## Optimality Conditions for DC Composite Optimization Problems with Conical Constraints

Hu Lingli,, Tian Liping,, Fang Donghui,

 基金资助: 国家自然科学基金.  11861033湖南省自然科学基金.  2020JJ4494吉首大学科研基金.  Jd20008吉首大学科研基金.  Jd20009

 Fund supported: the NSFC.  11861033the NSF of Hunan Province.  2020JJ4494the Scientific Research Fund of Jishou University.  Jd20008the Scientific Research Fund of Jishou University.  Jd20009

Abstract

In the case when the functions are not necessarily lower semicontinuous and the sets are not necessarily closed, we first define the dual problem for DC composite optimization problems with conical constraints by using convexification technique, then some optimality conditions and saddle point theorems are obtained, which extend the corresponding results in the previous papers.

Keywords： DC composite optimization problem ; Optimality condition ; Saddle point theorem

Hu Lingli, Tian Liping, Fang Donghui. Optimality Conditions for DC Composite Optimization Problems with Conical Constraints. Acta Mathematica Scientia[J], 2021, 41(4): 1079-1087 doi:

## 1 引言

$X$, $Y$, $Z$是实局部凸Hausdorff拓扑向量空间, $X^\ast$, $Y^\ast$, $Z^\ast$分别表示$X$, $Y$, $Z$的共轭空间且$Y $$Z 分别在闭凸锥 K\subseteq Y$$ S\subseteq Z$下有序. 记$Y^\bullet: = Y\cup \{\infty_Y\}$, $Z^\bullet: = Z\cup \{\infty_Z\}$, 其中$\infty_Y$, $\infty_Z$分别表示$Y $$Z 中关于偏序关系 \leq_K , \leq_S 下的最大元. 设 f:X\rightarrow \overline{{{\Bbb R}}}: = {{\Bbb R}}\cup\{+\infty\} 是真凸函数, h:X\to Z^\bullet 是真 S -凸函数, C$$ X$中的非空凸子集. 许多学者研究了如下经典的锥约束优化问题

$$$\label{eq00-1} (P_1)\quad\quad \begin{array}{ll} \mathrm{inf }&f(x)\\ \mathrm{s.t.}&x\in C, h(x)\in-S, \nonumber \end{array}$$$

## 2 预备知识

$X$, $Y$是实局部凸Hausdorff拓扑向量空间, $X^{*} $$Y^{*} 分别是 X$$ Y$的共轭空间, 分别赋予弱$^{*}$拓扑$\omega^{*}(X^{*}, X) $$\omega^{*}(Y^{*}, Y) . \langle x^\ast, x\rangle 表示泛函 x^\ast\in X^\ast 在点 x\in X 的值, 即 \langle x^\ast, x\rangle = x^\ast(x) . S$$ Y$中的闭凸锥, $Y $$S 所定义的序空间. 对于 Y 中的偏序 \le_S , 定义 Y 中的最大元为 \infty_Y . Y^\bullet = Y\cup\{\infty_Y\} . X 的子集 Z , \hbox{cl}Z$$ {\rm cone}Z$分别表示$Z$的闭包及凸锥包. 进一步, 若$Z $$X 的凸子集, Z^\oplus 表示 Z 的对偶锥, 即 N_Z(z_0) 表示 Z$$ z_0$点的法锥, 定义为

$\delta_Z$表示$Z$的示性函数, 定义为

$f:X\to \overline{{\Bbb R}}$是真凸函数, 分别定义$f$的有效定义域, 次微分, 共轭函数为

$$$N_{Z}(x) = \partial\delta_{Z}(x), \quad \forall x\in Z.$$$

$$$f(x)+f^\ast(x^\ast)\ge \langle x^\ast, x \rangle, \quad { \mathrm{ \forall } }(x, x^\ast)\in X\times X^\ast,$$$

$$$f(x)+f^*(x^*) = \langle x^*, x\rangle \Longleftrightarrow x^*\in \partial f(x) .$$$

$$$\partial f(a)+\partial h(a)\subseteq \partial (f+h)(a), \quad { \forall a\in {\rm{dom}}\, f\cap {\rm{dom}}\, h }.$$$

$$$\widehat{\partial}(\phi_{1}-\phi_{2})(x_{0})\subseteq \bigcap\limits_{u^{*}\in \widehat{\partial}\phi_{2}(x_{0})}(\widehat{\partial}\phi_{1}(x_{0})-u^{*}),$$$

