﻿ 求解半无限规划问题的一类新的精确罚函数方法 A New Exact Penalty Function Method for Solving Semi-Infinite Programming Problems

Operations Research and Fuzziology
Vol.07 No.04(2017), Article ID:22682,10 pages
10.12677/ORF.2017.74014

A New Exact Penalty Function Method for Solving Semi-Infinite Programming Problems

Yanping Zhang, Qian Liu

School of Mathematics and Statistics, Shandong Normal University, Jinan Shandong

Received: Oct. 21st, 2017; accepted: Nov. 3rd, 2017; published: Nov. 15th, 2017

ABSTRACT

For semi-infinite programming problems, we provide a new generalized exact penalty function, which contains many commonly used penalty functions as a special case. It is proved that the local optimal solution of the unconstrained optimization subproblem is also the local optimal solution of the original problem when the penalty parameter is suﬃciently large under some constraint qualification. Moreover, under suitable conditions, we also prove that the global optimal solution sequence of unconstrained optimization subproblem converges to the global optimal solution of the original problem.

Keywords:Semi-Infinite Programming, Exact Penalty Function, Penalty Function Algorithm

1. 引言

$\begin{array}{l}\mathrm{min}f\left(x\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{g}_{j}\left(x,\omega \right)\le 0,\forall \omega \in \Omega ,j=1,\cdots ,m\end{array}$ (1.1)

${f}_{\sigma }\left(x,\epsilon \right)=\left\{\begin{array}{l}f\left(x\right),\text{}\epsilon =0,{g}_{j}\left(x,\omega \right)\le 0\text{}\left(\omega \in \Omega \right),\\ f\left(x\right)+{\epsilon }^{-\alpha }\Delta \left(x,\epsilon \right)+\sigma {\epsilon }^{\beta },\text{}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon >0,\\ +\infty ,\text{}其他,\end{array}$

$\Delta \left(x,\epsilon \right)=\sum _{j=1}^{m}{\int }_{\Omega }{\left[\mathrm{max}\left\{0,{g}_{j}\left(x,\omega \right)-{\left(\epsilon \right)}^{\gamma }\right\}\right]}^{2}\text{d}\omega$

${f}_{\sigma }\left(x,\epsilon \right)=\left\{\begin{array}{l}f\left(x\right),\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon =0,{g}_{j}\left(x,\omega \right)\le 0\text{}\left(\omega \in \Omega \right),\\ f\left(x\right)-{\epsilon }^{-\alpha }\mathrm{log}\left(1-\Delta \left(x,\epsilon \right)\right)+\sigma {\epsilon }^{\beta },\text{}\epsilon >0,\Delta \left(x,\epsilon \right)<1,\\ +\infty ,\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise}.\end{array}$

2. 一种新的精确罚函数

${f}_{\sigma }\left(x,\epsilon \right)=\left\{\begin{array}{l}f\left(x\right),\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon =0,{g}_{j}\left(x,\omega \right)\le 0\text{}\left(\omega \in \Omega \right),\\ f\left(x\right)+{\epsilon }^{-\alpha }\varphi \left(\Delta \left(x,\epsilon \right)\right)+\sigma {\epsilon }^{\beta },\text{}\epsilon >0,\Delta \left(x,\epsilon \right)<1,\\ +\infty ,\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise}.\end{array}$

$\Delta \left(x,\epsilon \right)=\sum _{j=1}^{m}{\int }_{\Omega }{\left[\mathrm{max}\left\{0,{g}_{j}\left(x,\omega \right)-{\left(\epsilon \right)}^{\gamma }\right\}\right]}^{2}\text{d}\omega$

1) $\varphi$$\left[0,a\right)$ 上是凸函数，连续可微的，且 $\varphi \left(0\right)=0$

2) ${\varphi }^{\prime }\left(t\right)>0,\forall t\in \left[0,a\right)$

$\begin{array}{l}{\varphi }_{1}\left(t\right)=\frac{t}{{\left(1-qt\right)}^{\alpha }}\text{}\left(a={q}^{-1},\alpha \ge 1\right)\\ {\varphi }_{2}\left(t\right)=\mathrm{tan}\left(t\right)\text{}\text{ }\left(a=\frac{\text{π}}{2}\right)\\ {\varphi }_{3}\left(t\right)=-\mathrm{log}\left(1-{t}^{\alpha }\right)\text{}\text{ }\text{ }\left(a=1,\alpha \ge 1\right)\\ {\varphi }_{4}\left(t\right)=t\text{}\left(a=+\infty \right)\\ {\varphi }_{5}\left(t\right)={\text{e}}^{t}-1\text{}\text{ }\text{ }\text{ }\left(a=+\infty \right)\\ {\varphi }_{6}\left(t\right)=\frac{1}{2}\left(\sqrt{{t}^{2}+4}+t\right)-1\text{}\left(a=+\infty \right)\end{array}$

