## Approximate Farkas Lemma and Approximate Duality for Fractional Optimization Problems

Xie Feifei,, Fang Donghui,*

College of Mathematics and Statistics, Jishou University, Hunan Jishou 416000

 基金资助: 国家自然科学基金.  12261037国家自然科学基金.  11861033湖南省自然科学基金.  2020JJ4494吉首大学校级科研项目.  Jdy20064

Revised: 2022-09-20

 Fund supported: The NSFC.  12261037The NSFC.  11861033Natural Science Foundation of Hunan Province.  2020JJ4494Scientific Research Fund of Jishou University.  Jdy20064

Abstract

By using the infimal convolution of conjugate functions and the epigraph technique, we introduce some new constraint qualifications. Under those new constraint qualifications, approximate Farkas lemmas and approximate duality results of the fractional optimization problem with conic constraint are established, which extend the corresponding results in the previous papers.

Keywords： Fractional optimization problem ; Constraint qualification ; Approximate Farkas lemma ; Approximate duality

Xie Feifei, Fang Donghui. Approximate Farkas Lemma and Approximate Duality for Fractional Optimization Problems. Acta Mathematica Scientia[J], 2023, 43(1): 305-320 doi:

## 1 引言

$\begin{eqnarray*} (P)\quad\quad\quad\quad &&\inf \ \frac{f(x)}{g(x)}\\ \mbox{s.t. } &&x\in {C},h(x)\in -S, \end{eqnarray*}$

$D^\oplus := \left\{ x^{*} \in X^{*} : \langle x^{*},x\rangle\ge 0, \forall x\in D \right\},$
$\delta_D(x) := \left\{\begin{array}{ll} 0, \quad &{x\in D,}\\ +\infty, &\mbox{其它}. \end{array}\right.$

$f:X\rightarrow \overline{\Bbb R}$ 是真凸函数, 定义 $f$ 的有效定义域, 上图和共轭函数分别为

${\rm dom}\,f:=\{x\in X: f(x)<+\infty\},$
${\rm epi}\, f:=\{(x,r)\in X\times \Bbb R:\; f(x)\le r\},$
$f^\ast(x^\ast):=\sup\{\langle x^\ast,x\rangle-f(x):\; x\in X\}, \forall x^\ast\in X^\ast.$

${\rm cl}\,f$ 表示 $f$ 的下半连续包, 即

${\rm epi}\,({\rm cl}\,f)={\rm cl}\,({\rm epi}\,f).$

$f^*=({\rm cl}\,f)^*,\quad f^{**}:=(f^*)^*\leq{\rm cl}\,f\leq f,$

$$$\label{eq2.11} f(x)+f^\ast(x^\ast)\geq\langle x^\ast,x\rangle,\forall(x,x^\ast)\in X\times X^\ast.$$$

$g,h:X\rightarrow\overline{\Bbb R}$ 为真凸函数, 且满足 ${\rm dom}\,g\cap{\rm dom}\,h\neq\emptyset$, 则

${\rm epi}\,g^*+{\rm epi}\,h^*\subseteq{\rm epi}\,(g+h)^*,$
$$$\label{eq2.12} g\leq h\Rightarrow g^*\geq h^*\Leftrightarrow{\rm epi}\,g^*\subseteq{\rm epi}\,h^*.$$$

$\phi(tx_1+(1-t)x_2)\leq_S t\phi(x_1)+(1-t)\phi(x_2),$

$$$\label{eq3.1} \inf\limits_{x\in A}\{f(x)-\mu g(x)-\langle p,x\rangle\}=-(f-\mu g+\delta_A)^*(p),\forall p\in X^*.$$$

$$$\label{eq3.2} (p,r)\in {\rm epi}\, (f-\mu g+\delta_A)^*\Leftrightarrow v(P_{(\mu,p)})\geq -r.$$$

$$$\label{eq3.31} v(P_{(\mu,p)})\geq v(D_{(\mu,p)}), \forall p\in X^*.$$$

$\varepsilon\geq0.$ 下面给出问题 $(P_\mu)$$(D_\mu) 之间的近似对偶间隙性质和稳定近似对偶间隙性质的相关定义. 定义 3.1 (i) 若 v(P_\mu)-v(D_\mu)\leq\varepsilon, 则称问题 (P_\mu)$$(D_\mu)$ 之间的近似对偶间隙性质成立.

