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

Guaranteeing highly robust weakly efficient solutions for uncertain multi-objective convex programs

This paper deals with uncertain multi-objective convex programming problems, where the data of the objective function or the constraints or both are allowed to be uncertain within specified uncertainty sets. We present sufficient conditions for the existence of highly robust weakly efficient solutions, that is, robust feasible solutions which are weakly efficient for any possible instance of the objective function within a specified uncertainty set. This is done by way of estimating the radius of highly robust weak efficiency under linearly distributed uncertainty of the objective functions. In the particular case of robust quadratic multi-objective programs, we show that these sufficient conditions can be expressed in terms of the original data of the problem, extending and improving the corresponding results in the literature for robust multi-objective linear programs under ball uncertainty.This research was partially supported by the Australian Research Council, Discovery Project DP120100467 and the MINECO of Spain and ERDF of EU, Grants MTM2014-59179-C2-1-P and ECO2016-77200-P

Robust solutions to multi-objective linear programs with uncertain data

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a specified uncertainty set under affine data parametrization. We then present numerically tractable optimality conditions for minmax robust weakly efficient solutions, i.e., the weakly efficient solutions of the robust counterpart. We also consider highly robust weakly efficient solutions, i.e., robust feasible solutions which are weakly efficient for any possible instance of the objective matrix within a specified uncertainty set, providing lower bounds for the radius of highly robust efficiency guaranteeing the existence of this type of solutions under affine and rank-1 objective data uncertainty. Finally, we provide numerically tractable optimality conditions for highly robust weakly efficient solutions.This research was partially supported by the Australian Research Council, Discovery Project DP120100467, the MICINN of Spain, grant number MTM2011-29064-C03-02, and Generalitat Valenciana, grant number ACOMP/2013/062

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

Automating control system design via a multiobjective evolutionary algorithm

This chapter presents a performance-prioritized computer aided control system design (CACSD) methodology using a multi-objective evolutionary algorithm. The evolutionary CACSD approach unifies different control laws in both the time and frequency domains based upon performance satisfactions, without the need of aggregating different design criteria into a compromise function. It is shown that control engineers' expertise as well as settings on goal or priority for different preference on each performance requirement can be easily included and modified on-line according to the evolving trade-offs, which makes the controller design interactive, transparent and simple for real-time implementation. Advantages of the evolutionary CACSD methodology are illustrated upon a non-minimal phase plant control system, which offer a set of low-order Pareto optimal controllers satisfying all the conflicting performance requirements in the face of system constraints

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

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

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