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This paper consists of a study of differentiable stability in non convex and non differentiable mathematical programming. Vertically perturbed programs are studied and upper and lower bounds are estimated for the potential directional derivative of the perturbed objective function.
The Farkas lemma is examined in the context of point-to-set mappings. Some general non-linear inclusions are studied and the standard linear results are rederived in a strengthened and simplified form.
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...
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...
In this paper a rather general class of modified Lagrangians is described for which the main results of the duality theory hold. Within this class two families of modified Lagrangians are taken into special consideration. The elements of the first family are characterized by so-called stability of saddle points and the elements of the second family generate smooth dual problems. The computational...
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.
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.
In this paper, we examine the relationships between the fixed point set of a point-to-set map A(·), and the asymptotic properties of the sequences which may be iteratively generated by using the map A(·). Let L be the set of all limit points, and Q be the set of all cluster points of all sequences which may be iteratively generated by A(·). The consequences of various assumptions on the map A(·) and...
We consider a class of “generalized equations,” involving point-to-set mappings, which formulate the problems of linear and nonlinear programming and of complementarity, among others. Solution sets of such generalized equations are shown to be stable under certain hypotheses; in particular a general form of the implicit function theorem is proved for such problems. An application to linear generalized...
In this paper we consider the application of the recent algorithms that compute fixed points in unbounded regions to the nonlinear programming problem. It is shown that these algorithms solve the inequality constrained problem with functions that are not necessarily differentiable. The application to convex and piecewise linear problems is also discussed.
A general scheme, covering a wide class of two-level methods, is formulated. The possibilities for handling truncation errors and stopping rules on both levels are described. The convergence of the algorithm is proved under general assumptions, and computational results for an example of a minimax problem are given.
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...
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