (steps S1001 and S1003), the variable x that minimizes U (x, γ) is brought closer to the true value.
[0077]
(5) Explanation of design optimization device
FIG. 8 is a block diagram of a design optimizing apparatus having a function of automatically executing design optimization on electrical equipment and the like regarding thermal fluid, structure, electromagnetic field, and the like. This design optimization apparatus uses the above-described iterative approximate line search method. In the figure, 11 is a control computing device (computer), 12 is a CPU (Central Processing Unit), 13 is an input device, 14 is an output device, 15 is a memory, 16a is a control program, and 16b is a control. Data, 17 is I / F device, 18 is measuring deviceThe, 19 indicate control objects, respectively.
[0078]
The control calculation apparatus 11 stores a control program 16 a having a nonlinear optimum solution search method using the above-described repetitive approximate line search method in the memory 15. Further, the control calculation device 11 receives the measurement result of the control object 19 from the measurement device 18 and transmits the control object change instruction signal of the control program 16 a to the measurement device 18 via the I / F device 17. The measuring device 18 transmits the measurement data of the control target 19 to the control calculation device 11 and the measurement condition or control condition of the control target 19 according to the change instruction signal to the control target 19 received from the control calculation device 11. It has a function to change.
[0079]
According to the design optimization device using the approximate line search method, first, an objective function, a design variable, and a constraint condition based on (Equation 6) are input to the memory 15 by the input device of the control calculation device 11. Store. Next, the control program 16a stored in the memory 15 of the control computer 11 is executed. The control program 16a in the memory 15 of the control calculation device 11 in the figure uses the design variable in the memory 15 and the measurement result received from the I / F device 17 to calculate a candidate solution for the design variable that minimizes the objective function. Then, the control target change instruction signal is transmitted to the I / F device 17. The measuring device 18 that has received the signal via the I / F device 17 updates the control of the control object 19 based on the content and performs measurement again. This operation can be automatically repeated until the objective function is minimized.
[0080]
As described above, according to the first embodiment, the control program includes the nonlinear optimum solution search method that repeatedly uses the approximate line search method shown in FIG. 6 or FIG. The total number of iterations can be reduced by drastically reducing the number of iterations, and the variation in the number of iterations can be reduced by effectively gradually decreasing the increase / decrease coefficient M. The approximate best design solution that could not be obtained can be obtained stably in a short time.
[0081]
Embodiment 2. FIG.
FIG. 9 is a block diagram of a design optimizing apparatus having a function of automatically executing design optimization on electrical equipment and the like regarding the thermal fluid, structure, and electromagnetic field in the second embodiment. In this case, FIG. The optimization target is the model to be measured. This design optimization apparatus repeatedly uses the approximate straight line search method of the first embodiment described above. Components having the same reference numerals as those shown in FIG. 8 basically represent the same devices. In FIG. 9, 21 is a control computing device, 12 is a CPU (Central Processing Unit), 13 is an input device, 14 is an output device, 15 is a memory, 16a is a control program, and 16b is control data. 22a indicates measurement program, 22b indicates measurement data, 23a indicates analysis program, 23b indicates analysis data, and 24 indicates control target model data.
[0082]
The control calculation device 21 stores in the memory 15 a control program 16a having a nonlinear optimal solution search method that repeatedly uses an approximate line search method. In addition to the control program 16a, the memory 15 stores a measurement program 22a and measurement data 22b, an analysis program 23a and analysis data 23b, and controlled object model data 24. Here, the analysis program 23 a has a function of executing numerical analysis of the thermal fluid, structure, electromagnetic field, and the like based on the control target model data 24. In addition, the measurement program 22a has a data change function for acquiring the result of the analysis program 23a and determining the shape data, calculation conditions, and the like of the control target model data 24.
[0083]
In order to use the design optimization apparatus of FIG. 9, first, an objective function, a design variable, and a constraint condition based on (Equation 5) are input by the input device 13 of the control calculation device 21 and stored in the memory 15. Next, the control program 16a stored in the memory 15 of the control computing device 21 is executed. The control program 16a in the memory 15 of the control computing device 21 uses the design variable in the memory 15 and the measurement result received from the measurement program 22a to calculate a candidate solution of the design variable that minimizes the objective function, and the measurement program 22a A control object change instruction signal is transmitted to. The measurement program 22a that has received the signal updates the control target model data 24 based on the signal, instructs the analysis program 23a to execute, and acquires the analysis result of the analysis program 23a. This operation can be automatically repeated until the objective function is minimized.