(ii) 若对任意的 $p\in X^*$$v(P_{(\mu,p)})-v(D_{(\mu,p)})\leq\varepsilon, 则称问题 (P_\mu)$$(D_\mu)$ 之间的稳定近似对偶间隙性质成立.

$$$\label{eq3.3} {\rm epi}\,(f-\mu g+\delta_A)^*\subseteq {\rm epi}\,(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)-(0,\varepsilon),$$$

$\begin{eqnarray*} {\rm epi}\,(f+\delta_A)^*\subseteq {\rm epi}\,(f^*\Box h^\diamond\Box\delta_C^*)-(0,\varepsilon). \end{eqnarray*}$

$v(D_{(\mu,p)})=-(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)(p).$

$\begin{eqnarray*} & &(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)(p) \\ & =&\inf\limits_{(y^*,u^*,z^*)\in X^*\times X^*\times X^* } \{f^*(y^*)+(-{\mu}g)^*(u^*) +h^\diamond(z^*)+\delta_C^*(p-u^*-y^*-z^*)\}\\ &=&\inf\limits_{(y^*,u^*,z^*)\in X^*\times X^*\times X^* } \{f^*(y^*)+(-{\mu}g)^*(u^*)+\inf\limits_{\lambda\in S^\oplus}(\lambda h)^*(z^*)+\delta_C^*(p-u^*-y^*-z^*)\}\\ &=&\inf\limits_{(\lambda,y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^* } \{f^*(y^*)+(-{\mu}g)^*(u^*)+(\lambda h)^*(z^*)+\delta_C^*(p-u^*-y^*-z^*)\}\\ &=&-v(D_{(\mu,p)}). \end{eqnarray*}$

$\begin{eqnarray*} v(P_{(\mu,p)})-v(D_{(\mu,p)})\leq\varepsilon, \forall p\in X^*. \end{eqnarray*}$

$(p,r)\in{\rm epi}\,(f-\mu g+\delta_A)^*$. 由(3.2)式可知 $v(P_{(\mu,p)})\geq -r.$$v(D_{(\mu,p)})\geq-r-\varepsilon. 由命题 3.1 可知 $$\label{eq3.12} (f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)(p)\leq r+\varepsilon.$$ 因此, (p,r+\varepsilon)\in{\rm epi}\,(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*),$$(p,r)\in{\rm epi}\,(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)-(0,\varepsilon)$. 从而(3.4)式成立.

$(p,r)\in{\rm epi}\,(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)-(0,\varepsilon),$

(ii) 若对任意的 $p\in X^*$, $v(P_{(\mu,p)})=v(D_{(\mu,p)})$ 且问题 $(D_{(\mu,p)})$ 有最优解, 则称问题 $(P_\mu)$$(D_\mu) 之间的稳定强对偶成立. (iii) 若存在 ({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*, 使得 \begin{eqnarray*} v(P_{\mu}) \leq- f^*(y^*)-(-\mu g)^*(u^*)-(\lambda h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)+\varepsilon, \end{eqnarray*} 则称问题 (P_\mu)$$(D_\mu)$ 之间的近似强对偶成立.

(iv) 若对任意的 $p\in X^*$, 存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得

$$$\label{eq3.47} v(P_{(\mu,p)}) \leq- f^*(y^*)-(-\mu g)^*(u^*)-(\lambda h)^*(z^*) -\delta_C^*(p-y^*-u^*-z^*)+\varepsilon,$$$

$$$\label{eq3.13} {\rm epi}\,(f-\mu g+\delta_A)^*\subseteq \Lambda-(0,\varepsilon),$$$

$\begin{eqnarray*} {\rm epi}\,(f+\delta_A)^*\subseteq {\rm epi}\,f^*+\bigcup_{\lambda\in S^\oplus}{\rm epi}\,(\lambda h)^*+{\rm epi}\,\delta_C^*-(0,\varepsilon). \end{eqnarray*}$

