Abstract

This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form max∥T(x)∥, min∥x∥ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering.

Highlights

  • Speaking, a Pareto optimal solutions (POS) is a feasible solution such that, if any other feasible solution is more optimal at one objective function, it is less optimal at another objective function

  • Pareto optimal solutions are sometimes graphically displayed in Pareto charts (PC)

  • We provide a sufficient topological condition to guarantee the existence of Pareto optimal solutions

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Summary

Introduction

Multiobjective optimization problems (MOPs) appear quite often in all areas of pure and applied mathematics, for instance, in the geometry of Banach spaces [1,2,3], in operator theory [4,5,6,7], in lineability theory [8,9,10], in differential geometry [11,12,13,14], and in all areas of Experimental, Medical and Social Sciences [15,16,17,18,19,20]. The existence of a global solution that optimizes all the objective functions of an MOP at once is very unlikely. This is were Pareto optimal solutions (POS) come into play. Speaking, a POS is a feasible solution such that, if any other feasible solution is more optimal at one objective function, it is less optimal at another objective function. Pareto optimal solutions are sometimes graphically displayed in Pareto charts (PC). In this manuscript, we prove a characterization of POS by relying on orderings and equivalence relations. We provide a sufficient topological condition to guarantee the existence of Pareto optimal solutions

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