[0084]
As described above, according to the second embodiment, the control program for optimizing the design of the controlled object model uses a nonlinear optimal solution search method that repeatedly uses the approximate line search method shown in FIG. In addition to reducing the total number of iterations by greatly reducing the number of iterations per line search, and reducing the variation in the number of iterations by effectively decreasing the increase / decrease coefficient M. Therefore, the approximate best design solution that could not be obtained by the conventional design method can be obtained stably in a short time.
[0085]
【The invention's effect】
  As described above, according to the nonlinear optimal solution search device according to the present invention,Enclosure processing meansIs the vector of the search direction with one end point of the section as the reference point aLe d, Step in the search direction from the reference point aWidth α andGreater than 1 to increase or decrease the step width αMultipleIncrease / decrease coefficient MIn the storage meansFrom the point of adding the step width α to the reference point aUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α exceeds the function value f (a) of the reference point a for the first time,Function value above andThe search point that outputs the function value exceedingWhen the processing for obtaining the search point for one increase / decrease coefficient M is stored in the storage means and is a single enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andFor each increase / decrease coefficient M, the plurality of increase / decrease coefficients M held in the storage means are sequentially read in the direction in which the value gradually decreases.Repeated enclosing processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.The other end point b can be obtained, and the minimum solution search means can search for the minimum value existing in the obtained section [a, b], so that the number of iterations per linear search is greatly increased. The total number of iterations can be reduced by reducing it to, and an approximate best design solution that cannot be obtained by the conventional design method can be obtained in a short time.
[0086]
According to the nonlinear optimum solution searching apparatus according to the next invention, the enclosing processing means performs the step using the increase / decrease coefficient M so that the interval between the search points becomes gradually larger during each enclosing process. Since the width can be increased, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the approximate best design solution that could not be obtained by conventional design methods can be shortened in a short time. Can be sought.
[0087]
  According to the nonlinear optimum solution search apparatus according to the next invention, the enclosing processing means includes a search point x1 that first exceeds the same value as the function value f (a) of the reference point a, and a search point immediately before the search point x1. x2 andBetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g The point x3 where (x3) = f (a) is set as the starting point of the next search and finally obtained.searchSince the point x3 can be the other end point b, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the approximate best design that could not be obtained by the conventional design method. The solution can be obtained in a short time.
[0088]
According to the nonlinear optimal solution search apparatus according to the present invention, the minimum solution search means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and 1 of the quadratic approximate expression. Since it is possible to obtain an approximate minimum point at which the second order differential is 0, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the variation in the number of iterations can be reduced. Thus, an approximate best design solution that could not be obtained by the conventional design method can be obtained stably in a short time.
[0089]
According to the nonlinear optimal solution search apparatus according to the present invention, the minimum solution search means creates a quadratic approximate expression by the least square method from arbitrary three or more points in the interval [a, b], and the quadratic approximate expression. Since the approximate minimum point at which the first-order derivative of 0 can be obtained is 0, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the variation in the number of iterations can be reduced. Thus, the approximate best design solution that could not be obtained by the conventional design method can be obtained stably in a short time.
[0090]
According to the nonlinear optimum solution searching apparatus according to the present invention, the search direction vector d can be obtained by using the gradient method. Therefore, even when the reference point exists at the minimum neighboring point or even when α1 is reduced, f1 <f0. It is possible to avoid a situation where a solution cannot be obtained, such as when the above condition is not satisfied, and to obtain an approximate best design solution that could not be obtained by a conventional design method in a short time and stably. it can.
[0091]
According to the nonlinear optimum solution search apparatus according to the present invention, the objective function U (x, γ) to which the penalty coefficient γ is added is used instead of the function f (x), and the penalty coefficient γ is increased stepwise during the search. Therefore, by reducing the number of iterations per linear search, the total number of iterations can be greatly reduced, and the variation in the number of iterations can be reduced, which is not possible with conventional design methods. The approximate best design solution can be obtained stably in a short time.
