## Characterizations of Farkas Lemmas for a Class of Fractional Optimization with DC Functions

Feng Xinyi,, Sun Xiangkai,

 基金资助: 重庆市自然科学基金.  cstc2020jcyj-msxmX0016重庆市重点实验室开放课题.  KFJJ2019097重庆工商大学科研团队项目.  ZDPTTD201908重庆市巴渝学者青年学者项目

 Fund supported: the NSF of Chongqing.  cstc2020jcyj-msxmX0016the Open Research Platform of CTBU.  KFJJ2019097the Project of CTBU.  ZDPTTD201908the Education Committee Project Foundation of Chongqing for Bayu Young Scholar

Abstract

This paper deals with some new Farkas lemmas for a class of constraint fractional optimization with DC functions(the difference of convex functions). Following the idea due to Dinkelbach, we first associate the fractional optimization with a DC optimization problem. Then, by using the epigraph technique of the conjugate function, we introduce some new regularity conditions and establish the duality between the DC optimization problem and its Fenchel-Lagrange dual problem. Finally, we obtain some new Farkas lemmas for the fractional optimization problem. Furthermore, we also show that the results obtained in this paper extend and improve the corresponding results in the literature.

Keywords： Fractional optimization ; Regularity conditions ; Farkas lemmas

Feng Xinyi, Sun Xiangkai. Characterizations of Farkas Lemmas for a Class of Fractional Optimization with DC Functions. Acta Mathematica Scientia[J], 2021, 41(3): 827-836 doi:

## 1 引言

(ⅰ) ${\rm epi}(\sup\limits_{i\in I}{f_{i}}) = \bigcap\limits_{i\in I}{\rm epi }{f}_{i}.$

(ⅱ) $(\inf\limits_{i\in I}{f_{i}})^{*} = \sup\limits_{i\in I}{f_{i}^{*}};$ 因此, ${\rm epi}(\inf\limits_{i\in I}{f_{i}})^{*} = \bigcap\limits_{i\in I}{\rm epi }{f}_{i}^{*}.$

$p\in X^{*}, $$p 可看作 X 上的函数 p(x): = \langle p, x\rangle , \forall x\in X. 因此, 对任意 \alpha\in {{\Bbb R}} 以及任意函数 \phi: X\rightarrow \overline{{{\Bbb R}} }, $$(\phi+p+\alpha)^{*}(x^{*}) = \phi^{*}(x^{*}-p)-\alpha, \ \forall x^{*}\in X^{*}$$ $$\text{epi}(\phi+p+\alpha)^{*} = \text{epi}{\phi^{*}}+(p, -\alpha).$$ ### 3 正则性条件和Farkas引理 如上所述, 本文考虑如下分式优化问题 其中 A: = \{x\in C, h(x)\in -S\}. 本文中总是假设 g_{1}(x)-g_{2}(x)\geq 0 . 为刻画(P) 的一些新的Farkas引理, 利用Dinkelbach方法[13]将(P)转化为如下优化问题 其中 \mu\in {{\Bbb R}}. 为简便起见, 本文若无特殊说明, 优化问题(P)的最优值记为 {\rm val}(P) . 其它优化问题的最优值类似标记. 下面引理给出了(P)与(P _{\mu}) 最优值之间的一个等价关系. 引理3.1 不等式 {\rm val}(P)\geq\mu 成立的充分必要条件是不等式 {\rm val}(P_{\mu})\geq0. 接下来需要建立(P _{\mu}) 与其对偶问题的对偶关系, 从而得到(P)的一些新的Farkas引理. 因为(P _\mu) 的目标函数由 \mu 的符号确定, 即当 \mu\geq 0 时, (P _\mu) 的目标函数可以看作 f_{1}-\mu g_{1}$$ f_{2}-\mu g_{2}$ 这两个凸函数的差; 当$\mu < 0$ 时, (P$_\mu)$的目标函数可以看作$f_{1}+\mu g_{2} $$f_{2}+\mu g_{1} 这两个凸函数的差. 为此, 本文分以下两种情形对(P _\mu) 展开讨论. ### 3.1 参数 \mu 为非负数情形 \mu\geq0 , 则(P _\mu) 的目标函数可以看作 f_{1}-\mu g_{1}$$ f_{2}-\mu g_{2}$这两个凸函数的差. 因此我们可以借助DC优化问题的对偶方法建立(P$_\mu)$的对偶问题. 类似于文献[12, 16]方法, 易知, 其Fenchel-Lagrange对偶问题$(D_{\mu})$

