Abstract
In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of this work is to present the relationship between the three problems mentioned above. First, applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact. Next, using these mappings we analyze the structure of the Pareto sets. Some basic topological properties of the Pareto solutions sets in the general case and in the convex case are also discussed.
Highlights
The Berge Maximum Theorem, shortly the MaximumTheorem, has become one of the most useful and powerful theorems in optimization theory, mathematical economics and game theory
Applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact
We have shown an application of the Maximum Theorem to multi-criteria optimization for the construction of the Pareto-retract mappings and the role of these mappings to analyze the structure of the Pareto-optimal and the Pareto-front sets
Summary
Theorem, has become one of the most useful and powerful theorems in optimization theory, mathematical economics and game theory. The Maximum Theorem is often used in a special situation when the multifunction D is convex-valued and the function u is quasi-concave or concave in its second variable in addition to the hypotheses of Theorem 1. (a) m is a continuous function on X, and S is a compact-valued and upper semi-continuous multifunction on X. (c) If u(x, ) is strictly quasi-concave in y for each x X , and D is convex-valued, S is a continuous function on X. (d) If u is concave on X Y , and D has a convex graph, m is a concave function on X and S is a convex-valued multifunction on X. (e) If u is strictly concave on X Y , and D has a convex graph, m is a strictly concave and continuous function on X, and S is a continuous function on X. If S x 1 for all x X , m and S are two continuous function on X
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