[0092]
  According to the nonlinear optimal solution search device according to the next invention,Enclosure processing meansIs the vector of the search direction with one end point of the section as the reference point aLe d, Step in the search direction from the reference point aWidth α andGreater than 1 to increase or decrease the step width αMultipleIncrease / decrease coefficient MIn the storage meansFrom the point of adding the step width α to the reference point aUsing one of a plurality of increase / decrease coefficients M held in the storage meansWhen the function value calculated by increasing the step width α falls below the function value f (a) of the reference point a for the first time,Lower function value andThe search point that outputs the lower function valueWhen the processing for obtaining the search point for one increase / decrease coefficient M is stored in the storage means and is a single enclosing process,Each time the enclosing process is completed, the step width α is returned to the search point immediately before the search point, andFor each increase / decrease coefficient M, the plurality of increase / decrease coefficients M held in the storage means are sequentially read in the direction in which the value gradually decreases.Repeated enclosing processAnd the function value f at the reference point a ( a ) And the search point when the deviation from the function value at the search point obtained with respect to the increase / decrease coefficient M read last is less than or equal to a predetermined value.Since it can be obtained as the other end point b, and the maximum solution search means can search for the maximum value existing in the obtained section [a, b], the number of iterative calculations per linear search The total number of iterative calculations can be reduced by significantly reducing, and an approximate best design solution that cannot be obtained by the conventional design method can be obtained in a short time.
[0093]
According to the nonlinear optimum solution searching apparatus according to the next invention, the enclosing processing means performs the step using the increase / decrease coefficient M so that the interval between the search points becomes gradually larger during each enclosing process. Since the width can be increased, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the approximate best design solution that could not be obtained by conventional design methods can be shortened in a short time. Can be sought.
[0094]
  According to the non-linear optimal solution search apparatus according to the next invention, the enclosing processing means searches for the search point x1 that first becomes the same value as the function value f (a) of the reference point a, and the search immediately before the search point x1. Point x2 andBetweenThen, when the relationship f (x1) <f (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g The point x3 where (x3) = f (a) is set as the starting point of the next search and finally obtained.searchSince the point x3 can be the other end point b, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the approximate best design that could not be obtained by the conventional design method. The solution can be obtained in a short time.
[0095]
According to the nonlinear optimal solution search apparatus according to the present invention, the maximum solution search means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and 1 of the quadratic approximate expression. Since it is possible to find the approximate maximum point at which the second derivative is 0, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the variation in the number of iterations can be reduced. Thus, an approximate best design solution that could not be obtained by the conventional design method can be obtained stably in a short time.
[0096]
According to the nonlinear optimal solution search apparatus according to the present invention, the maximum solution search means creates a quadratic approximate expression by the least square method from any three or more points in the interval [a, b], and the quadratic approximate expression. Since the approximate maximum point at which the first-order derivative of 0 can be obtained is 0, the total number of iterations can be greatly reduced by reducing the number of iterations per line search, and the variation in the number of iterations can be reduced. Thus, the approximate best design solution that could not be obtained by the conventional design method can be obtained stably in a short time.
[0097]
According to the nonlinear optimum solution searching apparatus according to the present invention, the enclosing processing means can obtain the search direction vector d by using the gradient method. Therefore, when the reference point is present at the minimum neighboring point or when α1 is reduced. However, it is possible to avoid a situation where a solution cannot be obtained, such as when the condition of f1 <f0 is not satisfied.TheAn approximate best design solution that cannot be obtained by the conventional design method can be obtained stably in a short time.
[0098]
According to the nonlinear optimum solution search apparatus according to the present invention, the objective function U (x, γ) to which the penalty coefficient γ is added is used instead of the function f (x), and the penalty coefficient γ is increased stepwise during the search. Therefore, by reducing the number of iterations per linear search, the total number of iterations can be greatly reduced, and the variation in the number of iterations can be reduced, which is not possible with conventional design methods. The approximate best design solution can be obtained stably in a short time.
