arXiv subjects

Michel Thera

Publications and source records attributed to Michel Thera.

6 recordsLinked to original sources

arXiv2025

Complexity of Error Bounds for Systems of Linear Inequalities

Error bounds have been studied for more than seventy years, beginning with the seminal result of Hoffman (1952) [{\it J. Res. Natl. Bur. Standards}, 49 (1952), 263--265], which establishes an upper bound for the distance from an arbitrary point to the solution set of a linear system. Despite this long history, relatively little is known about the intrinsic computational complexity of error bounds. In this paper, we investigate the complexity of error bounds for systems of linear inequalities. We first reformulate the problem as a finite collection of min--max optimization problems defined on the unit sphere and associated with subsets of the rows of the given matrix. We then prove that the problem does not belong to the class {\bf P}, while it is {\bf co\mbox{-}NP}-complete. Furthermore, we establish the existence of a pseudo-polynomial-time algorithm for solving the complementary problem. In particular, the complement may be regarded as a number problem, although it is not {\bf NP}-complete in the strong sense unless {\bf P} = {\bf NP}.

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arXiv2025

Metric Subregularity of Multifunctions and Applications to Characterizations of Asplund Spaces

In this paper, we investigate metric subregularity of multifunctions between Asplund spaces. Using Mordukhovich normal cones and coderivatives, we introduce the limiting Basic Constraint Qualification (BCQ) associated with a given multifunction. This BCQ provides necessary dual conditions for the metric subregularity of multifunctions in the Asplund space setting. Furthermore, we establish characterizations of Asplund spaces in terms of the limiting BCQ condition implied by metric subregularity. By employing Frechet normal cones and coderivatives, we derive necessary dual conditions for metric subregularity expressed as fuzzy inclusions, and we also obtain characterizations of Asplund spaces via these fuzzy inclusions. As an application, we examine metric subregularity of the conic inequality defined by a vector-valued function and a closed (not necessarily convex) cone with a nontrivial recession cone. By using Mordukhovich and Frechet subdifferentials relative to the given cone, we establish necessary dual conditions for the metric subregularity of such inequalities in Asplund spaces. The results based on Mordukhovich subdifferentials characterize Asplund spaces, while those based on Frechet subdifferentials yield necessary or sufficient conditions for Asplund spaces. These conditions recover, as special cases, the known error-bound results for inequalities defined by extended-real-valued functions on Asplund spaces. Overall, this work highlights that the validity of necessary conditions formulated via normal cones and subdifferentials for error bounds of convex or nonconvex inequalities depends crucially on the Asplund property of the underlying space.

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arXiv2024

Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems

In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimun of the directional derivative at a reference point over the sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving the convex optimization/minimization problems (defined by directional derivatives) over the sphere.

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arXiv2015

On Extended Versions of Dancs- Hegedüs-Medvegyev's Fixed Point Theorem

In this article we establish some fixed point (known also as critical point, invariant point) theorems in quasi-metric spaces. Our results unify and further extend in some regards the fixed point theorem proposed by Dancs et al. (1983), the results given by Khanh and Quy (2010, 2011), the preorder principles established by Qiu (2014), and the results obtained by Bao et al. (2015). In addition, we provide examples to illustrate that the improvements of our results are significant.

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arXiv2013

Local nonsmooth Lyapunov pairs for first-order evolution differential inclusions

The general theory of Lyapunov's stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous work. This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.

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arXiv2013

Metric Regularity of the Sum of Multifunctions and Applications

In this work, we use the theory of error bounds to study metric regularity of the sum of two multifunctions, as well as some important properties of variational systems. We use an approach based on the metric regularity of epigraphical multifunctions. Our results subsume some recent results by Durea and Strugariu.

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