$(\Rightarrow): \ $$(0, \alpha)\in\Lambda. 则对任意 x^{*}\in\text{dom}(f_{2}-\mu g_{2})^{*}, 故存在 \lambda\in S^{*}, (y^{*} , \alpha_{1})\in\text{epi}{f_{1}^{*}} , (z^{*}, \alpha_{2})\in\text{epi}{(-g_{1})^{*}} , (p^{*}, \alpha_{3})\in\text{epi}{(\lambda h)^{*}} 以及 (q^{*}, \alpha_{4})\in\text{epi}{\delta_{C}^{*}}, 使得 $$x^{*} = y^{*}+\mu z^{*}+p^{*}+q^{*} ,$$ $$(f_{2}-\mu g_{2})^{*}(x^{*})+\alpha = \alpha_{1}+\mu\alpha_{2}+\alpha_{3}+\alpha_{4} .$$ 又因为 f_{1}^{*}(y^{*})\leq\alpha_{1} , (-g_{1})^{*}(z^{*})\leq\alpha_{2} , (\lambda h)^{*}(p^{*})\leq\alpha_{3} 以及 \delta_{C}^{*}(q^{*})\leq\alpha_{4} , 从而由(3.2)和(3.3)式可知 因此 (\Leftarrow): 假设 \text{val}(D_{\mu})\geq -\alpha, 以及对任意 x^{*}\in\text{dom}(f_{2}-\mu g_{2})^{*}, 存在 \lambda\in S^{*} , y^{*}\in\text{dom}{f_{1}^{*}} , z^{*}\in \text{dom}(-g_{1})^{*} 以及 p^{*}\in \text{dom}(\lambda h)^{*} , 使得 从而 $$(x^{*}-y^{*}-\mu z^{*}-p^{*}, (f_{2}-\mu g_{2})^{*}(x^{*})-f_{1}^{*}(y^{*})-\mu (-g_{1})^{*}(z^{*})-(\lambda h)^{*}(p^{*})+\alpha)\in\text{epi}{\delta_{C}^{*}}.$$ 又易知 $$0 = y^{*}+\mu z^{*}+p^{*}+(x^{*}-y^{*}-\mu z^{*}-p^{*})-x^{*},$$ 以及 \begin{eqnarray} \alpha& = &f_{1}^{*}(y^{*})+\mu (-g_{1})^{*}(z^{*})+(\lambda h)^{*}(p^{*})+\Big((f_{2}-\mu g_{2})^{*}(x^{*})\\ & &-f_{1}^{*}(y^{*})-\mu (-g_{1})^{*}(z^{*})-(\lambda h)^{*}(p^{*})+\alpha\Big)-(f_{2}-\mu g_{2})^{*}(x^{*}). \end{eqnarray} 从而由(3.4), (3.5)以及(3.6)式有 x^{*}\in\text{dom}(f_{2}-\mu g_{2})^{*} 的任意性可知 命题得证. 借助(3.1)式, 本文引入如下正则性条件. 定义3.1 设 \mu\geq0.$$ (f_{1}, f_{2}, g_{1}, g_{2}, \delta_{C}, h)$满足新的正则性条件$NRC$当且仅当

假设存在$\alpha\in{{\Bbb R}},$ 使得

$\begin{eqnarray} \text{val}{(P_{\mu})} & = &\mathop{\text{inf}}\limits_{x\in X}\{f_{1}(x)-f_{2}(x)+\mu(-g_{1})(x)-\mu(-g_{2})(x)+\delta_{A}\}{} \\ & = &-(f_{1}-f_{2}+\mu(-g_{1})-\mu(-g_{2})+\delta_{A})^{*}(0). \end{eqnarray}$

$\text{val}(P_{\mu})\geq-\alpha,$ 与假设矛盾. 因此$\text{val}(P_{\mu})\geq\text{val}(D_{\mu}).$

(ⅰ) $x\in C, \ h(x)\in -S\Longrightarrow\frac{f_{1}(x)-f_{2}(x)}{g_{1}(x)-g_{2}(x)}\geq\mu;$

(ⅱ) 对任意$x^{*}\in{\rm dom}(f_{2}+\mu g_{1})^{*},$

(ⅲ) 对任意$x^{*}\in{\rm dom}(f_{2}+\mu g_{1})^{*},$ 存在$\lambda\in S^{*}$, $y^{*}\in{\rm dom}{f_{1}^{*}}$, $z^{*}\in{\rm dom} (-g_{2})^{*}$以及$p^{*}\in{\rm dom}(\lambda h)^{*}$, 使得

(ⅰ) $x\in C, \ h(x)\in -S\Longrightarrow\frac{f_{1}(x)-f_{2}(x)}{g_{1}(x)-g_{2}(x)} < \mu;$

(ⅱ) 对任意$x^{*}\in{\rm dom}(f_{2}+\mu g_{1})^{*},$ 存在$\lambda\in S^{*}$, $y^{*}\in{\rm dom}{f_{1}^{*}}$, $z^{*}\in{\rm dom} (-g_{2})^{*}$以及$p^{*}\in{\rm dom}(\lambda h)^{*}$, 使得

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