DOI:10.12677/aam.2024.136261,PDF,HTML,XML,被引量下载: 46浏览: 91国家自然科学基金支持
作者:肖程凤^*,田超松：吉首大学师范学院，湖南 吉首
关键词:DC复合优化问题；最优性条件；凸化；DC Composite Optimization Problem；Optimality Condition；Convexification

文章引用：肖程凤, 田超松. 一元DC复合优化问题的最优性条件[J]. 应用数学进展, 2024, 13(6): 2726-2733. https://doi.org/10.12677/aam.2024.136261

1. 引言

约束优化问题在自动控制、图像处理、水处理、网络分析、工程设计中有着十分重要的应用，实际生活中的许多问题在一定条件下都可以看作或者转化为一个约束优化问题，例如，运输问题、分派问题、生产调度问题、设备更新年限问题等，因此无论从理论角度还是实际应用来看，约束优化问题的研究都具有非常重要的意义。对于经典的凸优化问题[1][2]、复合优化问题[3]-[6]，DC (两个凸函数的差)优化问题[7][8]，已有诸多学者利用内点条件、闭性条件、上图类条件等进行了研究，并得到了鞍点定理、对偶理论等系列结论。受此启发，本文拟对目标函数为复合函数，约束函数为DC函数的这一类一元DC复合优化问题进行研究。

2. 预备知识

设 $X,Y$是Banach空间， $X^{*},Y^{*}$分别是 $X,Y$的共扼空间。定义Y上的序为：若 $y-x\in -K$，则 $y{\le }_{K}x$。设Z是X的非空子集，记Z的凸锥包为 $\text{cone}Z$。非空集合Z的对偶锥和示性函数分别定义为：

$Z^{\oplus }:=\left\{x^{\ast }\in X^{\ast }:\langle x^{\ast },x\rangle \ge 0,\forall x\in Z\right\},$

${\delta }_Z\left(x\right):=\{\begin{array}{l}0,x\in Z,\\ +\infty ,其它.\end{array}$

定义凸子集Z在 $z_0\in Z$处的法锥为

$N\left(z_0;Z\right):=\left\{x^{\ast }\in X^{\ast }:\langle x^{\ast },z-z_0\rangle \le 0,\forall z\in Z\right\}.$

设f是X上的真函数，定义f的有效定义域和共轭函数分别为：

$\text{dom}f:=\left\{x\in X:f\left(x\right)<+\infty \right\},$

$f^{\ast }\left(x^{\ast }\right):=\sup \left\{\langle x^{\ast },x\rangle -f\left(x\right):x\in X\right\},\forall x^{\ast }\in X^{\ast },$

当f是凸函数时，f在点 $x\in \text{dom}f$的次微分定义为

$\partial f\left(x\right):=\left\{x^{\ast }\in X^{\ast }:f\left(x\right)+\langle x^{\ast },y-x\rangle \le f\left(y\right),\forall y\in X\right\}.$

设 $\varphi :X\to ℝ\cup \left\{\pm \infty \right\}$是实值延拓函数，点 $x_0\in \text{dom}\varphi$且满足 $\left|\varphi \left(x_0\right)\right|<+\infty$。由文[9]知，函数 $\varphi$在 $x_0$处ε-Fréchet次微分定义为 ${\hat{\partial }}_{\epsilon }\varphi \left(x_0\right):=\left\{x^{\ast }\in X^{\ast }:\underset{x\to x_0}{\lim \inf }\frac{\varphi \left(x\right)-\varphi \left(x_0\right)-\langle x^{\ast },x-x_0\rangle }{\Vert x-x_0\Vert }\ge -\epsilon \right\},\epsilon \ge 0$。当 $x_0\notin \text{dom}\varphi$时，

${\hat{\partial }}_{\epsilon }\varphi \left(x_0\right)=\varnothing$。当 $\epsilon =0$时， $\hat{\partial }\varphi \left(x_0\right):={\hat{\partial }}_{\epsilon }\varphi \left(x_0\right)$为 $\varphi$在 $x_0$处Fréchet次微分。特别地，当 $\varphi$是凸函数时， $\hat{\partial }\varphi \left(x_0\right):=\partial \varphi \left(x_0\right)$即为凸分析中经典的次微分。由定义有

$x_0是\varphi 的局部最优解\Rightarrow 0\in \hat{\partial }\varphi \left(x_0\right).$(2.1)

