Ciencias

In this paper, we consider a parametric family of convex inequality systems in the Euclidean space, with an arbitrary infinite index set, T, and convex constraints depending continuously on a parameter ranging in a separable metric space. No structure is assumed for T, and so ...

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.

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.

Abstract We consider the parametric space of all the linear semi-infinite programming problems with constraint systems having the same index set. Under a certain regularity condition, the so-called well-posedness with respect to the solvability, it is known from ...

In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.

ABSTRACT 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.

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.