Quasipolyhedral sets in linear semiinfinite inequality systems

AbstractThis paper provides an extension to linear semiinfinite systems of a well-known property of finite linear inequality systems, the so-called Weyl property, which characterizes the extreme points of the solution set as those solution points such that the gradient vectors of the active constraints form a complete set. A class of linear semiinfinite systems which satisfy this property is identified, the p-systems. It is also shown that any p-system contains an equivalent minimal subsystem

Stability of the Duality Gap in Linear Optimization

In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + ∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-01 and SECTyP-UNCuyo Res. 4540/13-R

Selected Applications of Linear Semi-Infinite Systems Theory

In this paper we, firstly, review the main known results on systems of an arbitrary (possibly infinite) number of weak linear inequalities posed in the Euclidean space Rn (i.e., with n unknowns), and, secondly, show the potential power of this theoretical tool by developing in detail two significant applications, one to computational geometry: the Voronoi cells, and the other to mathematical analysis: approximate subdifferentials, recovering known results in both fields and proving new ones. In particular, this paper completes the existing theory of farthest Voronoi cells of infinite sets of sites by appealing to well-known results on linear semi-infinite systems.This research was partially supported by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF); by CONICET, Argentina, Res D No 4198/17; and by Universidad Nacional de Cuyo, Secretaría de Investigación, Internacionales y Posgrado (SIIP), Res. 3922/19-R, Cod.06/D227, Argentina

The Voronoi inverse mapping

Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s , denoted by VT(s)VT(s), consists of all points closer to s than to any other site. The Voronoi mapping of s , denoted by ψsψs, associates to each set T∋sT∋s the Voronoi cell VT(s)VT(s) of s w.r.t. T . These Voronoi cells are solution sets of linear inequality systems, so they are closed convex sets. In this paper we study the Voronoi inverse problem consisting in computing, for a given closed convex set F∋sF∋s, the family of sets T∋sT∋s such that ψs(T)=Fψs(T)=F. More in detail, the paper analyzes relationships between the elements of this family, ψs−1(F), and the linear representations of F , provides explicit formulas for maximal and minimal elements of ψs−1(F), and studies the closure operator that assigns, to each closed set T containing s , the largest element of ψs−1(F), where F=VT(s)F=VT(s).This work has been supported by MINECO of Spain and ERDF from EC, Grants MTM2011-29064-C03-01, MTM2011-29064-C03-02, Australian Research Council’s Discovery Projects funding scheme (project number DP140103213), Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563), and SECTyP-UNCuyo, Argentina, Grant Res. 4540/13-R. The second author is affiliated to MOVE (Markets, Organizations and Votes in Economics)

Risk aversion, moral hazard, and the principal's loss

In their seminal paper on the principal-agent model with moral hazard, Grossman and Hart (1983) show that if the agent's utility function is $U(I,a)=-e^{-k(I-a)}$, then the loss to the principal from being unable to observe the agent's action is increasing in the agent's degree of absolute risk aversion. Their proof is restricted to the case where the number of observable outcomes is equal to two, and it uses an argument that is specific to that case. In this note, we provide an alternative proof that generalizes their result to any (finite) number of outcomes.Moral hazard, Principal-agent, Risk aversion.

On implicit active constraints in linear semi-infinite programs with unbounded coefficients

The concept of implicit active constraints at a given point provides useful local information about the solution set of linear semi-infinite systems and about the optimal set in linear semi-infinite programming provided the set of gradient vectors of the constraints is bounded, commonly under the additional assumption that there exists some strong Slater point. This paper shows that the mentioned global boundedness condition can be replaced by a weaker local condition (LUB) based on locally active constraints (active in a ball of small radius whose center is some nominal point), providing geometric information about the solution set and Karush-Kuhn-Tucker type conditions for the optimal solution to be strongly unique. The maintaining of the latter property under sufficiently small perturbations of all the data is also analyzed, giving a characterization of its stability with respect to these perturbations in terms of the strong Slater condition, the so-called Extended-Nürnberger condition, and the LUB condition.MICINN of Spain, Grant MTM2008-06695-C03-01, CONACyT of MX.Grant 55681 and SECYT-UNCuyo of Argentina, Grant Res. 1094/09-R

Linear conic and two-stage stochastic optimization revisited via semi-infinite optimization

In this paper, we update the theory of deterministic linear semi-infinite programming, mainly with the dual characterizations of the constraint qualifications, which play a crucial role in optimality and duality. From this theory, we obtain new results on conic and two-stage stochastic linear optimization. Specifically, for conic linear optimization problems, we characterize the existence of feasible solutions and some geometric properties of the feasible set, and we also provide theorems on optimality and duality. Analogously, regarding stochastic optimization problems, we study the semi-infinite reformulation of a problem-based scenario reduction problem in two-stage stochastic linear programming, providing a sufficient condition for the existence of feasible solutions as well as optimality and duality theorems to its non-combinatorial part.This research was partially supported by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF); and by Universidad Nacional de Cuyo, Secretaría de Investigación, Internacionales y Posgrado (SIIP), Res. 3032/2022-R, Cod. 06/L014-T1, Argentina

On coderivatives and Lipschitzian properties of the dual pair in optimization

In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem.The research of this author has been partially supported by MICINN Grant MTM2008-06695-C03-01 from Spain, and by ARC Project DP110102011 from Australia