由文[4]知，设函数 $\psi :Y\to \overline{ℝ}$，对任意的 $x,y\in Y$，若当 $y{\le }_{K}x$时有 $\psi \left(y\right)\le \psi \left(x\right)$，则称函数 $\psi$是K-增函数。设函数 $\phi :X\to Y$，对任意的 $x_1,x_2\in X,t\in \left[0,1\right]$，有

$\phi \left(t{x}_1+\left(1-t\right)x_2\right){\le }_{K}t\phi \left(x_1\right)+\left(1-t\right)\phi \left(x_2\right),$

则称函数 $\phi$是K-凸函数。对于任意的 $\lambda \in K^{\oplus }$，定义 $\left(\lambda \phi \right)(·):X\to \overline{ℝ}$为

$\left(\lambda \phi \right)\left(x\right):=\{\begin{array}{l}\langle \lambda ,\phi \left(x\right)\rangle ,x\in \text{dom}\phi ,\\ +\infty ,其它.\end{array}$

定义复合函数的复合形式为

$\left(f\circ \phi \right)\left(x\right):=\{\begin{array}{l}f\left(\phi \left(x\right)\right),x\in \text{dom}\phi ,\\ +\infty ,其它.\end{array}$

并且，对X上的系统 $\left\{S_t:t\in T\right\}$，约定 $\underset{t\in \varphi }{\cap }{S}_t=X$。

3. DC复合优化问题的最优性条件

设 $C\subseteq X$是非空凸子集，T是任意(可能无限)指标集， $\phi :X\to Y$是真K-凸映射， $f:Y\to \overline{ℝ}$是真凸K-增函数， $f_t,g_t:X\to \overline{ℝ},t\in T$是真凸函数。考虑一元DC复合优化问题

$\begin{array}{l}\text{Min}\left(f\circ \phi \right)\left(x\right)\\ \left(\mathbb{P}\right)\text{s}\text{. t}\text{.}{f}_t\left(x\right)-g_t\left(x\right)\le 0,t\in T,\\ x\in C.\end{array}$

本文将在函数不一定具有连续性，集合C不一定是闭集的情形下，利用函数次微分的相关性质，引进新的约束规范条件，建立问题 $\left(\mathbb{P}\right)$的局部和全局最优性条件的充分和必要条件，从而推广和改进前人的相关结论。

3.1. 局部最优性条件

设A表示系统 $\left\{x\in C;f_t\left(x\right)-g_t\left(x\right)\le 0,\forall t\in T\right\}$的解集，即 $A:=\left\{x\in C:f_t\left(x\right)-g_t\left(x\right)\le 0,\forall t\in T\right\}$。设x₀是问题 $\left(\mathbb{P}\right)$的局部极小点，则x₀也将是下面问题的局部极小点：

$\begin{array}{l}\text{Min}\left(f\circ \phi \right)\left(x\right)\\ \text{s}\text{. t}\text{.}{f}_t\left(x\right)-\langle u_t^{*},x\rangle +g_t^{*}\left(u_t^{*}\right)\le 0,t\in T,\\ x\in C,\end{array}$(3.1)

其中： $u_t^{*}\in \text{dom}{g}_t^{*}$。对任意的 $u_t^{*}\in \text{dom}{g}_t^{*}$，问题(3.1)是凸优化问题。用 $T\left(x_0\right)$， $T_{u_t^{*}}\left(x_0\right)$分别表示不等式系统 $\left\{x\in C;f_t\left(x\right)-g_t\left(x\right)\le 0,t\in T\right\}$与 $\left\{x\in C;f_t\left(x\right)-\langle u_t^{*},x\rangle +g_t^{*}\left(u_t^{*}\right)\le 0,t\in T\right\}$在点x₀的活动指标集，即

$T\left(x_0\right):=\left\{t\in T:f_t\left(x_0\right)-g_t\left(x_0\right)=0\right\},$

$T_{u_t^{*}}\left(x_0\right):=\left\{t\in T:f_t\left(x_0\right)-\langle u_t^{*},x_0\rangle +g_t^{*}\left(u_t^{*}\right)=0\right\}.$

由文([8]，(1.2)式)可以定义问题(3.1)的局部KKT条件：

x₀是问题(3.1)的局部极小点

$\begin{array}{l}\Rightarrow \forall u_t^{\ast }\in \text{dom}{g}_t^{\ast },\exists \lambda ={\left({\lambda }_t\right)}_{t\in T}\in {ℝ}_{+}^{\left(T\right)},\beta \in \partial f\left(\phi \left(x_0\right)\right),\\ \text{s}\text{. t}\text{.}0\in \partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T_{u_t^{\ast }}\left(x_0\right)}{\sum }{\lambda }_t}\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right).\end{array}$