(ii) 由文献[10]易知 $\Lambda\subseteq{\rm epi}\,(f-\mu g+\delta_A)^*$, 故当 $\varepsilon=0,$$(RCE_1) 条件转化为文献[10]中的 $$\label{eq3.14} {\rm epi}\,(f-\mu g+\delta_A)^*=\Lambda.$$ 命题 3.2 下面关系成立 (RCE_1)\mbox{条件}\Rightarrow (RCI_1)\mbox{条件}. 假设 (RCE_1) 条件成立. 设 (p,r)\in{\rm epi}\,(f-\mu g+\delta_A)^*, 由(3.7)式可知, 存在 {\lambda}\in S^\oplus, 使得 (p,r)\in{\rm epi}\,f^*+{\rm epi}\,(-\mu g)^*+{\rm epi}\,({\lambda} h)^*+{\rm epi}\,\delta_C^*-(0,\varepsilon). 于是, 由(2.3)和(2.2)式可知 (p,r)\in {\rm epi}\,(f^*\Box(-\mu g)^*\Box h^\diamond\Box\delta_C^*)-(0,\varepsilon). 因此, (RCI_1) 条件成立. 定理 3.3 考虑以下命题 (i) (RCE_1) 条件成立. (ii) 若对任意的 (p, x)\in X^\ast\times A$$r\in \Bbb R$$f(x)-\mu g(x)-\langle p,x\rangle\geq-r, 则存在 ({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*, 使得 $$\label{eq3.15} - f^*(y^*) -(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(p-y^*-u^*-z^*)\geq-r-\varepsilon.$$ (iii) 问题 (P_\mu)$$(D_\mu)$ 之间的稳定近似强对偶成立.

(iv) 若对任意的 $(p, x)\in X^\ast\times A$$\frac{f(x)-\langle p,x\rangle}{ g(x)}\geq\mu, 则存在 ({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*, 使得 $$\label{eq3.16} - f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(p-y^*-u^*-z^*)\geq-\varepsilon.$$ 则有 (i)\Leftrightarrow (ii)\Leftrightarrow(iii)\Rightarrow(iv). (i)\Rightarrow(ii). 假设 (RCE_1) 条件成立. 设 (p,x)\in X^*\times A 满足 f(x)-\mu g(x)-\langle p,x\rangle\geq-r, 则由(3.1)式可知 -(f-\mu g+\delta_A)^*(p)\geq-r. 于是 (p,r)\in{\rm epi}\,(f-\mu g+\delta_A)^* . 从而由(3.7)式可知 (p,r)\in\Lambda-(0,\varepsilon), 即存在 {\lambda}\in S^\oplus,(y^*,r_1)\in {\rm epi}\,f^*,(u^*,r_2)\in {\rm epi}\,(-\mu g)^*,(z^*,r_3)\in{\rm epi}\,({\lambda} h)^*,(w^*, r_4)\in{\rm epi}\,\delta_C^*, 使得 p=y^*+u^*+z^*+w^*, r=r_1+r_2+r_3+r_4-\varepsilon. 因为 f^*(y^*)\leq r_1,(-\mu g)^*(u^*)\leq r_2,({\lambda}h)^*(z^*)\leq r_3,\delta_C^*(w^*)\leq r_4, 所以由(3.11)和(3.12)式可知 r \geq f^*(y^*)+(-\mu g)^*(u^*)+({\lambda} h)^*(z^*) +\delta_C^*(p-y^*-u^*-z^*)-\varepsilon, 即(3.9)式成立. (ii)\Rightarrow(iii). 假设 (ii) 成立. 设 p\in X^*,-r:=v(P_{(\mu,p)})\in\Bbb R, 则对任意的 x\in A$$f(x)-\mu g(x)-\langle p,x\rangle\geq-r$. 从而由 (ii) 可知, 存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得(3.6)式成立. 故问题 $(P_\mu)$$(D_\mu) 之间的稳定近似强对偶成立. (iii)\Rightarrow(i). 假设 (iii) 成立. 设 (p,r)\in{\rm epi}\,(f-\mu g+\delta_A)^* , 则由(3.2)式可知 v(P_{(\mu,p)})\geq-r. 于是由问题 (P_\mu)$$(D_\mu)$ 之间的稳定近似强对偶成立可知, 存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得

