M. J. Cánovas

Stability and Well-Posedness in Linear Semi-Infinite Programming

This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.

Metric regularity of semi-infinite constraint systems

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems.

Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems

Abstract.In this paper we consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space, with a fixed and arbitrary (possibly infinite) index set. This parameter space is endowed with the topology of the uniform convergence of the coefficient vectors by means of an extended distance. Some authors, in a different context in which the index set is finite and, accordingly, the coefficients are bounded, consider the boundary of the set of consistent systems as the set of ill-posed systems. The distance from the nominal system to this boundary (‘distance to ill-posedness’), which constitutes itself a measure of the stability of the system, plays a decisive role in the complexity analysis of certain algorithms for finding a solution of the system. In our context, the presence of infinitely many constraints would lead us to consider separately two subsets of inconsistent systems, the so-called strongly inconsistent systems and the weakly inconsistent systems. Moreover, the possible unboundedness of the coefficient vectors of a system gives rise to a special subset formed by those systems whose distance to ill-posedness is infinite. Attending to these two facts, and according to the idea that a system is ill-posed when small changes in the system’s data yield different types of systems, now the boundary of the set of strongly inconsistent systems arises as the ‘generalized ill-posedness’ set. The paper characterizes this generalized ill-posedness of a system in terms of the so-called associated hypographical set, leading to an explicit formula for the ‘distance to generalized ill-posedness’. On the other hand, the consistency value of a system, also introduced in the paper, provides an alternative way to determine its distance to ill-posedness (in the original sense), and additionally allows us to distinguish the consistent well-posed systems from the inconsistent well-posed ones. The finite case is shown to be a meeting point of our linear semi-infinite approach to the distance to ill-posedness with certain results derived for conic linear systems. Applications to the analysis of the Lipschitz properties of the feasible set mapping, as well as to the complexity analysis of the ellipsoid algorithm, are also provided.

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set

Upper Semicontinuity of the Feasible Set Mapping for Linear Inequality Systems

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Distance to Solvability/Unsolvability in Linear Optimization

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Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions

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CALMNESS MODULUS OF LINEAR SEMI-INFINITE PROGRAMS ∗

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Solving Strategies and Well-Posedness in Linear Semi-Infinite Programming

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