因此，可类似定义问题 $\left(\mathbb{P}\right)$在点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$的局部KKT条件。

定义3.1.1设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$是问题 $\left(\mathbb{P}\right)$的局部极小点。若对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$， $t\in T$，存在 $\lambda ={\left({\lambda }_t\right)}_{t\in T}\in {ℝ}_{+}^{\left(T\right)}$和 $\beta \in \partial f\left(\phi \left(x_0\right)\right)$，使得

$0\in \partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t}\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right),$

则称系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足局部KKT条件。若系统在任意点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$均满足局部KKT条件，则称系统满足局部KKT条件。

为研究问题 $\left(\mathbb{P}\right)$的局部最优性条件，我们首先引进如下约束规范条件。

定义3.1.2设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$，若

$\hat{\partial }\left(f\circ \phi +{\delta }_A\right)\left(x_0\right)\subseteq \underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right),$(3.2)

则称系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足F-(BCQ)₁条件。若系统在任意点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$均满足F-(BCQ)₁条件，则称系统满足F-(BCQ)₁条件。

定理3.1.1设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。若系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足F-(BCQ)₁条件，则称此系统在点x₀满足局部KKT条件。

证明假设系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足F-(BCQ)₁条件。设x₀是问题 $\left(\mathbb{P}\right)$的局部极小点。由(3.2)及(2.1)式可知

$0\in \hat{\partial }\left(f\circ \phi +{\delta }_A\right)\left(x_0\right)\subseteq \underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right).$

故对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$， $t\in T$，存在 $\lambda ={\left({\lambda }_t\right)}_{t\in T}\in {ℝ}_{+}^{\left(T\right)}$和 $\beta \in \partial f\left(\phi \left(x_0\right)\right)$，使得

$0\in \partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t}\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right),$

故定理得证。

注3.1.1[4]由文[4]知，令 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$，若

$\partial \left(f\circ \phi +{\delta }_A\right)\left(x_0\right)\subseteq \underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+\text{cone}\left(\underset{t\in T\left(x_0\right)}{\cup }\partial f_t\left(x_0\right)\right),$

则称系统 $\left\{f,\phi ,{\delta }_C;f_t:t\in T\right\}$在点x₀满足(CBCQ)条件。若系统在任意点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$均满足(CBCQ)条件，则称系统满足(CBCQ)条件。显然，当 $g_t=0,t\in T$时F-(BCQ)₁条件即为文[4]中的(CBCQ)条件。

故由定理3.1.1及注3.1.1可直接得到下面推论。

推论3.1.1设 $g_t=0,t\in T$。设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$是问题 $\left(\mathbb{P}\right)$的局部极小点。若系统 $\left\{f,\phi ,{\delta }_C;f_t:t\in T\right\}$在点x₀满足(CBCQ)条件，则下面结论成立：

$0\in \underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\partial f_t}\left(x_0\right).$

设 $u_t^{\ast }\in \text{dom}{g}_t^{\ast }$， $t\in T$，定义凸函数 $F_t^{u_t^{\ast }}:X\to \overline{ℝ}$为

$F_t^{u_t^{\ast }}\left(x\right):=f_t\left(x\right)-\langle u_t^{\ast },x\rangle +g_t^{\ast }\left(u_t^{\ast }\right),\forall x\in X.$

由文([10]，定理2.4.2(vi))知

$\partial F_t^{u_t^{\ast }}\left(x\right)=\partial f_t\left(x\right)-u_t^{\ast },\forall x\in \text{dom}{f}_t.$(3.3)

用 $A_{u_t^{\ast }}$表示不等式系统 $\left\{x\in C;F_t^{u_t^{\ast }}\left(x\right)\le 0,t\in T\right\}$的解集，即 $A_{u_t^{\ast }}:=\left\{x\in C:F_t^{u_t^{\ast }}\left(x\right)\le 0,t\in T\right\}$。由文[10]的Young-Fenchel不等式知

$f_t-g_t\le F_t^{u_t^{\ast }},\forall t\in T,$(3.4)

所以

$A_{u_t^{\ast }}\subseteq A.$(3.5)