[Brief description of the drawings]
FIG. 1 is a flowchart showing a processing procedure of an enclosing method according to a first embodiment.
FIG. 2 is a conceptual diagram showing α1 correction processing.
FIG. 3 is a conceptual diagram of processing of a minimum solution search method using a quadratic approximate expression or a parabolic approximate expression by a least square method.
FIG. 4 is a flowchart showing a processing procedure of a minimum solution search method using a quadratic approximate expression.
5 is a line search for searching for a minimum / maximum solution using a quadratic approximation formula, which combines the improved enclosing method shown in FIG. 1 and the minimum solution search method using a quadratic approximation formula shown in FIG. It is a flowchart of a method.
6 is a flowchart showing processing of an optimum solution search method combining the gradient method and the line search method shown in FIG. 5 when there is no constraint condition.
7 is a flowchart showing processing of an optimal solution search method that combines the gradient method and the straight line search method shown in FIG. 5 when there is a constraint condition. FIG.
FIG. 8 is a block diagram of a design optimization apparatus having a function of automatically executing design optimization on electrical equipment such as a thermal fluid, structure, and electromagnetic field in the first embodiment.
FIG. 9 is a block diagram of a design optimization apparatus having a function of automatically executing design optimization on electrical equipment such as thermal fluid, structure, and electromagnetic field in Embodiment 2.
10A is a flowchart of processing of a conventional enclosing method, and FIG. 10B is an explanatory diagram of processing concept of the conventional enclosing method.
11A is a flowchart of processing of the golden section method, and FIG. 11B is an explanatory diagram of a processing concept of the golden section method.
FIG. 12 is a flowchart of a conventional straight line search method.
[Explanation of symbols]
11 control computer, 12 CPU, 13 input device, 14 output device, 15 memory, 16a control program, 16b control data, 17 I / F device, 18 measuring device, 19 control object, 21 control computing device, 22a measurement Program, 22b Measurement data, 23a Analysis program, 23b Analysis data, 24 Control target model.

Claims (14)

And enclosure processing means asking you to interval including minimum point of the function number f, and the minimum solution search means for obtaining the minimum value of the function f in the interval determined by said enclosure processing means, at least said enclosure processing means and said minimum a storage means for holding a control program that serves as a solution search means, anda control means for controlling programmed computer operations held in the storage means, nonlinear optimization in the function f based on the control of the control means In a nonlinear optimal solution search device for searching for a solution,
The enclosure processing means includes:
The one end point as a reference point a of the section, search direction vector d, 1 is greater than a plurality of decrease coefficient for increasing or decreasing the step size α and the step size α in the search direction from the reference point a M Is stored in the storage means ,
Function value calculated by increasing the step size α with one of the reference point a to the step width α a plurality of decrease coefficient M, which is held in the storage means from the point of addition is, When the function value f (a) of the reference point a is exceeded for the first time, the function value that exceeds and the search point that outputs the function value that exceeds the value are held in the storage means,
When the process for obtaining the search point for one increase / decrease coefficient M is a single enclosing process , the step width α is returned to the search point immediately before the search point every time the enclosing process is completed. And, while sequentially reading out a plurality of increase / decrease coefficients M held in the storage means in a direction in which the value gradually decreases, the enclosing process is repeatedly executed for each increase / decrease coefficient M ,
The search point when the deviation between the function value f ( a ) at the reference point a and the function value at the search point obtained for the last read increase / decrease coefficient M is equal to or less than a predetermined value is designated as the other end point b. determined as,
The minimum solution search means includes:
Nonlinear optimal solution search apparatus according to claim and Turkey be searched minimum value present in the determined was the interval [a, b].   The enclosing processing means increases the step width by using an increase / decrease coefficient M so that the search point interval gradually increases during each enclosing process. The nonlinear optimal solution search apparatus described.   The enclosing processing means sets f (x1) <f () between the search point x1 that first exceeds the same value as the function value f (a) of the reference point a and the search point x2 immediately before the search point x1. a) When a relationship of <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g (x3) = f (a) The non-linear optimal solution search apparatus according to claim 1, wherein the search point x 3 is a start point of the next search, and the finally obtained search point x 3 is the other end point b.   