## 3 复合优化问题的最优性条件与鞍点定理

$X$, $Y$, $Z$是实局部凸Hausdorff拓扑向量空间, $C $$X 中的非空凸子集, K$$ S$分别为$Y $$Z 中的闭凸锥, h:X\to Z^\bullet 是真 S -凸函数, f_1:Y^{\bullet}\rightarrow \overline{{{\Bbb R}}} 是真凸 K -增函数, f_2:X\rightarrow Y^\bullet 是真 K - 凸函数, g_1:Z\rightarrow \overline{{{{\Bbb R}}}} 是真凸 S -增函数, g_2:X\rightarrow Z^{\bullet} 是真 S -凸函数. 本文主要研究如下带锥约束的DC复合优化问题 A 表示问题 ({P}) 的解集, 即 A: = \{x\in C:\;h(x)\in -S\}. 为刻画问题 (P) 的KKT类最优性条件, 我们引入以下的约束规范条件, 详见文献[15]. 定义3.1 设 x_0\in A\cap f_{2}^{-1}({\rm{dom}} f_{1}) , 若 $$\partial(f_1\circ f_2+\delta_{A})(x_{0})\subseteq\bigcup\limits_{\mu\in\partial f_{1}(f_{2}(x_{0}))} \partial(\mu f_{2})(x_{0})+N_{C}(x_{0})+\bigcup\limits_{\lambda\in S^{\oplus}\atop (\lambda h)(x_{0}) = 0 }\partial(\lambda h)(x_{0}),$$ 则称系统 \{f_1, f_2, \delta_{C};\lambda h:\lambda\in S^{\oplus}\}$$ x_{0}$点满足$(CBCQ)$条件.

$$$x^{*}\in \partial(\mu f_{2})(x_{0})+N_{C}(x_{0})+\partial(\lambda h)(x_{0}).$$$

$\begin{eqnarray} (\omega g_{2})(x)-g_{1}^{*}(\omega)&\leq&-f_{1}^{*}(\mu)+(\mu f_{2})(x)+(\lambda h)(x)+\delta_{C}(x)+r+\epsilon, \forall x\in X. \end{eqnarray}$

$$$-f_{1}^{*}(\mu)+(\mu f_{2})(x)\leq f_{1}(f_{2}(x)) , \forall x\in X.$$$

$g_{1}$为下半连续函数, 因此

(ⅱ) $f_{1}(f_{2}(x_{0}))-g_{1}(g_{2}(x_{0})) = L_{\omega}(x_{0}, \bar\lambda, \bar\mu) = \inf\limits_{x\in C} L_{\omega}(x, \bar\lambda, \bar\mu).$

(ⅲ) $f_{1}(f_{2}(x_{0}))-g_{1}(g_{2}(x_{0}))\le L_{\omega}(x_{0}, \bar\lambda, \bar\mu) = \inf\limits_{x\in C} L_{\omega}(x, \bar\lambda, \bar\mu).$

(ⅳ) $(x_{0}, \bar\lambda, \bar\mu)$是Lagrange函数$L_{\omega}$的一个鞍点.

设${\omega}\in \partial g_{1}(g_{2}(x_0))$满足$v({D}) = v({D}^{\omega})$.

(ⅰ)$\Rightarrow$(ⅱ). 令$(\bar\lambda, \bar\mu)\in S^{\oplus}\times {\rm{dom}} f_{1}^{*}$使得(ⅰ)成立, 则

$$$f_{1}(f_{2}(x_{0}))-g_{1}(g_{2}(x_{0})) = v({P}) = v({D}^{\omega}) = \inf\limits_{x\in C} L_{\omega}(x, \bar\lambda, \bar\mu)\leq L_{\omega}(x_{0}, \bar\lambda, \bar\mu).$$$

$\omega\in \partial g_{1}(g_{2}(x_0)),$

$$$g_{1}^{*}(\omega)-(\omega g_{2})(x_{0}) = -g_{1}(g_{2}(x_0)).$$$

(ⅱ)$\Rightarrow$(ⅰ). 令$(\bar\lambda, \bar\mu)\in S^{\oplus}\times {\rm{dom}} f_{1}^{*}$使得(ⅱ)成立, 则

$$$v({P})\leq f_{1}(f_{2}(x_{0}))-g_{1}(g_{2}(x_{0})) = \inf\limits_{x\in C} L_{\omega}(x, \bar\lambda, \bar\mu)\leq v({D}^{\omega}).$$$

$g_{1}$为下半连续函数, 则由命题3.1可知问题$({P})$和问题$({D})$之间的弱对偶成立, 因此$v({P})\geq v({D}) = v({D}^{\omega}),$从而(3.9)式的不等号全变为等号, 故结论(ⅰ)成立.