令 $T_{u_t^{\ast }}\left(x_0\right):=\left\{t\in T:F_t^{u_t^{\ast }}\left(x_0\right)=0\right\}$，故对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$，由文[10]的Young等式知

$F_t^{u_t^{\ast }}\left(x_0\right)=f_t\left(x_0\right)-g_t\left(x_0\right),\forall t\in T,$(3.6)

因此，

$T\left(x_0\right)=T_{u_t^{\ast }}\left(x_0\right),\forall u_t^{\ast }\in \partial g_t\left(x_0\right).$(3.7)

定理3.1.2设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。若对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right),t\in T$，系统 $\left\{f,\phi ,{\delta }_C;F_t^{u_t^{\ast }}:t\in T\right\}$点x₀满足(CBCQ)条件，则系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$点x₀满足局部KKT条件。

证明设x₀是问题 $\left(\mathbb{P}\right)$的局部极小点，任取 $u_t^{\ast }\in \partial g_t\left(x_0\right)$。由(3.4)~(3.6)式知，x₀也是下面问题的局部极小点：

$\begin{array}{l}\text{Min}\left(f\circ \phi \right)\left(x\right)\\ \text{s}\text{. t}\text{.}x\in A_{u_t^{\ast }}.\end{array}$(3.8)

由于 $f\circ \phi$是凸函数且 $A_{u_t^{\ast }}$是X上的凸集可知，x₀也是问题(3.8)的局部极小点，故结合([10]，定理2.5.7)有

$0\in \partial \left(f\circ \phi +{\delta }_{A_{u_t^{\ast }}}\right)\left(x_0\right).$

又对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$，系统 $\left\{f,\phi ,{\delta }_C;F_t^{u_t^{\ast }}:t\in T\right\}$在点x₀满足(CBCQ)条件，故

$0\in \underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+\text{cone}\left(\underset{t\in T_{u_t^{\ast }}\left(x_0\right)}{\cup }\partial F_t^{u_t^{\ast }}\left(x_0\right)\right).$

再结合(3.3)和(3.7)式可得，对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$，存在 $\lambda ={\left({\lambda }_t\right)}_{t\in T}\in {ℝ}_{+}^{\left(T\right)}$和 $\beta \in \partial f\left(\phi \left(x_0\right)\right)$，使得

$0\in \partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t}\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right),$

因此定理得证。

当函数 $g_t=0,t\in T$时，设 $x_0\in A$，由文[11]知，若

$N_A\left(x_0\right)=N_C\left(x_0\right)+\text{cone}\left(\underset{t\in T\left(x_0\right)}{\cup }\partial f_t\left(x_0\right)\right),$

则称系统 $\left\{{\delta }_C;f_t:t\in T\right\}$在点x₀满足(BCQ)条件。

推论3.1.2设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。假设 $f,\phi$分别在点 $\phi \left(x_0\right)$与x₀处连续。若对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$， $t\in T$，系统 $\left\{{\delta }_C;F_t^{u_t^{\ast }}:t\in T\right\}$在点x₀满足(BCQ)条件，则系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$点x₀满足局部KKT条件。

证明设 $u_t^{\ast }\in \partial g_t\left(x_0\right)$。由定理3.1.2知，欲证此推论，只需证明系统 $\left\{f,\phi ,{\delta }_C;F_t^{u_t^{\ast }}:t\in T\right\}$在点x₀满足(CBCQ)条件，即证

$\partial \left(f\circ \phi +{\delta }_{A_{u_t^{\ast }}}\right)\left(x_0\right)\subseteq \underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+\text{cone}\left(\underset{t\in T_{u_t^{\ast }}\left(x_0\right)}{\cup }\partial F_t^{u_t^{\ast }}\left(x_0\right)\right).$(3.9)

注意到，函数 $f,\phi$分别在点 $\phi \left(x_0\right)$与x₀处连续， $f\circ \phi$是凸函数且 $A_{u_t^{\ast }}$是凸集，故

$\partial \left(f\circ \phi +{\delta }_{A_{u_t^{\ast }}}\right)\left(x_0\right)\subseteq \underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_{A_{u_t^{\ast }}}\left(x_0\right).$(3.10)

由(3.6)式及 $x_0\in A$可得 $x_0\in A_{u_t^{\ast }}$。又系统 $\left\{{\delta }_C;F_t^{u_t^{\ast }}:t\in T\right\}$在点x₀满足(BCQ)条件，故

