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The validity of Zangwill’s general algorithm for finding a point of a subset of a set is given here with weakened hypotheses. In particular the closedness of the point-to-set map used in the algorithm is not needed.
A general structure is established that allows the comparison of various conditions that are sufficient for convergence of algorithms that can be modeled as the recursive application of a point-to-set map. This structure is used to compare several earlier sufficient conditions as well as three new sets of sufficient conditions. One of the new sets of conditions is shown to be the most general in that...
The study of the convergence of algorithms of optimization obtained by composition or union, taken in sense of the relaxation, is done. After having recalled the Zangwill’s theorem and given two extensions we study the obtainment of generalized fixed points in the framework of the composition or the union of algorithms obtained in a free steering way for, firstly functions having a unique maximum...
A new approach for synthetizing optimization algorithms is presented. New concepts for the continuity of point-to-set maps are given in terms of families of maps. These concepts are well adapted to construct fixed point theorems that are widely useful for synthetizing optimization methods. The general algorithms already published are shown to be particular applications and illustrations in the field...
In this paper dual programs of convex optimization problems having a parametric objective function and a fixed linear feasible set are studied. By using some properties of the primal problem the continuity of the dual optimal solution set is proved. Two examples show the necessity of the suppositions.
This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower...
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