Optimality and duality in nonsmooth composite vector optimization and applications
type: Item
instance of: scholarly article
title: Optimality and duality in nonsmooth composite vector optimization and applications
zbMATH Open document ID: 1464.49009
author: Thai Doan Chuong
published in: Annals of Operations Research
publication date: 2021-05-05T00:00:00Z
review text: Let \(X,W,Y,V,Z\) be finite-dimensional spaces, let \(K\subset Y\) be a closed, convex, and pointed cone with nonempty interior, and \(S\subset Z\) be a closed convex cone. Let let \(F:X\to W\), \(f:W\to Y\), \(G:X\to V\), \(g:V\to Z\) be Lipschitz functions and consider the (composite vector/multiobjective optimization) problem \[ \min_K\{(f\circ F)(x) : x\in C\}, \] where \[ C=\{ x\in X: (g\circ G)(x)\in -S\}. \] A point \(\bar x\) is said to be a weakly efficient solution if \[ \forall x \in C, \quad (f \circ F)(x) - ( f \circ F)(\bar x) \not\in -\mathrm{int } K. \] The author presents first, in full generality, necessary conditions for \(\bar x\) to be a weakly efficient solution of the form \(0\) belongs to the sum of suitable limiting/Mordukhovich subdifferentials involving the above maps, together with a complementary slackness property. Second, a point satisfying necessary conditions is proved to be weakly efficient if a constraint qualification and a suitable generalized convexity (i.e., a separation condition) are assumed. Then weak and strong formulation of duality properties are proposed and proved, together with some corollaries that express the above results for the particular case of linear maps. The relevance of most assumptions is illustrated through counterexamples. The authors point out that they do not use scalarization arguments.
reviewed by: Giovanni Colombo
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 49N15
zbMATH DE Number: 7344086
zbMATH Keywords: necessary conditions
zbMATH Keywords: sufficient conditions
zbMATH Keywords: duality
zbMATH Keywords: composite vector optimization
zbMATH Keywords: generalized convexity
zbMATH Keywords: limiting/Mordukhovich subdifferential
MaRDI profile type: Publication
full work available at URL: s10479-019-03349-1
OpenAlex ID: W2965909857
cites work: Optimality conditions for weak efficiency to vector optimization problems with composed convex functions
cites work: Conjugate duality for multiobjective composed optimization problems
cites work: A survey of recent developments in multiobjective optimization
cites work: Linear Matrix Inequality Conditions and Duality for a Class of Robust Multiobjective Convex Polynomial Programs
cites work: Optimality and duality in nonsmooth conic vector optimization
cites work: Optimality conditions and duality in nonsmooth multiobjective optimization problems
cites work: Nonsmooth semi-infinite multiobjective optimization problems
cites work: A class of nonsmooth fractional multiobjective optimization problems
cites work: Nondifferentiable minimax programming problems with applications
cites work: Optimization and nonsmooth analysis
cites work: Equivalence of a Ky Fan type minimax theorem and a Gordan type alternative theorem
cites work: Variational methods in partially ordered spaces
cites work: Q3141900
cites work: Q4461359
cites work: Convex composite multi-objective nonsmooth programming
cites work: Q4669751
cites work: On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems
cites work: Solving fractional multicriteria optimization problems with sum of squares convex polynomial data
cites work: Lagrange multipliers saddle points and scalarizations in composite multiobjective nonsmooth programming
cites work: Generalized convex composite multi-objective nonsmooth programming and conditional proper efficiency
cites work: Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space
cites work: Q3325486
cites work: Variational Analysis and Applications
cites work: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming
cites work: Composite nonsmooth multiobjective programs with \(V\)-\(\rho\)-invexity
cites work: Convex Analysis
cites work: Generalized convexity and nonsmooth problems of vector optimization
cites work: Generalized higher-order cone-convex functions and higher-order duality in vector optimization
cites work: Optimality conditions for a class of composite multiobjective nonsmooth optimization problems
cites work: A duality theorem for non-linear programming
DOI: 10.1007/S10479-019-03349-1
description: scientific article; zbMATH DE number 7344086