Pareto Subdifferential Calculus for Convex Vector Mappings and Applications to Vector Optimization

This paper deals with the subdifferential of convex analysis defined in the Pareto sense, from a point of view of nonvacuity, characterizations, and calculus rules and their applications to the vector optimization, the convex maps being vector-valued in a finite- or infinite-dimensional ordered vector space. Subdifferentiability is characterized under conditions of Attouch-Brezis type. Formulations by derivatives, when they exist, are provided. Concerning the calculus rules, the first main result gives the gap between the Pareto subdifferential and the ordinary one, allowing thus the computation of the one from the other. Next, as central results, Pareto subdifferentials of the sum and/or composition of two convex vector mappings are developed. The formulas are obtained under Moreau-Rockafellar or Attouch-Brezis-type conditions, revealing, strangely, the presence of the ordinary subdifferential. These formulas actually allow the extension of the indicator function technique to the vector case, so that Pareto optimality (efficiency) conditions are easily derived and weakened with qualification conditions of the Attouch-Brezis kind. Finally, the gap between efficient and optimal sets is also deduced.

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