The minimum solution search means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and obtains an approximate minimum point at which the first derivative of the quadratic approximate expression is zero. The nonlinear optimal solution search apparatus according to any one of claims 1 to 3.   The minimum solution search means creates a quadratic approximate expression by a least square method from any three or more points in the interval [a, b], and determines an approximate minimum point at which the first derivative of the quadratic approximate expression is zero. The non-linear optimal solution search apparatus according to claim 1, wherein the non-linear optimal solution search apparatus is obtained.   The non-linear optimal solution search apparatus according to claim 1, wherein the enclosing processing unit obtains a search direction vector d using a gradient method.   7. The objective function U (x, γ) to which a penalty coefficient γ is added instead of the function f (x), and the penalty coefficient γ is increased stepwise during the search. The nonlinear optimal solution search apparatus as described in any one of these. And enclosure processing means asking you to sections including the maximum point of the function number f, and the maximum solution search means for obtaining a maximum value of the function f in the interval determined by said enclosure processing means, at least said enclosure processing means and said maximum comprising storage means for holding a control program that serves as a solution search means, and control means for controlling programmed computer operations held in the storage means, that put the function f based on the control of the control means In a nonlinear optimal solution search device that searches for a nonlinear optimal solution,
The enclosure processing means includes:
The one end point as a reference point a of the section, search direction vector d, 1 is greater than a plurality of decrease coefficient for increasing or decreasing the step size α and the step size α in the search direction from the reference point a M Is stored in the storage means ,
Function value calculated by increasing the step size α with one of the reference point a to the step width α a plurality of decrease coefficient M, which is held in the storage means from the point of addition is, When the function value f (a) of the reference point a is first reduced, the storage function stores the lower function value and the search point for outputting the lower function value ,
When the process for obtaining the search point for one increase / decrease coefficient M is a single enclosing process , the step width α is returned to the search point immediately before the search point every time the enclosing process is completed. And, while sequentially reading out a plurality of increase / decrease coefficients M held in the storage means in a direction in which the value gradually decreases, the enclosing process is repeatedly executed for each increase / decrease coefficient M ,
The search point when the deviation between the function value f ( a ) at the reference point a and the function value at the search point obtained for the last read increase / decrease coefficient M is equal to or less than a predetermined value is designated as the other end point b. determined as,
The maximum solution search means includes:
Nonlinear optimal solution search apparatus according to claim and Turkey to explore the maximum value present in the determined was the interval [a, b].   9. The enclosure processing means increases the step width by using an increase / decrease coefficient M so that the interval between search points gradually increases during each enclosing process. The nonlinear optimal solution search apparatus described.   The enclosing processing means sets f (x1) <f between the search point x1 that first becomes the same value as the function value f (a) of the reference point a and the search point x2 immediately before the search point x1. When the relationship of (a) <f (x2) is obtained, a linear expression g (x) is created from x1, x2, f (x1), f (x2), and g (x3) = f (a) 10. The nonlinear optimal solution search apparatus according to claim 8 or 9, wherein the search point x3 that becomes is set as the start point of the next search and the finally obtained search point x3 is set as the other end point b.   The maximum solution searching means creates a quadratic approximate expression by parabolic approximation from arbitrary three points in the interval [a, b], and obtains an approximate maximum point at which the first derivative of the quadratic approximate expression becomes zero. The nonlinear optimal solution search apparatus according to any one of claims 8 to 10.   The maximum solution searching means creates a quadratic approximate expression by a least square method from any three or more points in the interval [a, b], and determines an approximate maximum point at which the first derivative of the quadratic approximate expression is zero. The non-linear optimal solution search apparatus according to claim 8, wherein the non-linear optimal solution search apparatus is obtained.   The non-linear optimal solution search apparatus according to any one of claims 8 to 12, wherein the enclosing processing means obtains a search direction vector d using a gradient method.   14. The objective function U (x, γ) to which a penalty coefficient γ is added instead of the function f (x), and the penalty coefficient γ is increased stepwise during the search. The nonlinear optimal solution search apparatus as described in any one of these.

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