$\left({P}_{\sigma }\right)\text{}\mathrm{min}\text{}{f}_{\sigma }\left(x,\epsilon \right)\text{s}\text{.t}\text{.}\left(x,\epsilon \right)\in {R}^{n}×\left[0,+\infty \right)$

${S}_{\epsilon }=\left\{\left(x,\epsilon \right)\in {R}^{n}×{R}_{+}:{g}_{j}\left(x,\omega \right)\le {\epsilon }^{\gamma },\forall \omega \in \Omega ,j=1,\cdots ,m\right\}$ (2.1)

$\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\notin {S}_{{\epsilon }^{k,\ast }}$

$\frac{\partial {f}_{{\sigma }_{k}}\left({x}^{\ast },{\epsilon }^{\ast }\right)}{\partial \epsilon }=0$

${g}_{j}\left({x}^{k,\ast },\omega \right)\le {\left({\epsilon }^{k,\ast }\right)}^{\gamma },\forall \omega \in \Omega ,j=1,\cdots ,m$

$\begin{array}{l}\frac{\partial {f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}{\partial \epsilon }\\ =-\alpha {\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)-2\gamma \cdot {\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\cdot \sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\cdot {\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega +\beta \sigma {\left({\epsilon }^{k,\ast }\right)}^{\beta -1}\end{array}$ (2.2)

${\int }_{\Omega }\sum _{j=1}^{m}{\phi }_{j}\left(\omega \right)\frac{\partial {g}_{j}\left(\overline{x},\omega \right)}{\partial x}=0$

$\begin{array}{l}\frac{\partial {f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}{\partial x}\\ =\frac{\partial f\left({x}^{k,\ast }\right)}{\partial x}+2{\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\frac{\partial {g}_{j}\left({x}^{k,\ast },\omega \right)}{\partial x}\text{d}\omega \\ =0\end{array}$ (2.3)

$\begin{array}{l}\frac{\partial {f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}{\partial \epsilon }\\ =-\alpha {\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)-2\gamma {\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\cdot \sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\cdot {\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega +\beta {\sigma }_{k}{\left({\epsilon }^{k,\ast }\right)}^{\beta -1}\\ =0\end{array}$ (2.4)

$\begin{array}{l}-\alpha {\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\end{array}$ $-2\gamma {\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\cdot {\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega$ $=-{\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\left\{\alpha \varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)+2\gamma {\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)$ $-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\text{d}\omega \right\}$

$\frac{{\left({\epsilon }^{k,\ast }\right)}^{\alpha }}{{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}\cdot \frac{\partial f\left({x}^{k,\ast }\right)}{\partial x}+2\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\frac{\partial {g}_{j}\left({x}^{k,\ast },\omega \right)}{\partial x}\text{d}\omega =0$

$2\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{\ast },\omega \right)\right\}\frac{\partial {g}_{j}\left({x}^{\ast },\omega \right)}{\partial x}=0$

$\mathrm{max}\left\{0,{g}_{j}\left({x}^{\ast },\omega \right)\right\}=0$

${g}_{j}\left({x}^{\ast },\omega \right)\le 0,\forall \omega \in \Omega ,j=1,\cdots ,m$ ，所以 ${x}^{\ast }$ 为问题(P)的可行解。

$\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\to \Delta \left({x}^{\ast },{\epsilon }^{\ast }\right)=0$

${f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\to {f}_{{\sigma }_{k}}\left({x}^{\ast },0\right)=f\left(x\ast \right)$

${\nabla }_{\left(x,\epsilon \right)}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\to {\nabla }_{\left(x,\epsilon \right)}{f}_{{\sigma }_{k}}\left({x}^{\ast },0\right)=\left(\nabla f\left({x}^{\ast }\right),0\right)$

$\begin{array}{l}\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left\{f\left({x}^{k,\ast }\right)+{\left({\epsilon }^{k,\ast }\right)}^{-\alpha }\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)+{\sigma }_{k}{\left({\epsilon }^{k,\ast }\right)}^{\beta -1}\right\}\\ =f\left(x\ast \right)\end{array}$

$\begin{array}{c}\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{{\left({\epsilon }^{k,\ast }\right)}^{\alpha }}=\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}\cdot \frac{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}{{\left({\epsilon }^{k,\ast }\right)}^{\alpha }}\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}\cdot \frac{\sum _{j=1}^{m}{\int }_{\Omega }{\left[\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\right]}^{2}\text{d}\omega }{{\left({\epsilon }^{k,\ast }\right)}^{\alpha }}\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}\cdot \sum _{j=1}^{m}{\int }_{\Omega }{\left[\mathrm{max}\left\{0,\frac{{g}_{j}\left({x}^{k,\ast },\omega \right)}{{\left({\epsilon }^{k,\ast }\right)}^{\delta }}\cdot {\left({\epsilon }^{k,\ast }\right)}^{\delta -\frac{\alpha }{2}}-\frac{{\left({\epsilon }^{k,\ast }\right)}^{\gamma }}{{\left({\epsilon }^{k,\ast }\right)}^{\frac{\alpha }{2}}}\right\}\right]}^{2}\text{d}\omega \\ =0\end{array}$

$\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}{\nabla }_{\left(x,\epsilon \right)}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)=\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}{\left[{\nabla }_{x}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\text{ }{\nabla }_{\epsilon }{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right]}^{\text{T}}$

$\begin{array}{l}\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}{\nabla }_{x}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left\{\frac{\partial f\left({x}^{k,\ast }\right)}{\partial x}+2{\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\cdot \frac{\partial {g}_{j}\left({x}^{k,\ast },\omega \right)}{\partial x}\text{d}\omega \right\}\\ ={\nabla }_{x}f\left({x}^{\ast }\right)+\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}2{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }{\left({\epsilon }^{k,\ast }\right)}^{-\alpha }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\cdot \frac{\partial {g}_{j}\left({x}^{k,\ast },\omega \right)}{\partial x}\text{d}\omega \\ ={\nabla }_{x}f\left({x}^{\ast }\right)+\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}2{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,\frac{{g}_{j}\left({x}^{k,\ast },\omega \right)}{{\left({\epsilon }^{k,\ast }\right)}^{\delta }}\cdot {\left({\epsilon }^{k,\ast }\right)}^{\delta -\alpha }-{\left({\epsilon }^{k,\ast }\right)}^{\gamma -\alpha }\right\}\cdot \frac{\partial {g}_{j}\left({x}^{k,\ast },\omega \right)}{\partial x}\text{d}\omega \\ ={\nabla }_{x}f\left(x\ast \right)\end{array}$

$\begin{array}{l}\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}{\nabla }_{\epsilon }{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left\{-\alpha {\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)-2\gamma {\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega +\beta {\sigma }_{k}{\left({\epsilon }^{k,\ast }\right)}^{\beta -1}\right\}\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left\{\left(-\alpha \right)\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{{\left({\epsilon }^{k,\ast }\right)}^{\alpha +1}}-2\gamma {\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\frac{\sum _{j=1}^{m}{\int }_{\Omega }\left\{{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega }{{\left({\epsilon }^{k,\ast }\right)}^{\alpha }}\right\}\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left(-\alpha \right)\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{{\left({\epsilon }^{k,\ast }\right)}^{\alpha +1}}\\ =\underset{{e}^{k,\ast }\to {\epsilon }^{\ast }=0}{\mathrm{lim}}\left(-\alpha \right)\frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}.\frac{\sum _{j=1}^{m}{\int }_{\Omega }{\left[{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right]}^{2}\text{d}\omega }{{\left({\epsilon }^{k,\ast }\right)}^{\alpha +1}}=0\end{array}$

${\epsilon }^{k,\ast }\to {\epsilon }^{\ast }=0,{x}^{k,\ast }\to {x}^{\ast },\left({x}^{\ast },{\epsilon }^{\ast }\right)\in {S}_{0}$

$\begin{array}{l}{\left({\epsilon }^{k,\ast }\right)}^{1-\beta }\left[\left(-\alpha \right){\left({\epsilon }^{k,\ast }\right)}^{-\alpha -1}\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)-2\gamma {\left({\epsilon }^{k,\ast }\right)}^{-\alpha }{\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\\ \cdot \text{\hspace{0.17em}}\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma -1}\text{d}\omega +\beta {\sigma }_{k}{\left({\epsilon }^{k,\ast }\right)}^{\beta -1}\right]=0\end{array}$

$\begin{array}{l}{\left({\epsilon }^{k,\ast }\right)}^{-\alpha -\beta }\left\{\alpha \varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)+2\gamma {\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\\ \cdot \text{ }\text{\hspace{0.17em}}\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\text{d}\omega \right\}=\beta {\sigma }_{k}\end{array}$

$\begin{array}{l}\alpha \frac{\varphi \left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)}{\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)}\frac{\sum _{j=1}^{m}{\int }_{\Omega }{\left[\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}\right]}^{2}\text{d}\omega }{{\left({\epsilon }^{k,\ast }\right)}^{\alpha +\beta }}\\ +\text{\hspace{0.17em}}2\gamma {\varphi }^{\prime }\left(\Delta \left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\right)\frac{\sum _{j=1}^{m}{\int }_{\Omega }\mathrm{max}\left\{0,{g}_{j}\left({x}^{k,\ast },\omega \right)-{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\right\}{\left({\epsilon }^{k,\ast }\right)}^{\gamma }\text{d}\omega }{{\left({\epsilon }^{k,\ast }\right)}^{\alpha +\beta }}=\beta {\sigma }_{k}\end{array}$ (2.5)

