Calculating Radius of Robust Feasibility of Uncertain Linear Conic Programs via Semi-definite Programs

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise, the robust counterpart of an uncertain program is not well defined as a robust optimization problem. In this paper, we address a key fundamental question of robust optimization: How to compute the radius of robust feasibility of uncertain linear conic programs, including linear programs? We first provide computable lower and upper bounds for the radius of robust feasibility for general uncertain linear conic programs under the commonly used ball uncertainty set. We then provide important classes of linear conic programs where the bounds are calculated by finding the optimal values of related semi-definite linear programs, among them uncertain semi-definite programs, uncertain second-order cone programs and uncertain support vector machine problems. In the case of an uncertain linear program, the exact formula allows us to calculate the radius by finding the optimal value of an associated second-order cone program.This research was partially supported by the Australian Research Council, Discovery Project DP120100467 and the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22

Dual characterizations of set containments with strict convex inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.MCYT of Spain and FEDER of UE, Grant BMF2002-04114-CO201

Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

The radius of robust feasibility of a convex program with uncertain constraints gives a value for the maximal ‘size’ of an uncertainty set under which robust feasibility can be guaranteed. This paper provides an upper bound for the radius for convex programs with uncertain convex polynomial constraints and exact formulas for convex programs with SOS-convex polynomial constraints (or convex quadratic constraints) under affine data uncertainty. These exact formulas allow the radius to be computed by commonly available software.The first author would like to thank the University of New South Wales for its support during his stay in November/December 2014. This research was partially supported by the Australian Research Council, Discovery Project DP120100467, the MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-1-P

The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees feasibility of a robust counterpart of a mathematical program with uncertain constraints. The objective of this review of the state-of-the-art in this field is to present this useful tool of robust optimization to its potential users and to avoid undesirable overlapping of research works on the topic as those we have recently detected. In this paper we overview the existing literature on the radius of robust feasibility in continuous and mixed-integer linearly constrained programs, linearly constrained semi-infinite programs, convexly constrained programs, and conic linearly constrained programs. We also analyze the connection between the radius of robust feasibility and the distance to ill-posedness for different types of uncertain mathematical programs.This research was partially supported by the Australian Research Council, Discovery Project grant and the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22

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

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 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

Necessary and sufficient conditions for stable conjugate duality

The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.C

Conditions for global optimality of quadratic minimization problems with LMI constraints

In this paper we present sufficient conditions for global optimality of non-convex quadratic programs involving linear matrix inequality (LMI) cnstraints. Our approach makes use of the concept of a quadratic subgradient. We develop optimality conditions for quadratic programs with LMI constraints by using Lagrangian function and by examining conditions which minimizes a quadratic subgradient of the Lagrangian function over simple bounding constraints. As applications, we obtain sufficient optimality condition for quadratic programs with LMI and box constraints by minimizing a quadrtic subgradient over box constraints. We also give optimality conditions for quadratic minimization involving LMI and binary constraints. © World Scientific Publishing Co. & Operational Research Society of Singapore.C