P. Jiménez Guerra

Duality of nonscalarized multiobjective linear programs: dual balance, level sets, and dual clusters of optimal vectors

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.

Density theorems for ideal points in vector optimization

Abstract Several results are established concerning the density of the set of ideal points in the set of minimal solutions of positive support functionals of sets in normed linear spaces. The above ideal points are defined and several characterizations and sufficient conditions for their existence are also stated.

Sensitivity analysis in convex programming

The object of this paper is to perform an analysis of the sensitivity for convex vector programs with inequality constraints by examining the quantitative behavior of a certain set of optima according to changes of right-hand side parameters included in the program. The results in the paper prove that the sensitivity of the program depends on the solution of a dual program and its sensitivity.

Sensitivity in Multiobjective Programming by Differential Equations Methods. The case of Homogeneous Functions

The purpose of this paper is to characterize for convex multiobjective programming, the situations in which the sensitivity with respect to the right side vector of the constraints can be obtained as a solution of a dual program.

Sensitivity analysis in multiobjective differential programming

In this paper we deal with sensitivity analysis in multiobjective differential programs with equality constraints. We analyze the quantitative behavior of the optimal solutions according to changes of right-hand side values included in the original optimization problem. One of the difficulties lies in the fact that the efficient solution in multiobjective optimization in general becomes a set. If the preference of the decision maker is represented by a scalar utility which transforms optimal solutions in optima, we may apply existing methods of sensitivity analysis. However, when dealing with a subset of optimal points, the existence of a Frechet differentiable selection of such a set-valued map is usually assumed. The aim of the paper is to investigate the derivative of certain set-valued maps of efficient points. We show that the sensitivity depends on a set-valued map associated to the T-Lagrange multipliers and a projection of its sensitivity.

The balance space approach in optimization with Riesz spaces valued objectives. An application to financial markets

The balance set approach, first introduced in [1, pp. 138–140], is developed for optimization problems with objective functions taking values in Rn. As pointed out in [2,3], balance points have important economical interpretations. Since the theory of Riesz spaces and Banach lattices become more and more the natural setting for general equilibrium and dynamic economic models, see for instance [4], we propose here an extension of the balance space approach of [3], to models with objective functions taking values in Riesz spaces. As an application, we present an optimization problem with an objective function valued in an L2-space. It describes the process of an agent maximizing the profit coming from an arbitrage portfolio in a financial market.

Some geometrical aspects of the efficient line in vector optimization

Several relations are established between the proper ideal points and the extreme points, and also between the proper ideal points and the denting points. A necessary and sufficient geometrical condition is stated for a point to be a proper ideal point.

Measuring the Arbitrage Opportunities in an Intertemporal Dynamic Asset Pricing Model

This paper gives a measure of the arbitrage opportunities in an intertemporal dynamic asset pricing model. The measure is introduced by means of a stochastic process, and extends previous results obtained for static models in the literature. Two interesting applications of this theory can be considered. First, the measure of the cross-market arbitrage gives information about the integration between two or more financial markets. Second, dynamic asset pricing models with transaction costs can be analyzed.

Orthogonality in multiobjective optimization

Properties of nonlinear multiobjective problems implied by the Karush-Kuhn-Tucker necessary conditions are investigated. It is shown that trajectories of Lagrange multipliers corresponding to the components of the vector cost function are orthogonal to the corresponding trajectories of vector deviations in the balance space (to the balance set for Pareto solutions).

The mass-transfer vector problem

Abstract The mass-transference problem is studied for Banach valued cost functions and operator-valued measures. The solvability of the primal problem is stated under certain natural conditions, for general measurable functions and measures of bounded variation. The continuous case is also studied, and the solvability and the absence of a duality gap are established for continuous vector functions and regular operator valued measures.