On approximate solutions for robust semi-infinite multi-objective convex symmetric cone optimization
type: Item
instance of: scholarly article
title: On approximate solutions for robust semi-infinite multi-objective convex symmetric cone optimization
author: Amira Achouak Oulha
author: Baha M. Alzalg
published in: Positivity
publication date: 2022-10-31T00:00:00Z
review text: A robust semi-infinite multi-objective symmetric programming problem with uncertain da ta (RSIMP) is investigated according to \(\min (f_1(x),\dots,f_K(x))\) s.t. \[ x\in \mathcal{F_P}=\left\{x\in \mathbb{R}^m \mid a_0^{(t)}+\sum_{i=1}^m x_i a_i^{(t)}\in \mathcal{K_J}\;(\text{def. }\succcurlyeq 0),\; a_i^{(t)}\in \mathcal{V}_i^{(t)},\; i=1,2,\dots,m,\; t\in T \right\} \] where \(a_i^{(t)}\) belong to an uncertain subset \(\mathcal{V}_i^{(t)}\) of an \(n\)-dimensional Jordan algebra \([\mathcal{J},\circ ]\), \(\mathcal{K_J}\) is the cone of \(a\circ a\) of all \(a\in \mathcal{J}\) and \(T\) is an arbitrary, may be, infinite index set. Initially, the unusual \(\epsilon\)-approximate solutions \(\bar x\in \mathcal{F_P}\) according to the coordinate functions of the objective \(f_k(x)\ge f_k(\bar x)-\epsilon, k=1,\dots,K\) for all \(x \in \mathcal{F_P}\) (known as \(\epsilon\) utopia point) are considered. However, from Prop. 2.4 the solution concept seems to be changed to \(E^6 \;\epsilon\)-efficiency of \textit{D. J. White} [J. Optim. Theory Appl. 49, 319--337 (1986; Zbl 0573.90085)], originally for max-problems. The authors use a sufficient condition of \textit{K. Yokoyama} [J. Math. Anal. Appl. 203, No. 1, 142--149 (1996; Zbl 0858.90114), Lem. 4.2], \(\epsilon\)-Efficiency \(E_1\) for such \(E^6\;\epsilon\)-efficiency as \(\epsilon\)-efficiency definition in the further context. Some main role in the paper plays the robust characteristic cone \(\mathcal D\) inconveniently defined in the paper. If \(\mathcal{V}_i^{(t)}\) is some linear map of a compact convex set of \(\mathbb{R}^l\rightarrow \mathcal{J}\) and some Slater condition holds then \(\mathcal{D}\) is a convex and closed cone in \(\mathbb{R}^{m+1}\). Using a Farkas like lemma and subgradients of the convex assumed functions \(f_k,k\in I\) the \(\epsilon\)-efficiency for (RSIMP) is equivalent to an \(\epsilon\)-subgradient relation (Theorem 3.1) which should tend to Pareto efficiency whenever \(\epsilon \righ…
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 90C17
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 90C34
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C15
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 17C90
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 7610120
zbMATH Keywords: robust symmetric cone optimization
zbMATH Keywords: semi-infinite programming
zbMATH Keywords: multi-objective programming
zbMATH Keywords: approximate optimality conditions
zbMATH Keywords: approximate duality theorems
zbMATH Keywords: Jordan algebra
zbMATH Keywords: cone of squares
MaRDI profile type: Publication
full work available at URL: s11117-022-00952-8
OpenAlex ID: W4307927247
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DOI: 10.1007/S11117-022-00952-8
description: scientific article; zbMATH DE number 7610120