279 research outputs found
Robust optimality and duality for composite uncertain multiobjective optimization in Asplund spaces with its applications
This article is devoted to investigate a nonsmooth/nonconvex uncertain
multiobjective optimization problem with composition fields
((\hyperlink{CUP}{\mathrm{CUP}}) for brevity) over arbitrary Asplund spaces.
Employing some advanced techniques of variational analysis and generalized
differentiation, we establish necessary optimality conditions for weakly robust
efficient solutions of (\hyperlink{CUP}{\mathrm{CUP}}) in terms of the
limiting subdifferential. Sufficient conditions for the existence of (weakly)
robust efficient solutions to such a problem are also driven under the new
concept of pseudo-quasi convexity for composite functions. We formulate a
Mond-Weir-type robust dual problem to the primal problem
(\hyperlink{CUP}{\mathrm{CUP}}), and explore weak, strong, and converse
duality properties. In addition, the obtained results are applied to an
approximate uncertain multiobjective problem and a composite uncertain
multiobjective problem with linear operators.Comment: arXiv admin note: substantial text overlap with arXiv:2105.14366,
arXiv:2205.0114
Optimality conditions in convex multiobjective SIP
The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization problem with finitely many convex objective functions and infinitely many convex constraints. To do this, we introduce new and already known data qualifications (conditions involving the constraints and/or the objectives) in order to get optimality conditions which are expressed in terms of either Karusk–Kuhn–Tucker multipliers or a new gap function associated with the given problem.This research was partially cosponsored by the Ministry of Economy and Competitiveness (MINECO) of Spain, and by the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P
Second-order optimality conditions for interval-valued functions
This work is included in the search of optimality conditions for solutions to the scalar
interval optimization problem, both constrained and unconstrained, by means of
second-order optimality conditions. As it is known, these conditions allow us to reject
some candidates to minima that arise from the first-order conditions. We will define
new concepts such as second-order gH-derivative for interval-valued functions,
2-critical points, and 2-KKT-critical points. We obtain and present new types of
interval-valued functions, such as 2-pseudoinvex, characterized by the property that
all their second-order stationary points are global minima. We extend the optimality
criteria to the semi-infinite programming problem and obtain duality theorems.
These results represent an improvement in the treatment of optimization problems
with interval-valued functions.Funding for open access publishing: Universidad de Cádiz/CBUA. The research has been supported by MCIN through
grant MCIN/AEI/PID2021-123051NB-I00
Robust Solutions of MultiObjective Linear Semi-Infinite Programs under Constraint Data Uncertainty
The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.This research was partially supported by the Australian Research Council, Discovery Project DP120100467, the MICINN of Spain, grant MTM2011-29064-C03-02, and Generalitat Valenciana, grant ACOMP/2013/062
Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions
This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [5] from our viewpoint of robust Lipschitzian stability. We present meaningful interpretations and practical examples of such models. The main results establish necessary optimality conditions for a broad class of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Robust Solutions to Uncertain Multiobjective Programs
Decision making in the presence of uncertainty and multiple conflicting objec-tives is a real-life issue, especially in the fields of engineering, public policy making, business management, and many others. The conflicting goals may originate from the variety of ways to assess a system’s performance such as cost, safety, and affordability, while uncertainty may result from inaccurate or unknown data, limited knowledge, or future changes in the environment. To address optimization problems that incor-porate these two aspects, we focus on the integration of robust and multiobjective optimization. Although the uncertainty may present itself in many different ways due to a diversity of sources, we address the situation of objective-wise uncertainty only in the coefficients of the objective functions, which is drawn from a finite set of scenarios. Among the numerous concepts of robust solutions that have been proposed and de-veloped, we concentrate on a strict concept referred to as highly robust efficiency in which a feasible solution is highly robust efficient provided that it is efficient with respect to every realization of the uncertain data. The main focus of our study is uncertain multiobjective linear programs (UMOLPs), however, nonlinear problems are discussed as well. In the course of our study, we develop properties of the highly robust efficient set, provide its characterization using the cone of improving directions associated with the UMOLP, derive several bound sets on the highly robust efficient set, and present a robust counterpart for a class of UMOLPs. As various results rely on the polar and strict polar of the cone of improving directions, as well as the acuteness of this cone, we derive properties and closed-form representations of the (strict) polar and also propose methods to verify the property of acuteness. Moreover, we undertake the computation of highly robust efficient solutions. We provide methods for checking whether or not the highly robust efficient set is empty, computing highly robust efficient points, and determining whether a given solution of interest is highly robust efficient. An application in the area of bank management is included
Relative Pareto Minimizers to Multiobjective Problems: Existence and Optimality Conditions
In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers to general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces
Set-based Robust Optimization of Uncertain Multiobjective Problems via Epigraphical Reformulations
In this paper, we study a method for finding robust solutions to
multiobjective optimization problems under uncertainty. We follow the set-based
minmax approach for handling the uncertainties which leads to a certain set
optimization problem with the strict upper type set relation. We introduce,
under some assumptions, a reformulation using instead the strict lower type set
relation without sacrificing the compactness property of the image sets. This
allows to apply vectorization results to characterize the optimal solutions of
these set optimization problems as optimal solutions of a multiobjective
optimization problem. We end up with multiobjective semi-infinite problems
which can then be studied with classical techniques from the literature
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