$f\left(x\right)={f}_{{\sigma }_{k}}\left(x,0\right)\ge {f}_{{\sigma }_{k}}\left({x}^{\ast },0\right)=f\left(x\ast \right)$

${f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\le {f}_{{\sigma }_{k}}\left({x}^{\ast },0\right)=f\left({x}^{\ast }\right),\text{ }\forall k\ge 1$ (2.6)

${f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\le {f}_{{\sigma }_{k}}\left({x}^{k+1,\ast },{\epsilon }^{k+1,\ast }\right)\le {f}_{{\sigma }_{k+1}}\left({x}^{k+1,\ast },{\epsilon }^{k+1,\ast }\right)$

$\underset{k\to +\infty }{\mathrm{lim}}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\le f\left(x\ast \right)$

$\underset{k\to +\infty }{\mathrm{lim}}{f}_{{\sigma }_{k}}\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)\ge \underset{k\to +\infty }{\mathrm{lim}}f\left({x}^{k,\ast }\right)=f\left(\overline{x}\right)\ge f\left(x\ast \right)$

$f\left(\overline{x}\right)=f\left({x}^{\ast }\right)$ ，这表明聚点 $\overline{x}$ 为原问题(P)的全局最优解。

3. 罚函数算法

1) 令 ${\sigma }_{0}=1,\text{\hspace{0.17em}}{\epsilon }_{0}=0.1,\text{\hspace{0.17em}}{\sigma }^{\ast }={10}^{5},\text{\hspace{0.17em}}{\epsilon }^{\ast }={10}^{-9}$ ，设初始点为 $\left({x}_{0},{\epsilon }_{0}\right)$ ，迭代指数 $k=0$ ，选择合适的参数 $\alpha ,\beta ,\gamma$ ，其中参数的选择依赖于相应的原问题(P)。

2) 求解罚问题 $\left({P}_{{\sigma }_{k}}\right)$ ，令 $\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)$ 为其得到的最优解。

3) 如果 ${\epsilon }^{k,\ast }>{\epsilon }^{\ast },\text{\hspace{0.17em}}{\sigma }_{k}<{\sigma }^{\ast }$ ，令 ${\sigma }_{k+1}=10×{\sigma }_{k},k:=k+1$ ，返回到步骤(2)中，其中 $\left({x}^{k,\ast },{\epsilon }^{k,\ast }\right)$ 作为新的初始解进行。否则，令 ${\epsilon }^{k,\ast }:={\epsilon }^{\ast }$ ，进行步骤(4)。

4) 检验 ${x}^{k,\ast }$ 是否可行。如果 ${x}^{k,\ast }$ 是一可行解，则它是原问题(P)的局部最优解，否则进行步骤(5)。

5) 调整参数 $\alpha ,\beta ,\gamma$ ，令 $k:=0$ ，返回步骤(2)。

$\begin{array}{l}\mathrm{min}\text{}{\left({x}_{1}\right)}^{2}+{\left({x}_{2}-3\right)}^{2}\\ \text{s}\text{.t}\text{.}\text{ }{x}_{2}-2+{x}_{1}\mathrm{sin}\left(\frac{t}{{x}_{2}-\omega }\right)\le 0,\forall t\in \left[0,\text{π}\right],\\ \text{ }\text{ }-1\le {x}_{1}\le 1,\text{\hspace{0.17em}}0\le {x}_{2}\le 2\end{array}$

$\Delta \left(x,\epsilon \right)={\int }_{0}^{\text{π}}{\left[\mathrm{max}\left\{0,{x}_{2}-2+{x}_{1}\mathrm{sin}\left(\frac{t}{{x}_{2}-\omega }\right)-{\epsilon }^{\gamma }{W}_{j}\right\}\right]}^{2}\text{d}t$

Table 1. Numerical results of Example 1

Table 2. Numerical results of Example 2

$\begin{array}{l}\mathrm{min}\text{}{\left({x}_{1}+{x}_{2}-2\right)}^{2}+{\left({x}_{1}-{x}_{2}\right)}^{2}+30{\left[\mathrm{min}\left\{0,{x}_{1}-{x}_{2}\right\}\right]}^{2}\\ \text{s}\text{.t}\text{.}\text{ }{x}_{1}\mathrm{cos}t+{x}_{2}x\mathrm{sin}t-1\le 0,\forall t\in \left[0,\text{π}\right]\end{array}$

A New Exact Penalty Function Method for Solving Semi-Infinite Programming Problems[J]. 运筹与模糊学, 2017, 07(04): 138-147. http://dx.doi.org/10.12677/ORF.2017.74014

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