Catálogo de publicaciones - libros

The aim of the paper is to suggest a sequential method for generating the set of all efficient points of a bicriteria problem where the feasible region is a polytope and whose criteria are a linear function and a concave function which is the sum of a linear and the reciprocal of an affine function. The connectedness of and some theoretical properties of allow to give a finite simplex-like algorithm based on a suitable post-optimality analysis carried on a scalar parametric problem where the linear criteria plays the role of a parametric constraint.

The notions of a convex arc and piecewise convex curve in the plane generalize the notion of a convex curve, the latter is usually defined as the boundary of a planar compact convex set with nonempty interior. The integral representation of a piecewise convex curve through a Riemann-Stieltjes integral with a corresponding one-dimensional measure is studied. It is shown that the Minkowski operations known from the convex sets can be generalized to piecewise convex curves. It is shown that the decomposition of the measure in the integral representation of the piecewise convex curve leads to a decomposition of the piecewise convex curve into a sum of corresponding piecewise convex curves. On this base, applying the natural decomposition of the one-dimensional measure into an absolutely continuous function, a jump function, and a singular function, the structure of a piecewise convex curve is investigated. As some curious consequences, the existence of polygons with infinitely many sides and no vertices, and polygons with infinitely many vertices and no sides is shown.

Two classes of functions encompassing the cone of convex functions and the space of strictly differentiable functions are presented and compared. Related properties for sets and multimappings are dealt with.

There is a recent surge of interest for the representation of monotone operators by convex functions. It can explained by the success of convex analysis in obtaining the fundamental results about maximal monotone operators. Convex analysis can also be combined with variational analysis to get new convergence results. Here we take another direction and connect such a stream with the concept of duality in a general framework, heavily using order methods.

We consider two new classes of generalized relaxed -monotone and semimonotone functions and using the KKM technique we prove the existence of solutions for variational-like inequalities relative to these types of mappings in Banach spaces. Several examples and special cases are also considered.

This paper considers n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. For such games it is shown that there are two-point Nash equilibria in them, that is equilibria in players’ strategies with support consisting of at most two points. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.

This paper presents some first and second order conditions necessary for the Pareto optimality of box-constrained multi-objective optimization problems. These necessary conditions are related to the spectrum of a matrix defined via the gradient vectors and the Hessian matrices of the objective functions. These necessary conditions are used to develop two algorithms. The first one is built taking into account the first order necessary conditions and determines some critical points for the multi-objective problems considered. The second one is based on the second order necessary conditions and discards the critical points that do not belong to the local Pareto optimal front. Some numerical results are shown.