$-r \leq- f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(p-y^*-u^*-z^*)+\varepsilon,$

$\delta_C^*(p-y^*-u^*-z^*)\leq r-f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*)+\varepsilon.$

$\begin{eqnarray*} (p,r)&=&(y^*,f^*(y^*))+(u^*,(-\mu g)^*(u^*))+(z^*,({\lambda} h)^*(z^*))\\ & & +(p-y^*-u^*-z^*, r-f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*)+\varepsilon)-(0,\varepsilon)\\ & \in& {\rm epi}\,f^*+{\rm epi}\,(-\mu g)^*+{\rm epi}\,({\lambda} h)^*+{\rm epi}\,\delta^*_C-(0,\varepsilon). \end{eqnarray*}$

(iii)$\Rightarrow$(iv). 设 $(p,x)\in X^*\times A$ 满足 $\frac{f(x)-\langle p,x\rangle}{g(x)}\geq\mu$, 则由引理 3.1 可知 $v(P_{(\mu,p)})\geq0.$ 于是由 (iii) 可知, 存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得(3.10)式成立. 证毕.

(i) ${\rm epi}\,(f-\mu g+\delta_A)^*\cap(\{0\}\times \Bbb R)\subseteq \Lambda \cap(\{0\}\times \Bbb R)-(0,\varepsilon).$

(ii) 若对任意的 $x\in A$$r\in\Bbb R$$f(x)-\mu g(x)\geq-r$, 则存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得

$\begin{eqnarray*} - f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)\geq-r-\varepsilon. \end{eqnarray*}$

(iii) 问题 $(P_\mu)$$(D_\mu) 之间的近似强对偶成立. (iv) 若对任意的 x\in A$$\frac{f(x)}{ g(x)}\geq\mu$, 则存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得

$\begin{eqnarray*} - f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)\geq-\varepsilon. \end{eqnarray*}$

$f(x) := \left\{\begin{array}{ll} x+2, \quad &{x>0},\\ \frac{5}{2},&{x=0},\\ +\infty, &x<0, \end{array}\right.\quad g(x) := \left\{\begin{array}{ll} x+3, \quad &{x\geq0},\\ +\infty, &x<0. \end{array}\right.$

$A=[0,+\infty)$.$\mu\leq0$ 时,

$$$\label{q1}(f-\mu g+\delta_A)(x) = \left\{\begin{array}{ll} (1-\mu)x+2-3\mu, \quad &{x>0},\\ \frac{5}{2}-3\mu, \quad &{x=0},\\ +\infty, &x<0. \end{array}\right.$$$

$(f-\mu g+\delta_A)^*(x^*) = \left\{\begin{array}{ll} 3\mu-2, \quad &{x^*\leq 1-\mu},\\ +\infty, &x^*>1-\mu. \end{array}\right.$

${\rm epi}\,(f-\mu g+\delta_A)^*=(-\infty,1-\mu]\times [3\mu-2, +\infty).$

$f^*(x^*) = \left\{\begin{array}{ll} -2, \quad &{x^*\leq1},\\ +\infty, &x^*>1, \end{array}\right.\quad (-\mu g)^*(x^*) = \left\{\begin{array}{ll} 3\mu, \quad &{x^*\leq -\mu},\\ +\infty, &x^*>-\mu, \end{array}\right.$
$\delta_C^*(x^*) = \left\{\begin{array}{ll} 0, \quad &{x^*=0},\\ +\infty, &x^*\neq0, \end{array}\right.\quad (\lambda h)^*(x^*) = \left\{\begin{array}{ll} 0, \quad &{x^*=-\lambda},\\ +\infty, &x^*\neq-\lambda, \end{array}\quad\forall \lambda\ge 0.\right.$

$\varepsilon\geq0$.$\mu \le -1$ 时, 由

${\rm epi}\,f^*+{\rm epi}\,(-\mu g)^*+\bigcup_{\lambda\in S^*}{\rm epi}\,(\lambda h)^*+{\rm epi}\,\delta_C^*-(0,\varepsilon)=(-\infty,2]\times [-5-\varepsilon,+\infty).$

$- f^*(y^*) -(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)=5\geq r-\varepsilon.$

$\frac{f(x)}{g(x)}=\left\{\begin{array}{ll} 1-\frac{1}{x+3}, \quad &{x>0},\\ \frac{5}{6}, \quad &{x=0},\\ +\infty, &x<0. \end{array}\right.$

(iv) 对任意的 $(p, x)\in X^\ast\times A$, $\frac{f(x)-\langle p,x\rangle}{ g(x)}\geq\mu$ 成立当且仅当存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得(3.14)式成立.

