Alejandro Jofré

Sub differential Monotonicity as Characterization of Convex Functions

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.

Variational Inequalities and Economic Equilibrium

Variational inequality representations are set up for a general Walrasian model of consumption and production with trading in a market. The variational inequalities are of functional rather than geometric type and therefore are able to accommodate a wider range of utility functions than has been covered satisfactorily in the past. They incorporate Lagrange multipliers for budget constraints, which are shown to lead to an enhanced equilibrium framework with features of collective optimization. Existence of such an enhanced equilibrium is confirmed through a new result about solutions to nonmonotone variational inequalities over bounded domains. Truncation arguments with specific estimates, based on the data in one economic model, are devised to transform the unbounded variational inequality that naturally comes up into a bounded one having the same solutions.

A nonconvex separation property in Banach spaces

Abstract. We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our property with the extremal principle of Mordukhovich.

Operations Research challenges in forestry: 33 open problems

Forestry has contributed many problems to the Operations Research (OR) community. At the same time, OR has developed many models and solution methods for use in forestry. In this article, we describe the current status of research on the application of OR methods to forestry and a number of research challenges or open questions that we believe will be of interest to both researchers and practitioners. The areas covered include strategic, tactical and operational planning, fire management, conservation and the use of OR to address environmental concerns. The paper also considers more general methodological areas that are important to forestry including uncertainty, multiple objectives and hierarchical planning.

Linking strategic and tactical forestry planning decisions

In this paper a two-level model and optimization algorithms areintroduced to assist forestry companies in simultaneously considering strategicinvestment and tactical planning decisions. A procedure to reduce thediscrepancy produced in the aggregation and disaggregation process used to linkthese two-level decisions is also presented. This procedure is based onboth cluster analysis over economic parameters defined on “the standmacro-units to be harvested” and on the information transmittedbottom-up and top-down between the strategic and tactical models. Wealso show a new approach for solving the tactical mixed integer model.

A nonconvex separation property and some applications

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multipliers in Optimization.

A Variational Inequality Scheme for Determining an Economic Equilibrium of Classical or Extended Type

The existence of an equilibrium in an extended Walrasian economic model of exchange is confirmed constructively by an iterative scheme. In this scheme, truncated variational inequality problems are solved in which the agents’ budget constraints are relaxed by a penalty representation. Epi-convergence arguments are employed to show that, in the limit, a virtual equilibrium is obtained, if not actually a classical equilibrium. A number of technical hurdles are, in this way, surmounted.

Variational convergence of bivariate functions: lopsided convergence

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, of solutions to inclusions, the primal and dual solutions of convex optimization problems and of zero-sum games. These applications will be dealt with in a couple of accompanying papers.

Continuity Properties of Walras Equilibrium Points

We explore convergence notions for bivariate functions that yield convergence and stability results for their max/inf points. The results are then applied to obtain continuity results for Walras equilibrium points under perturbations of the utility functions of the agents.

Continuity and uniqueness of equilibria for linear exchange economies

The purpose of this paper is to study the continuity and uniqueness properties of equilibria for linear exchange economies. We characterize the sets of utility vectors and initial endowments for which the equilibrium price is unique and respectively the set for which the equilibrium allocation is unique. We show that the equilibrium allocation correspondence is continuous with respect to the initial endowments and we characterize the set of full measure where the equilibrium allocation correspondence with respect to the initial endowments and utility vectors is continuous.