Chuong, Thai Doan

Kim, Do Sang

Kyoto University Research Information Repository

TitleApproximate Solutions of Multiobjective OptimizationProblems (Nonlinear Analysis and Convex Analysis)Author(s)Kim, Do Sang; Chuong, Thai DoanCitation数理解析研究所講究録 (2016), 2011: 1-7Issue Date2016-12URL http://hdl.handle.net/2433/231588RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityApproximate Solutions of Multiobjective optimizationProblemsDo Sang Kim1 and Thai Doan Chuong21 Department of Applied MathematicsPukyong National University, Republic of Korea2 School of Mathematics and StatisticsUniversity of New South Wales, Australia1 Introduction and PreliminariesIn this paper, we employ the limiting subdifferential and the Mordukhovich nor‐mal cone (cf. [7]) to examine approximate Pareto solutions of a multiobjectiveoptimization problem. More precisely, we establish Fritz‐John type necessary con‐ditions for  $\epsilon$-( weakly) Pareto solutions and  $\epsilon$-quasi-( weakly) Pareto solutions of amultiobjective optimization problem involving nonsmooth/nonconvex functions.With the help of generalized convex functions defined in terms of the limit‐ing subdifferential and the Mordukhovich normal cone, the obtained necessaryconditions for approximate Pareto solutions of the considered problem becomesuficient ones. In this way, we are able to explore completely duality relations forapproximate Pareto solutions between multiobjective optimization problems suchas strong duality and converse duality.Throughout the paper we use the standard notation of variational analysis;see e.g., [7]. Unless otherwise specified, all spaces under consideration are Asplundspaces whose norms are always denoted by \Vert The canonical pairing betweenspace  X and its dual X^{*} is denoted by \{\cdot , The symbol  B_{X} stands for the closedunit ball in X . As usual, the polar cone of  $\Omega$\subset X is the set$\Omega$^{\mathrm{o}}:=\{x^{*}\in X^{*}|\{x^{*}, x\}\leq 0 \forall x\in $\Omega$\} . (1.1)Also, we denote by \mathbb{R}_{+}^{m} the nonnegative orthant of \mathbb{R}^{m} , where m\in \mathrm{N} :=\{1 , 2, . .数理解析研究所講究録第2011巻 2016年 1-71Given a set‐valued mapping F:X=X^{*} between X and its dual X^{*} , wedenote by\displaystyle \mathrm{L}\mathrm{i}\mathrm{m}\sup_{x\rightarrow\overline{x}}F(x):=\{x^{*}\in X^{*}| \exists sequences  x_{n}\rightarrow\overline{x} and x_{n}^{*}\rightarrow x^{*}w^{*}with x_{n}^{*}\in F(x_{n}) for all n\in \mathrm{N}\}the sequential Painleve‐Kuratowski upper/outer limit of F as x\rightarrow\overline{x} . Here the\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}1\rightarrow^{w^{*}} indicates the convergence in the \mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}^{*} topology of X^{*}.A set  $\Omega$\subset X is called closed around \overline{x}\in $\Omega$ if there is a neighborhood  U of \overline{x}such that  $\Omega$\cap \mathrm{c}1U is closed. We say that  $\Omega$ is locally closed if  $\Omega$ is closed around x for every  x\in $\Omega$ . Let  $\Omega$\subset X be closed around \overline{x}\in $\Omega$.The Préchet normal cone to  $\Omega$ at \overline{x}\in $\Omega$ is defined by\displaystyle \hat{N}(\overline{x}; $\Omega$) :=\{x^{*}\in X^{*}|\lim_{ $\Omega$,x\rightarrow}\sup_{\overline{x}}\frac{\langle x^{*},x-\overline{x}\}}{\Vert x-\overline{x}||}\leq 0\} , (1.2)where  x\rightarrow\overline{x} $\Omega$ means that  x\rightarrow\overline{x} with  x\in $\Omega$ . If  x\not\in $\Omega$ , we put \hat{N}(x; $\Omega$) :=\emptyset.The limiting/Mordukhovich normal cone N(\overline{x}; $\Omega$) to  $\Omega$ at \overline{x}\in $\Omega$ is obtainedfrom Fréchet normal cones by taking the sequential