(i) ${\rm epi}\,(f-\mu g+\delta_A)^*\cap(\{0\}\times \Bbb R)= \Lambda\cap(\{0\}\times \Bbb R).$

(ii) 若对任意的 $x\in A$$r\in\Bbb R$$f(x)-\mu g(x)\geq-r$, 则存在 $({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*,$ 使得

$\begin{eqnarray*} - f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)\geq-r. \end{eqnarray*}$

(iii) 问题 $(P_\mu)$$(D_\mu) 之间的强对偶成立. (iv) 对任意的 x\in A, \frac{f(x)}{ g(x)}\geq\mu 成立当且仅当存在 ({\lambda},y^*,u^*,z^*)\in S^\oplus\times X^*\times X^*\times X^*, 使得 \begin{eqnarray*} - f^*(y^*)-(-\mu g)^*(u^*)-({\lambda} h)^*(z^*) -\delta_C^*(-y^*-u^*-z^*)\geq0. \end{eqnarray*} 则有 (i)\Leftrightarrow(ii)\Leftrightarrow(iii)\Rightarrow(iv). 注 3.3 在假设 f,g 为真凸下半连续函数, C 为闭凸集, h$$S$ - 上图闭函数的情形下, 文献[11]利用闭性条件

$(CC)_1 \quad{\rm epi}\,f^*+ {\rm epi}\,(-{\mu}g)^*+\bigcup_{\lambda\in S^{\oplus}}{\rm epi}\,(\lambda h)^*+{\rm epi}\,\delta^*_C\quad\mbox{是弱}^\ast\mbox{闭凸集},$

### 3.2 $\mu>0$ 的情形

$\mu>0,p\in X^*$. 考虑带线性扰动的分式优化问题 $(P_p)$ 和约束优化问题 $(P_{(\mu,p)})$.$\mu>0$ 时, 问题 $(P_{(\mu,p)})$ 的目标函数为 DC 函数. 利用凸化技巧, 定义问题 $(P_{(\mu,p)})$ 的 Fenchel-Lagrange 对偶问题如下

$\begin{eqnarray*} (\overline{D_{(\mu,p)}}) \inf\limits_{u^*\in {\rm dom}\,g^*}\sup_{(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^* } \{({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)\}, \end{eqnarray*}$

$\begin{eqnarray*} (\overline{D_{(\mu,p)}^{u^*}}) \quad \sup_{(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^* }\{({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)\}. \end{eqnarray*}$

$\begin{eqnarray*} (\overline{D_{\mu}}) \quad \inf\limits_{u^*\in {\rm dom}\,g^*} \sup_{(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^* }\{({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)\}, \end{eqnarray*}$

$\begin{eqnarray*} (\overline{D_{\mu}^{u^*}}) \quad\quad \sup_{(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^* } \{({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)\}. \end{eqnarray*}$

$v(\overline{D_{\mu}})$, $v(\overline{D_{\mu}^{u^*}})$, $v(\overline{D_{(\mu,p)}})$ 以及 $v(\overline{D_{(\mu,p)}^{u^*}})$ 分别表示问题 $(\overline{D_{\mu}})$, $(\overline{D_{\mu}^{u^*}})$, $(\overline{D_{(\mu,p)}})$ 以及 $(\overline{D_{(\mu,p)}^{u^*}})$ 的最优值. 若 $v(P_{\mu})\geq v(\overline{D_{\mu}}),$ 则称问题 $(P_{\mu})$$(\overline{D_{\mu}}) 之间的弱对偶成立. 若对任意的 p\in X^*$$v(P_{(\mu,p)})\geq v(\overline{D_{(\mu,p)}}),$ 则称问题 $(P_{\mu})$$(\overline{D_{\mu}}) 之间的稳定弱对偶成立. 为刻画 (P_{\mu})$$(\overline{D_{\mu}})$ 之间的弱对偶, 引入如下约束规范条件

$$$\label{eq3.19} {\rm epi}\,(f-\mu g+\delta_A)^*= {\rm epi}\,(f-{\rm cl}\,(\mu g)+\delta_A)^*.$$$

$$$\label{eq3.20} {\rm epi}\,(f-\mu g+\delta_A)^*\subseteq \bigcap_{u^*\in {\rm dom}\,g^*} \bigg({\rm epi}\,(f^*\Box h^\diamond\Box\delta_C^*)-(u^*,(\mu g)^*(u^*))\bigg)-(0,\varepsilon),$$$

$v(\overline{D_{(\mu,p)}})\leq v(P_{(\mu,p)})\leq v(\overline{D_{(\mu,p)}})+\varepsilon,\forall p\in X^*.$

$(p,r)\in{\rm epi}\,(f-\mu g+\delta_A)^*$, 由(3.2)式可知 $v(P_{(\mu,p)})\geq -r.$ 于是 $v(\overline{D_{(\mu,p)}})\geq-r-\varepsilon.$ 故由 $v(\overline{D_{(\mu,p)}})$ 的定义可知, 对任意的 $u^*\in{\rm dom}\,g^*$, 存在 $(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^*,$ 使得

$(\mu g)^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)\geq-r-\varepsilon,$

$$$\label{eq3.48} \inf\limits_{(\lambda,y^*,z^*)\in S^\oplus\times X^*\times X^*}\{f^*(y^*)+(\lambda h)^*(z^*) +\delta_C^*(p+u^*-y^*-z^*)\}\leq r+(\mu g)^*(u^*)+\varepsilon.$$$

${\rm epi}(f\!-\!\mu g\!+\!\delta_A)^*\cap(\{0\}\!\times\! \Bbb R)\subseteq \bigcap_{u^*\in {\rm dom}\,g^*} ({\rm epi}(f^*\Box h^\diamond\Box\delta_C^*)\!-\!(u^*,(\mu g)^*(u^*)))\cap(\{0\}\times \Bbb R)\!-\!(0,\varepsilon).$

(ii) 若对任意的 $p\in X^*$, $v({P_{(\mu,p)}})= v(\overline{D_{(\mu,p)}})$, 且对任意满足 $v(\overline{D_{(\mu,p)}^{u^*}})=v(\overline{D_{(\mu,p)}})$${u^*\in {\rm dom}\,g^*}, (\overline{D_{(\mu,p)}^{u^*}}) 有最优解, 则称问题 (P_\mu)$$(\overline{D_\mu})$ 之间的稳定强对偶成立.

(iii) 若 $v(\overline{D_{\mu}})\leq v({P_{\mu}})\leq v(\overline{D_{\mu}})+\varepsilon$, 且对任意满足 $v(\overline{D_{\mu}^{u^*}})=v(\overline{D_\mu})$${u^*\in {\rm dom}\,g^*}, 存在 ({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*, 使得 \begin{eqnarray*} v(P_{\mu}) \leq {({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)}+\varepsilon, \end{eqnarray*} 则称问题 (P_\mu)$$(\overline{D_\mu})$ 之间的近似强对偶成立.

(iv) 若对任意的 $p\in X^*$, $v(\overline{D_{(\mu,p)}})\leq v({P_{(\mu,p)}})\leq v(\overline{D_{(\mu,p)}})+\varepsilon$, 且对任意满足 $v(\overline{D_{(\mu,p)}^{u^*}})=v(\overline{D_{(\mu,p)}})$${u^*\in {\rm dom}\,g^*}, 存在 ({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*, 使得 $$\label{eq3.22} v({P_{(\mu,p)}}) \leq({\mu g})^*(u^*)-f^*(y^*)-(\lambda h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)+\varepsilon,$$ 则称问题 (P_\mu)$$(\overline{D_\mu})$ 之间的稳定近似强对偶成立.