(ⅱ)$\Leftrightarrow$(ⅲ). 由于问题$({P})$和问题$({D})$之间的弱对偶成立, 因此

(ⅱ)$\Rightarrow$(ⅳ). 令$(\bar\lambda, \bar\mu)\in S^{\oplus}\times {\rm{dom}} f_{1}^{*}$使得(ⅱ)成立, 则由前面的证明可得结论(ⅰ)成立, 故对任意的$(x, \lambda, \mu)\in C\times S^{\oplus}\times {\rm{dom}}f_{1}^{*}$

(ⅳ)$\Rightarrow$(ⅱ). 设$(x_{0}, \bar\lambda, \bar\mu) $$L_{\omega} 的鞍点, 则对任意的 x\in C$$ L_{\omega}(x_{0}, \bar\lambda, \bar\mu)\leq L_{\omega}(x, \bar\lambda, \bar{\mu}),$

$g_{2}$为单位算子时, 问题$({P})$的Lagrange函数$L_{\omega}$和Lagrange对偶问题$(D)$分别转化为

$\begin{eqnarray*} \overline L_{\omega}(x, \lambda, \mu) = g_{1}^{*}(\omega)-\langle\omega, x\rangle-f_{1}^{*}(\mu)+(\mu f_{2})(x)+(\lambda h)(x), \nonumber\\ \forall (x, \lambda, \mu)\in C\times S^{\oplus}\times {\rm{dom}} f_{1}^{*}, \end{eqnarray*}$

$$$( \overline{D})\quad\quad \begin{array}{ll} &\inf\limits _{ { \omega\in {\rm{dom}} g_{1}^{*}}\atop } \sup\limits _{ { \lambda\in S^\oplus}\atop {\mu \in{\rm{dom}}f^{*}_{1}} }\inf\limits_{x\in C} \overline L_{\omega}(x, \lambda, \mu). \nonumber \end{array}$$$

$\omega\in {\rm{dom}} g_{1}^{*}$, 定义$( \overline{D})$的子问题为

(ⅰ) $x_{0}\times(\bar\lambda, \bar\mu)\in S({P})\times S({\overline D}^{\omega}) $$v({P}) = v({\overline D}) . (ⅱ) f_{1}(f_{2}(x_{0}))-g_{1}(x_{0}) = \overline L_{\omega}(x_{0}, \bar\lambda, \bar\mu) = \inf\limits_{x\in C} \overline L_{\omega}(x, \bar\lambda, \bar\mu). (ⅲ) f_{1}(f_{2}(x_{0}))-g_{1}(x_{0})\le \overline L_{\omega}(x_{0}, \bar\lambda, \bar\mu) = \inf\limits_{x\in C} \overline L_{\omega}(x, \bar\lambda, \bar\mu). (ⅳ) (x_{0}, \bar\lambda, \bar\mu) 是Lagrange函数 \overline L_{\omega} 的一个鞍点. 注3.2 注意到, 推论3.1即为文献[15]中的定理5.1, 因此本文的定理3.2推广了文献[15]中的相关结论. 例3.1 设 X = Y = Z = C: = {{\Bbb R}} , K = S: = [0, +\infty) , f_{2}$$ g_{2} $${{\Bbb R}} 中的单位算子, g_1 = h: = \delta_{(-\infty, 0]} 且对任意的 x\in{{\Bbb R}}, f_1 是真凸 K -增函数, f_2 是真 K -凸函数, g_1 是真凸 S -增函数, g_2$$ h$是真$S$ -凸函数. 易知$A: = \{x\in{{\Bbb R}}:h(x)\in -S\} = (-\infty, 0]$, $S(P) = (-\infty, 0)$, $f_1\circ f_2-g_1\circ g_2+\delta_{A} = f_1$. 任取$x_{0}\in S(P),$则有$\partial (f_{1}\circ f_{2}+\delta_{A})(x_{0}) = \{0\}$

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