A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

Abstract

Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals defined by Hiriart-Urruty, Drummond and Svaiter, and Gerstewitz. We completely determine their relationships. In particular, it is shown that the class of the functionals by Gerstewitz is minimal in this sense. Furthermore, we define a new (and larger) class of scalarizing functionals that are not necessarily convex, but rather quasidifferentiable and positively homogeneous. We show that our results are connected with some of the set relations in set optimization.

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Correspondence to Christiane Tammer.

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Dedicated to Alexander Ioffe in honor of his 80th birthday.

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Bouza, G., Quintana, E. & Tammer, C. A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization. Vietnam J. Math. 47, 683–713 (2019). https://doi.org/10.1007/s10013-019-00359-1

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Keywords

  • Scalarization
  • Vector optimization
  • Quasidifferentiability
  • Separation theorem
  • Dual spaces

Mathematics Subject Classification (2010)

  • 90C26
  • 90C29
  • 90C30
  • 90C48