$\Omega:=\bigcap_{u^*\in{\rm dom}\,g^*}\bigg({\rm epi}\,f^*+\bigcup_{\lambda\in S^\oplus}{\rm epi}\,(\lambda h)^*+{\rm epi}\,\delta_C^*-(u^*,(\mu g)^*(u^*))\bigg).$

$$$\label{eq3.23} {\rm epi}\,(f-\mu g+\delta_A)^*\subseteq \Omega-(0,\varepsilon),$$$

(ii) 假设(3.16)式成立, 由文献[10]可知 $\Omega\subseteq{\rm epi}\,(f-\mu g+\delta_A)^*$, 故当 $\varepsilon=0$ 时, $(RCE_2)$ 条件转化为文献[10]中的

$$$\label{eq3.24} {\rm epi}\,(f-\mu g+\delta_A)^*= \Omega.$$$

$(p,r)\in{\rm epi}\,f^*+{\rm epi}\,({\lambda} h)^*+{\rm epi}\,\delta_C^*-(u^*,(\mu g)^*(u^*))-(0,\varepsilon).$

$(p+u^*,r+(\mu g)^*(u^*)+\varepsilon) \in{{\rm epi}\,f^*+{\rm epi}\,({\lambda} h)^*+{\rm epi}\,\delta_C^*} \subseteq {\rm epi}\,(f^*\Box h^\diamond\Box\delta_C^*).$

$(p,r)\in\bigcap_{u^*\in {\rm dom}\,g^*} ({\rm epi}\,(f^*\Box h^\diamond\Box\delta_C^*)-(u^*,(\mu g)^*(u^*)))-(0,\varepsilon),$

(ii) 设 $p\in X^*$. 假设 $(RCE_2)$ 条件成立. 若对任意的 $x\in A$$\frac{f(x)-\langle p,x\rangle}{ g(x)}\geq\mu, 则对任意的 u^*\in{\rm dom}\,g^*, 存在 ({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*, 使得 $$\label{eq3.27} (\mu g)^*(u^*)-f^*(y^*)-({\lambda} h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)\geq-\varepsilon.$$ (i) 设 p\in X^*. 由引理 3.2 可知 v(P_{(\mu,p)})\geq v(\overline{D_{(\mu,p)}}). 下证 v({P_{(\mu,p)}})\leq v(\overline{D_{(\mu,p)}})+\varepsilon, 且对任意满足 v(\overline{D_{(\mu,p)}^{u^*}})=v(\overline{D_{(\mu,p)}})$${u^*\in {\rm dom}\,g^*}$, 存在 $({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*,$ 使得(3.19)式成立. 若 $v({P_{(\mu,p)}})=-\infty,$ 则结论自然成立. 下设 $v({P_{(\mu,p)}})=-r\in \Bbb R$. 由(3.2)和(3.20)式可知 $(p,r)\in\Omega-(0,\varepsilon).$ 于是, 由命题 3.5 可知 $v(\overline{D_{(\mu,p)}})\geq-r-\varepsilon$, 且对任意满足 $v(\overline{D_{(\mu,p)}^{u^*}})=v(\overline{D_{(\mu,p)}})$${u^*\in {\rm dom}\,g^*}, 存在 ({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*, 使得(3.19)式成立. 因此, 问题 (P_\mu)$$(\overline{D_\mu})$ 之间的稳定近似强对偶成立.

(ii) 设 $x\in A$.$\frac{f(x)-\langle p,x\rangle}{ g(x)}\geq\mu$, 则由引理 3.1 可知 $v(P_{(\mu,p)})\geq0.$ 于是由(3.2)和(3.20)式可知 $(p,0)\in\Omega-(0,\varepsilon).$ 从而由命题 3.5 可知, 对任意的 $u^*\in{\rm dom}\,g^*$, 存在 $({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*,$ 使得(3.23)式成立.证毕.

$$$\label{eq3.28} {\rm epi}\,(f-\mu g+\delta_A)^*\cap(\{0\}\times \Bbb R)\subseteq \Omega\cap(\{0\}\times \Bbb R)-(0,\varepsilon),$$$

