An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets

Affiliation(s)

Varna Free University, Varna, Bulgaria.

The George Washington University, Washington DC, USA.

Varna Free University, Varna, Bulgaria.

The George Washington University, Washington DC, USA.

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.

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.

Keywords

Multi-Criteria Optimization; Maximum Theorem; Pareto-Retract Mapping; Pareto-Optimal; Pareto-Front

Multi-Criteria Optimization; Maximum Theorem; Pareto-Retract Mapping; Pareto-Optimal; Pareto-Front

Cite this paper

Z. Slavov and C. Evans, "An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1415-1422. doi: 10.4236/am.2012.330199.

Z. Slavov and C. Evans, "An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets,"

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