$N_{A_{u_t^{\ast }}}\left(x_0\right)=N_C\left(x_0\right)+\text{cone}\left(\underset{t\in T_{u_t^{\ast }}\left(x_0\right)}{\cup }\partial F_t^{u_t^{\ast }}\left(x_0\right)\right).$(3.11)

结合(3.10)、(3.11)式，可得(3.9)式，故结论得证。

3.2. 全局最优性条件

本节主要对问题 $\left(\mathbb{P}\right)$的全局最优性条件进行研究，为此考虑下面问题：

$\begin{array}{l}\text{Min}\left(f\circ \phi \right)\left(x\right)-\langle p,x\rangle \\ \left({\mathbb{P}}_p\right)\text{s}\text{. t}\text{.}{f}_t\left(x\right)-g_t\left(x\right)\le 0,t\in T,\\ x\in C.\end{array}$

从而，我们可以得出以下定义。

定义3.2.1设 $p\in X^{\ast }$， $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。若x₀是问题 $\left({\mathbb{P}}_p\right)$的全局极小点当且仅当对任意的 $u_t^{\ast }\in \partial g_t\left(x_0\right)$， $t\in T$，存在 $\lambda ={\left({\lambda }_t\right)}_{t\in T}\in {ℝ}_{+}^{\left(T\right)}$和 $\beta \in \partial f\left(\phi \left(x_0\right)\right)$，使得

$p\in \partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t}\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right),$

则称系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$点x₀满足全局稳定KKT条件。特别地，当 $p=0$时，称系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$点x₀满足全局KKT条件。若系统在任意点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$均满足全局稳定KKT条件(或全局KKT条件)，则称系统满足全局稳定KKT条件(或全局KKT条件)。

特别地，当 $p=0$时，问题 $\left({\mathbb{P}}_p\right)$即为本文所研究的问题 $\left(\mathbb{P}\right)$，故由文([10]，定理2.5.7)知，系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足全局KKT条件当且仅当

$0\in \partial \left(f\circ \phi +{\delta }_A\right)\left(x_0\right)\iff 0\in \underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right).$

定义3.2.2设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。若

$\partial \left(f\circ \phi +{\delta }_A\right)\left(x_0\right)=\underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right),$

则称系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足(BCQ)₁条件。若系统在任意点 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$均满足(BCQ)₁条件，则称系统满足(BCQ)₁条件。

定理3.2.1设 $x_0\in \text{dom}\left(f\circ \phi \right)\cap A$。系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足(BCQ)₁条件当且仅当对任意的 $p\in X^{\ast }$，系统在点x₀满足全局稳定KKT条件。

证明对任意的 $p\in X^{\ast }$，系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$在点x₀满足全局稳定KKT条件当且仅当下面的等价关系成立：

$0\in \partial \left(f\circ \phi -p+{\delta }_A\right)\left(x_0\right)\iff p\in \underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right),$

即

$p\in \partial \left(f\circ \phi +{\delta }_A\right)\left(x_0\right)\iff p\in \underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right).$

故

$\partial \left(f\circ \phi +{\delta }_A\right)\left(x_0\right)=\underset{u_t^{\ast }\in \partial g_t\left(x_0\right)}{\cap }\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }\left(\underset{\beta \in \partial f\left(\phi \left(x_0\right)\right)}{\cup }\partial \left(\beta \phi \right)\left(x_0\right)+N_C\left(x_0\right)+{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\left(\partial f_t\left(x_0\right)-u_t^{\ast }\right)}\right),$

结论成立。

由定理3.2.1可直接得到下面推论。

推论3.2.1若系统 $\left\{f,\phi ,{\delta }_C;f_t,g_t:t\in T\right\}$满足(BCQ)₁条件，则此系统满足全局KKT条件。

注3.2.1当 $\phi$为单位算子， $g_t=0,t\in T$时，问题 $\left(\mathbb{P}\right)$变成文[1]中的问题 $\left(P_f\right)$，(BCQ)₁条件变成

$\partial \left(f+{\delta }_A\right)\left(x_0\right)=\partial f\left(x_0\right)+N_C\left(x_0\right)+\underset{\lambda \in {ℝ}_{+}^{\left(T\right)}}{\cup }{\displaystyle \underset{t\in T\left(x_0\right)}{\sum }{\lambda }_t\partial f_t}\left(x_0\right),$

即文[1]中的(BCQ)_f条件，故本文的结论是对文[1]中相关结论的推广。

基金项目

国家自然科学基金项目(11861033)。

NOTES

^*通讯作者。

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