$\begin{eqnarray*} (\mu g)^*(u^*)-f^*(y^*)-({\lambda} h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)\geq-\varepsilon. \end{eqnarray*}$

$f(x) := \left\{\begin{array}{ll} x+2, \quad &{x\geq0},\\ +\infty, &x<0, \end{array}\right.\quad g(x) := \left\{\begin{array}{ll} x+1, \quad &{x\geq0},\\ +\infty, &x<0. \end{array}\right.$

$A=[0,+\infty)$.$\varepsilon\geq0$, $\mu> 0$. 则有

$$$\label{q2}(f-\mu g+\delta_A)(x) = \left\{\begin{array}{ll} (1-\mu)x+2-\mu, \quad &{x\geq0},\\ +\infty, &x<0. \end{array}\right.$$$

$(f-\mu g+\delta_A)^*(x^*) = \left\{\begin{array}{ll} \mu-2, \quad &{x^*\leq 1-\mu},\\ +\infty, &x^*>1-\mu. \end{array}\right.$

${\rm epi}\,(f-\mu g+\delta_A)^*=(-\infty,1-\mu]\times [\mu-2,+\infty).$

$f^*(x^*) = \left\{\begin{array}{ll} -2, \quad &{x^*\leq1},\\ +\infty, &x^*>1, \end{array}\right.\quad (\mu g)^*(x^*) = \left\{\begin{array}{ll} -\mu, \quad &{x^*\leq \mu},\\ +\infty, &x^*>\mu, \end{array}\right.$
$\delta_C^*(x^*) = \left\{\begin{array}{ll} 0, \quad &{x^*=0},\\ +\infty, &x^*\neq0, \end{array}\right.\quad (\lambda h)^*(x^*) = \left\{\begin{array}{ll} 0, \quad &{x^*=-\lambda},\\ +\infty, &x^*\neq-\lambda, \end{array}\right.\quad \forall \lambda\ge 0.$

$\begin{eqnarray*} \Omega-(0,\varepsilon)&= & \bigcap_{u^*\in{\rm dom}\,g^*}\bigg({\rm epi}\,f^*+\bigcup_{\lambda\in S^\oplus}{\rm epi}\,(\lambda h)^*+{\rm epi}\,\delta_C^*-(u^*,(\mu g)^*(u^*))\bigg) \\ & = & (-\infty,0]\times [-1-\varepsilon,+\infty), \end{eqnarray*}$

$(\mu g)^*(u^*)- f^*(y^*) -({\lambda} h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)=1\geq-\varepsilon.$

$$$\label{eq3.42} (\mu g)^*(u^*)-f^*(y^*)-({\lambda} h)^*(z^*) -\delta_C^*(p+u^*-y^*-z^*)\geq0,$$$

$\varepsilon=0$ 时, 由注 3.5(ii), 定理 3.7 和 3.8 以及命题 3.6 可得以下推论, 其中推论 3.4 即为文献 [10,定理 3.15].

(i) 问题 $(P_\mu)$$(\overline{D_{\mu}}) 之间的稳定强对偶成立. (ii) 对任意的 (p,x )\in X^*\times A, \frac{f(x)-\langle p,x\rangle}{ g(x)}\geq\mu 成立当且仅当对任意的 u^*\in{\rm dom}\,g^*, 存在 ({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*, 使得(3.26)成立. 推论 3.4 假设 {\rm epi}\,(f-\mu g+\delta_A)^*\cap(\{0\}\times \Bbb R)= \Omega\cap(\{0\}\times \Bbb R), 以下命题成立 (i) 问题 (P_\mu)$$(\overline{D_{\mu}})$ 之间的强对偶成立.

(ii) 对任意的 $x\in A$, $\frac{f(x)}{ g(x)}\geq\mu$ 成立当且仅当对任意的 $u^*\in{\rm dom}\,g^*$, 存在 $({\lambda},y^*,z^*)\in S^\oplus\times X^*\times X^*,$ 使得

$\begin{eqnarray*} (\mu g)^*(u^*)-f^*(y^*)-({\lambda} h)^*(z^*) -\delta_C^*(u^*-y^*-z^*)\geq0. \end{eqnarray*}$

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