Lipschitz lower semicontinuity moduli for linear inequality systems

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 Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings

Robust and continuous metric subregularity for linear inequality systems

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 continuity behavoir of the (always finite) metric subregularity modulus leads us to consider the aforementioned properties. After characterizing both of them, the radius of robust metric subregularity is computed and some insights on the radius of continuous metric subregularity are provided.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research has been partially supported by Grant PGC2018-097960-B-C2(1,2) from MICINN, Spain, and ERDF, “A way to make Europe”, European Union, and Grant PROMETEO/2021/063 from Generalitat Valenciana, Spain

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 T. The parameter space of admissible perturbations under consideration is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l∞ type. By using advanced tools of variational analysis and exploiting specific features of linear infinite systems, we establish complete characterizations of robust Lipschitzian stability entirely via their initial data with computing the exact bound of Lipschitzian moduli. A crucial part of our analysis addresses the precise computation of the coderivative of the feasible set mapping and its norm. The results obtained are new in both semi-infinite and infinite frameworks. (A correction to the this article has been appended at the end of the pdf file.)This research was partially supported by grants MTM2005-08572-C03 (01-02) from MEC (Spain) and FEDER (EU), MTM2008-06695-C03 (01-02) from MICINN (Spain), and ACOMP/2009/047&133 from Generalitat Valenciana (Spain); National Science Foundation (USA) under grant DMS-0603846

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

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. Optim., 20 (2009), pp. 1504–1526] from the viewpoint of robust Lipschitzian stability. The main results establish necessary optimality conditions for broad classes of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming. We illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs.This research was partially supported by grants MTM2008-06695-C03 (01-02) from MICINN (Spain), ACOMP/2009/047&133, and ACOMP/2010/269 from Generatitat Valenciana (Spain)

Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming

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 attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given

Critical objective size and calmness modulus in linear programming

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 perturbations of all coefficients. Our starting point is an upper bound on this modulus given in \cite{CHPTmp}. In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds

Calmness modulus of linear semi-infinite programs

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 space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.This research has been partially supported by grants MTM2011-29064-C03 (02-03) from MINECO, Spain, ACOMP/2013/062 from Generalitat Valenciana, Spain, grant C10E08 from ECOS-SUD, and grant DP110102011 from the Australian Research Council

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.This research was partially supported by grants PB95-0687 and SAB 95-0311 from DGES and by grants GV-2219/94 and GV-C-CN-10-067-96 from Generalitat Valenciana

Distance to Solvability/Unsolvability in Linear Optimization

In this paper we measure how much a linear optimization problem, in Rn, has to be perturbed in order to lose either its solvability (i.e., the existence of optimal solutions) or its unsolvability property. In other words, if we consider as ill-posed those problems in the boundary of the set of solvable ones, then we can say that this paper deals with the associated distance to ill-posedness. Our parameter space is the set of all the linear semi-infinite programming problems with a fixed, but arbitrary, index set. In this framework, which includes as a particular case the ordinary linear programming, we obtain a formula for the distance from a solvable problem to unsolvability in terms of the nominal problem's coefficients. Moreover, this formula also provides the exact expression, or a lower bound, of the distance from an unsolvable problem to solvability. The relationship between the solvability and the primal-dual consistency is analyzed in the semi-infinite context, underlining the differences with the finite case.This research has been partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), and GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain)

La cronobiología: una herramienta de apoyo a la docencia

[SPA] En esta comunicación se propone la realización de un taller teórico-práctico, en el que alumnos de ESO podrán reforzar y adquirir nuevos conocimientos de Ciencias de la naturaleza, Biología y Geología, y Ciencias sociales, recogidos en el Anexo II del Real Decreto 1631/2006, mediante la realización de determinados experimentos y observaciones de situaciones cotidianas relacionadas con la Cronobiología, trabajando en distintos grupos, en función del curso al que pertenezcan. La Cronobiología es una ciencia cuyo objetivo es el estudio de los ritmos biológicos. Éstos son oscilaciones que sufren determinadas variables biológicas de forma regular y con un periodo determinado. Además, con esta innovadora forma de trabajo se pretenden alcanzar importantes objetivos como son la familiarización con las características básicas del trabajo científico, la interpretación de información de carácter científico y utilización de dicha información para formarse una opinión propia. Además, se pretende que los alumnos aprendan a valorar las aportaciones de las ciencias de la naturaleza para dar respuesta a las necesidades de los seres humanos y por supuesto la utilización de fuentes secundarias de información para la realización de pequeñas investigaciones.[ENG] The aim of this communication is to propose a theoretical and practical workshop to be carried out with Compulsory Secondary Education (ESO) students, where their Natural Science, Biology and Geology curriculum under Annex II of RD 1631/2006 would be deepened and further developed. This would be achieved through a number of experiments and through the study and recording of everyday situations related to Chronobiology. These experiences would be developed by classifying the students in different groups according to their academic level. Chronobiology is the field of study that examines biological rhythms as biological patterns which are subject to a periodic or cyclic rhythm. Furthermore, by means of this innovative approach significant objectives may be attained, such as furthering the students’ awareness of the basic features of scientific research, helping them interpret scientific data and teaching them how to adequately use such information to develop a personal view on each subject. Students would acquire also a better knowledge of the contribution of Natural Sciences towards a better answer for the humans’ needs in all fields and, of course, they would learn how to use secondary information sources necessary to carry our basic research