Painlevé‐Kuratowski upperlimits as: N(\displaystyle \overline{x}; $\Omega$) :=\mathrm{L}\mathrm{i}\mathrm{m} \sup_{ $\Omega$,x\rightarrow\overline{x}}\hat{N}(x; $\Omega$) . (1.3)If  x\not\in $\Omega$ , we put  N(x; $\Omega$) :=\emptyset.For an extended real‐valued function  $\varphi$ :  X\rightarrow\overline{\mathbb{R}}:=[-\infty, \infty] , we set\mathrm{d}\mathrm{o}\mathrm{m} $\varphi$:=\{x\in X| $\varphi$(x)<\infty\}, \mathrm{e}\mathrm{p}\mathrm{i} $\varphi$:=\{(x,  $\mu$)\in X\times \mathbb{R}| $\mu$\geq $\varphi$(x)\}.The limiting/Mordukhovich subdifferential of  $\varphi$ at \overline{x}\in X with | $\varphi$(\overline{x})|<\infty isdefined by\partial $\varphi$(x) :=\{x^{*}\in X^{*}|(x^{*}, -1)\in N((\overline{x},  $\varphi$(\overline{x})) ; epi  $\varphi$ (1.4)If | $\varphi$(\overline{x})|=\infty , then one puts \partial $\varphi$(\overline{x}) :=\emptyset.2Considering the indicator function  $\delta$  $\Omega$ ) defined by  $\delta$(x; $\Omega$) :=0 for  x\in $\Omega$and by  $\delta$(x; $\Omega$) :=\infty otherwise, we have (see [7, Proposition 1.79]): N(\overline{x}; $\Omega$)=\partial $\delta$(\overline{x}; $\Omega$) \forall\overline{x}\in $\Omega$ . (1.5)The nonsmooth version of Fermats rule (see, e.g., [7, Proposition 1.114]) isformulated as follows: If \overline{x} is a local minimizer for  $\varphi$ , then 0\in\partial $\varphi$(\overline{x}) . (1.6)For a function  $\varphi$ locally Lipschitz at \overline{x} with modulus \ell>0 , it holds that (see[7, Corollary 1.81])||x^{*}||\leq P\forall x^{*}\in\partial $\varphi$(\overline{x}) . (1.7)2 Optimality Conditions for Approximate So‐lutionsThis section is devoted to presenting optimality conditions for approximate solu‐tions in multiobjective optimization prolems. Let  $\Omega$ be a nonempty closed subsetof  X , and let K :=\{1, 2, m\} , and I:=\{1, 2, p\} be index sets. Suppose thatf :=(f_{k}) , k\in K , and g :=(g_{i}) , i\in I are vector functions with locally Lipschitzcomponents defined on X.We focus on the following constrained multiobjective optimization problem(P):\displaystyle \min_{\mathbb{R}_{+}^{m}}\{f(x)|x\in C\} , (2.8)where C is the feasible set given byC:=\{x\in $\Omega$|g_{i}(x)\leq 0, i\in I\} . (2.9)Definition 2.1 ([5, 6]) Let  $\epsilon$:=( $\epsilon$ 1, \ldots, $\epsilon$_{m})\in \mathbb{R}_{+}^{m}.(i) We say that \overline{x}\in C is an  $\epsilon$‐Pareto solution of problem (2.8) iff there is no  x\in Csuch thatf_{k}(x)+$\epsilon$_{k}\leq f_{k}(\overline{x}) , k\in K (2.10)3with at least one strict inequality.(ii) A point \overline{x}\in C is called an  $\epsilon$‐quasi‐Pareto solution of problem (2.8) iff there isno  x\in C such thatf_{k}(x)+$\epsilon$_{k}||x-x \leq f_{k}(\overline{x}) , k\in K (2.11)with at least one strict inequality.If all the inequalities in (2.10) (resp., (2.11)) are strict, then one has the defini‐tion for  $\epsilon$ ‐weakly Pareto solution (resp.,  $\epsilon$‐quasi‐weakly Pareto solution) of problem(2.8). We denote the set of  $\epsilon$‐Pareto solutions (resp.,  $\epsilon$‐weakly Pareto solutions, $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}‐Pareto solutions, and  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}‐weakly Pareto solutions) of problem (2.8) by $\epsilon$-S(P) (resp.,  $\epsilon$-\mathcal{S}^{w}(P),  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}s\mathrm{i}-S(P) , and  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}s\mathrm{i}-\mathcal{S}^{w}(P) ). Note that we alwaysassume hereafter that  $\epsilon$\in \mathbb{R}_{+}^{m}\backslash \{0\}.To simplify the statements concerning problem (2.8), for fixed \overline{x}\in X and $\epsilon$\in \mathbb{R}_{+}^{m}\backslash \{0\} we define (cf. [3]) a real‐valued function  $\psi$ on  X as follows: $\psi$(x) :=kK,i\displaystyle \in I\max_{\in}\{f_{k}(x)-f_{k}(\overline{x})+ $\epsilon$ k, g_{i}(x)\}, x\in X . (2.12)Theorem 2.1 Let \overline{x}\in $\epsilon$-S^{w}(P) . For any  $\nu$>0 , there exist  x_{ $\nu$}\in $\Omega$ and  $\lambda$_{k}\geq 0, k\in K, $\mu$_{i}\geq 0, i\in I with \displaystyle \sum_{k\in K}$\lambda$_{k}+\sum_{i\in I}$\mu$_{i}=1_{f} such that ||x_{ $\nu$}-x \leq $\nu$ and 0\displaystyle \in\sum_{k\in K}$\lambda$_{k}\partial f_{k}(x_{ $\nu$})+\sum_{i\in I}$\mu$_{i}\partial g_{i}(x_{ $\nu$})+\frac{\max_{k\in K}\{ $\epsilon$ k\}}{ $\nu$}B_{X^{*}}+N(x_{ $\nu$}; $\Omega$) ,$\lambda$_{k}[f_{k}(x_{ $\nu$})-f_{k}(\overline{x})+$\epsilon$_{k}- $\psi$(x_{ $\nu$})]=0, k\in K,$\mu$_{i}[g_{i}(x_{ $\nu$})- $\psi$(x_{ $\nu$})]=0, i\in I,where the function  $\psi$ was defined in (2.12).The forthcoming theorem presents a FYitz‐John type necessary condition for $\epsilon$‐quasi (weakly) Pareto solutions of problem (2.8) with the help of Ekeland Vari‐ational Principle [2].4Theorem 2.2 Let \overline{x}\in $\epsilon$-quasi-S^{w}(P) . Then there exist $\lambda$_{k}\geq 0, k\in K , and$\mu$_{i}\geq 0, i\in I with \displaystyle \sum_{k\in K}$\lambda$_{k}+\sum_{i\in I}$\mu$_{i}=1 , such that0\displaystyle \in\sum_{k\in K}$\lambda$_{k}\partial f_{k}(\overline{x})+\sum_{i\in I}$\mu$_{i}\partial g_{i}(\overline{x})+\sum_{k\in K}$\lambda$_{k}$\epsilon$_{k}B_{X^{*}}+N(\overline{x}; $\Omega$) , (2.13)$\mu$_{i}g_{i}(\overline{x})=0, i\in I.Remark 2.1 According to Theorem 2.2, if \overline{x} is an  $\epsilon$‐quasi (weakly) Pareto so‐lution of problem (2.8), then the approximate (KKT) condition defined above isguaranteed by the following constraint qualification (CQ) due to [1](\mathrm{f}\mathrm{o}\mathrm{r} specialcases, one can see [4, 7, 8 One says that condition (CQ) is satisfied at \overline{x}\in C ifthere do not exist $\mu$_{i}\geq 0, i\in I(\overline{x}) not all zero, such that0\displaystyle \in\sum_{i\in I(\overline{x})}$\mu$_{i}\partial g_{i}(\overline{x})+N(\overline{x}; $\Omega$) , (2.14)where I(\overline{x}) :=\{i\in I|g_{i}(\overline{x})=0\}.Theorem 2.3 Let \overline{x}\in C satisfy the  $\epsilon$‐approximate (KKT) condition.(i) If  f and g are generalized convex on  $\Omega$ at \overline{x} , then \overline{x}\in $\epsilon$-quasi-\mathcal{S}^{w}(P) .(ii) If f is strictly generalized convex and g is generalized convex on  $\Omega$ at \overline{x},then \overline{x}\in $\epsilon$-quasi-\mathcal{S}(P) .3 Duality for Approximate SolutionsFor z\in X,  $\lambda$:=($\lambda$_{k}) , $\lambda$_{k}\geq 0, k\in K, and  $\mu$ :=($\mu$_{i}) , $\mu$_{i}\geq 0, i\in I , let us denotea vector Lagrangian function L byL(z,  $\lambda$,  $\mu$):=f(z)+\{ $\mu$, g(z)\rangle e,where e :=(1, \ldots, 1)\in \mathbb{R}^{m} . In connection with the constrained multiobjectiveoptimization problem (P) formulated in (2.8) and a given  $\epsilon$ :=($\epsilon$_{1}, \ldots, $\epsilon$_{m})\in\mathbb{R}_{+}^{m}\backslash \{0\} , we consider a multiobjective dual problem in the following form (D):\displaystyle \max_{\mathbb{R}_{+}^{m}}\{L(z,  $\lambda$,  $\mu$)|(z,  $\lambda$,  $\mu$)\in C_{D}\} . (3.15)5Here the feasible set C_{D} is defined byC_{D}:=\displaystyle \{(z,  $\lambda$,  $\mu$)\in $\Omega$\times(\mathbb{R}_{+}^{m}\backslash \{0\})\times \mathbb{R}_{+}^{p}|0\in\sum$\lambda$_{k}\partial f_{k}(z)+\sum$\mu$_{i}\partial g_{i}(z)k\in K i\in I+\displaystyle \sum_{k\in K}$\lambda$_{k}$\epsilon$_{k}B_{X}*+N(z; $\Omega$) , \sum_{k\in K}$\lambda$_{k}=1\}(3.16)Definition 3.1 Let L :=(L_{1}, \ldots, L_{m}) , and let  $\epsilon$:=($\epsilon$_{1}, \ldots, $\epsilon$_{m})\in \mathbb{R}_{+}^{m}\backslash \{0\}.We say that (\overline{z},\overline{ $\lambda$},\overline{ $\mu$})\in C_{D} is an  $\epsilon$‐quasi‐Pareto solution of problem (3.15) iff thereis no (z,  $\lambda$,  $\mu$)\in C_{D} such thatL_{k}(z,  $\lambda$,  $\mu$)\geq L_{k}(\overline{z},\overline{ $\lambda$},\overline{ $\mu$})+ $\epsilon$ k||(\overline{z},\overline{ $\lambda$},\overline{ $\mu$})-(z,  $\lambda$,  $\mu$ k\in K (3.17)with at least one strict inequality.If all the inequalities in (3.17) are strict, then one has the definition for  $\epsilon$-quasi‐weakly Pareto solution of problem (3.15). Also, the set of  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}‐Paretosolutions (resp.,  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}‐weakly Pareto solutions) of problem (3.15) is denoted by $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-\mathrm{S}(D) (resp.,  $\epsilon$-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}-\mathcal{S}^{w}(D) ).Theorem 3.1 (Duality) Let \overline{x}\in $\epsilon$-quasi-S^{w}(P) be such that the (CQ) defined in(2.14) is satisfied at this point. Then there exist \overline{ $\lambda$}:=(\overline{ $\lambda$}_{k}) , \overline{ $\lambda$}_{k}\geq 0, k\in K , not allzero, and \overline{ $\mu$}:=(\overline{ $\mu$}_{i}) , \overline{ $\mu$}_{i}\geq 0, i\in I , such that (\overline{x},\overline{ $\lambda$},\overline{ $\mu$})\in C_{D} and f(\overline{x})=L(\overline{x},\overline{ $\lambda$},  $\mu$In addition,(i) If  f and g are generalized convex on  $\Omega$ at any  z\in $\Omega$ , then (\overline{x},\overline{ $\lambda$}, \overline{ $\mu$})\in $\epsilon$‐quasi‐\mathcal{S}^{w}(D) .(ii) If f is strictly generalized convex and g is generalized convex on  $\Omega$ at any z\in $\Omega$ , then (\overline{x},\overline{ $\lambda$},\overline{ $\mu$})\in $\epsilon$-quasiS(D) .Theorem 3.2 (Converse Duality) Let (\overline{x},\overline{ $\lambda$},\overline{ $\mu$})\in C_{D} such that f(\overline{x})=L(\overline{x},\overline{ $\lambda$},  $\mu$(i) If \overline{x}\in C and f and g are generalized convex on  $\Omega$ at \overline{x} , then \overline{x}\in $\epsilon$-quasi-S^{w}(P) .(ii) If \overline{x}\in C and f is strictly generalized convex and g is generalized convex on  $\Omega$at \overline{x} , then \overline{x}\in $\epsilon$-quasi-S(P) .6References[1] T. D. Chuong, D. S. Kim, Optimality conditions and duality in nonsmoothmultiobjective optimization problems, Ann. Oper. Res. 217 (2014), 117‐136.[2] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324‐353.[3] C. Gutiérrez, B. Jiménez, V. Novo,  $\epsilon$‐Pareto optimality conditions for convexmultiobjective programming via \displaystyle \max function. Numer. Funct. Anal. Optim.27 (2006), 57‐70.[4] S. P. Han, O. L. Mangasarian, Exact penalty functions in nonlinear program‐ming, Math. Programming 17 (1979), no. 3, 251‐269.[5] J. C. Liu,  $\epsilon$‐duality theorem of nondifferentiable nonconvex multiobjectiveprogramming, J. Optim. Theory Appl. 69 (1991), no. 1, 153‐167.[6] P. Loridan,  $\epsilon$‐solutions in vector minimization problems, J. Optim. TheoryAppl. 43 (1984), 265‐276.[7] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I:Basic Theory, Springer, Berlin, 2006.[8] R. T. Rockafellar, R. J‐B. Wets, Variational Analysis. Springer, Berlin, 1998.7

Approximate Solutions of Multiobjective Optimization Problems (Nonlinear Analysis and Convex Analysis)

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/231588/1/2011-01.pdf

Approximate Solutions of Multiobjective Optimization Problems (Nonlinear Analysis and Convex Analysis)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository