## About

76

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863

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Citations since 2017

Introduction

I am interested in parametric optimization (mainly linear, semi-infinite linear, and convex), as well as in several notions of distance to ill-posedness, and also in variational analysis (mainly Lipschitz properties, calmness and Hoffman constants).

Additional affiliations

May 2012 - present

## Publications

Publications (76)

This paper introduces two new variational properties, robust and continuous metric subregularity, for finite linear inequality systems under data perturbations. The motivation of this study goes back to the seminal work by Dontchev, Lewis, and Rockafellar (2003) on the radius of metric regularity. In contrast to the metric regularity, the unstable...

The present paper deals with uncertain linear optimization problems where the objective function coefficient vector belongs to a compact convex uncertainty set and the feasible set is described by a linear semi-infinite inequality system (finitely many variables and possibly infinitely many constrainsts), whose coefficients are also uncertain. Pert...

In this paper, we use a geometrical approach to sharpen a lower bound given in [5] for the Lipschitz modulus of the optimal value of (finite) linear programs under tilt perturbations of the objective function. The key geometrical idea comes from orthogonally projecting general balls on linear subspaces. Our new lower bound provides a computable exp...

This work is focussed on computing the Lipschitz upper semicontinuity modulus of the argmin mapping for canonically perturbed linear programs. The immediate antecedent can be traced out from Camacho J et al. [2022. From calmness to Hoffman constants for linear semi-infinite inequality systems. Available from: https://arxiv.org/pdf/2107.10000v2.pdf]...

We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter. Such structured perturbations provide a unified framework for different parametric models in the literature, as block, directional and/or partial perturbations of both inequalities and equal...

In this paper we deal with (finite) linear inequality systems parameterized by their right-hand side. We show that the sharp Hoffman constant at a given nominal parameter may be expressed as the maximum of the calmness moduli of the feasible set mapping at finitely many feasible points (extreme points when the nominal feasible set contains no lines...

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in \({\mathbb {R}}^{n}\). To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of \({\ma...

The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the L...

The paper concerns multiobjective linear optimization problems in \(\mathbb {R}^{n}\) that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific direction...

This paper generalizes and unifies different recent results as well as provides a new methodology concerning vector optimization problems involving composite mappings in locally convex Hausdorff topological vector spaces. The Lagrangian and weak Lagrangian dual problems are proposed. Characterizations of strong duality results are proved at the sam...

The paper concerns multiobjective linear optimization problems in R^n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is...

This paper analyzes the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in R^n. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of R^(n+1). In thi...

The present paper is devoted to the computation of the Lipschitz modulus of the optimal value function restricted to its domain in linear programming under different types of perturbations. In the first stage, we study separately perturbations of the right-hand side of the constraints and perturbations of the coefficients of the objective function....

We deal with the feasible set mapping of linear inequality systems under right-hand side perturbations. From a version of Farkas lemma for difference of convex functions, we derive an operative relationship between calmness constants for this mapping at a nominal solution and associated neighborhoods where such constants work. We also provide illus...

In this paper we develop point-based formulas for the calmness modulus of the feasible set mapping in the context of linear inequality systems with a fixed abstract constraint and (partially) perturbed linear constraints. The case of totally perturbed linear systems was previously analyzed in [Cánovas MJ, López MA, Parra J, et al. Calmness of the f...

The present paper deals with uncertain linear inequality systems viewed as nonempty closed coefficient sets in the (n+ 1) -dimensional Euclidean space. The perturbation size of these uncertainty sets is measured by the (extended) Hausdorff distance. We focus on calmness constants—and their associated neighborhoods—for the feasible set mapping at a...

The final goal of the present paper is computing/estimating the calmness modulifrom below and above of the optimal value function restricted to the set of solvable linear problems.Roughly speaking, these moduli provide measures of the maximum rates of decrease and increaseof the optimal value under perturbations of the data (provided that solvabili...

From a computational point of view, this paper provides a significant advance in the study of the calmness property of ordinary (finite) linear programs under canonical perturbations (i.e., perturbations of the objective function coefficient vector and the right-hand side of the constraint system). In the recent literature we find, for both the fea...

This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and pertu...

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-function...

This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the...

Our main goal is to compute or estimate the calmness modulus of the argmin
mapping of linear semi-infinite optimization problems under canonical
perturbations, i.e., perturbations of the objective function together with
continuous perturbations of the right-hand-side of the constraint system (with
respect to an index ranging in a compact Hausdorff...

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides...

This paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key too...

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is...

This paper is firstly concerned with the modulus of metric regularity of intersection mappings. We consider a finite collection of set-valued mappings and analyze the relationship between the regularity moduli of these mappings (specifically, the maximum of them) and the regularity modulus of the associated intersection mapping. As an application w...

This article extends some results of Cánovas et al. [M.J. Cánovas, M.A. López, J. Parra, and F.J. Toledo, Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems, Math. Prog. Ser. A 103 (2005), pp. 95–126.] about distance to ill-posedness (feasibility/infeasibility) in three directions: from individual perturb...

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of...

This paper concerns parameterized convex infinite (or semi-infinite)
inequality systems whose decision variables run over general
infinite-dimensional Banach (resp. finite-dimensional) spaces and that are
indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand
side of the inequalities are measurable and bounded, and thus the...

