Regularity conditions in vector optimization

Giancarlo Bigi and Massimo Pappalardo


In the field of scalar Optimization regularity is conceived as a condition which guarantees the existence of generalized Karush-Kuhn-Tucker (in short KKT) multipliers such that the one corresponding to the objective function is nonzero. In the field of Vector Optimization, the analysis is more difficult; in fact, since there are many objective functions, different concepts of regularity can be considered. In a recent paper, new concepts of regularity and total regularity for linear separation between a set and a convex cone have been studied. Here, we aim to apply the results of that paper to the study of regularity in Vector Optimization. Most of the known optimality conditions can be expressed in terms of the impossibility of generalized systems; following the image space approach, the impossibility of such systems can be turned into the disjunction between a set and a convex cone and studied by means of linear separation tools. Besides, the gradient of a separating hyperplane is a vector of KKT multipliers. We prove that regularity conditions for linear separation provide regularity conditions for vector optimization problems. Moreover, we show that total regularity for linear separation is equivalent to very general Mangasarian-Fromovitz type conditions, which can be employed to achieve additional properties of KKT multipliers. Besides, we also show that many well-known regularity conditions and constraint qualifications are less general than the ones developed in our scheme and they can be recovered as particular cases.

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