In this paper, we introduce the concepts of linear regularity and equirregularity for an arbitary family of set-valued mappings
between (extended) metric spaces. The concept of linear regularity is inspired in the same property for a family of sets.
Then we analyze the relationship between the (metric) regularity moduli of the mappings in the famil...

We extend some recent developments on metric regularity in convex semi-infinite optimization from the case of a compact metric index set (for the constraint system) to the case of a compact Hausdorff one. The latter is the continuous setting where different contributions on stability in semi-infinite programming were developed since the 1980s (see...

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [Cánovas et al., SIAM J...

We are concerned with the Lipschitz modulus of the optimal set mapping
associated with canonically perturbed convex semi-infinite optimization
problems. Specifically, the paper provides a lower and an upper bound for
this modulus, both of them given exclusively in terms of the problem's data.
Moreover, the upper bound is shown to be the exact modul...

This paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi-infinite optimization problem subject to canonical perturbations. We provide a sufficient condition for isolated calmness of this mapping. This sufficient condition characterizes the strong uniqueness of minimizers, under the Slater constraint qual...

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 stabilit...

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations
of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints
is assumed to be a compact metric space and all coefficients depend continuously on th...

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of
the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context,
Cánovas et al. (SIAM J. Optim. 18:717–732, [2007]) introduced a sufficient condition (called ENC i...

We relate the Lipschitz modulus of the optimal set mapping of a canonically perturbed convex optimization problem in ℝ n with the Lipschitz moduli of suitable subproblems with exactly n constraints. This approach allows us to simplify other expressions for the modulus given in [M. J. Cánovas, A. Hantoute, M. A. López and J. Parra, Lipschitz modulus...

In this paper we analyze the connections among different paramet- ric settings in which the stability theory for linear inequality systems may be developed. Our discussion is focussed on the existence, or not, of an index set (possibly infinite). For some stability approaches it is not convenient to have a fixed set indexing the constraints. This i...

We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mappingE The main goal is to determine the regularity modulu...

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...

In this article, some sensitivity analysis of the dual optimal value in linear semi-infinite optimization is carried out via the notion of primal/dual asymptotic solution. The sensitivity results are then applied to derive some Hoffman-type inequalities (error bounds). Like in [Renegar, J., 1994, Some perturbation theory for linear programming. Mat...

In this article we consider the parameter space of all the linear constraint systems, in the n-dimensional Euclidean space, whose inequality constraints are indexed by an arbitrary, but fixed, set T (possibly infinite) and the number of equations is m≤ n. This parameter space is endowed with the topology of the uniform convergence of the coefficien...

This paper deals with the so-called total ill-posedness of linear optimization problems with an arbitrary (possibly infinite) number of constraints. We say that the nominal problem is totally ill-posed if it exhibits the highest unstability in the sense that arbitrarily small perturbations of the problem’s coefficients may provide both, consistent...

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 optim...

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 Cánovas et al. [2] that the optimal value function is Lipschitz continuous around the nominal pr...

We consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space and with a fixed index set, endowed with the topology of the uniform convergence of the coefficient
vectors. A system is ill-posed with respect to the consistency when arbitrarily small perturbations yield both consistent
and inconsistent syst...

We characterize those linear optimization problems that are ill-posed in the sense that arbitrarily small perturbations of the problem’s data may yield both, solvable and unsolvable problems. Thus, the ill-posedness is identified with the boundary of the set of solvable problems. The associated concept of well-posedness turns out to be equivalent t...

In this paper we measure how much a linear optimization problem, in Rn, has to be perturbed in order to loose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed problems those in the boundary of the set of solvable ones, this paper deals with the associated di...

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 su...

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 con...

In this paper, we propose a parametric approach to the stability theory for the solution set of a semi-infinite linear inequality system in the n-dimensional Euclidean space
\mathbb Rn{\mathbb R}^{n}
. The main feature of this approach is that the coefficient perturbations are modeled through the so-called mapping of parametrized systems, which as...

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in n. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing th...

In this paper we study the (Berge) upper semicontinuity of a generic multifunction assigning to each parameter, in a metric
space, a closed convex subset of the n-dimensional Euclidean space. A relevant particular case arises when we consider the feasible set mapping associated with
a parametric family of convex semi-infinite programming problems....

In this paper we characterize the upper semicontinuity of the feasible set mapping at a consistent linear semi-infinite system (LSIS, in brief). In our context, no standard hypothesis is required in relation to the set indexing the constraints and, consequently, the functional dependence between the linear constraints and their associated indices h...

In this paper we consider a parametrized family of linear inequality systems whose coefficients depend continuously on a parameter ranging in an arbitrary metric space. We analyze the lower semicontinuity (lsc) of the feasible set mapping in terms of the so-called carrier index set, consisting of those indices whose associated inequalities are sati...

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 forT, and so the dependence of the constraints on the index has no particular property....

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 a...

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 propert...

Abstract This paper,is devoted,to quantify,the Lipschitzian behavior,of the optimal,solutions set in linear optimization,under,perturbations,of the objective function and,the right hand,side of the constraints,(inequalities).