Nonconvex scalarization in set optimization with set-valued maps

By Elvira Hernández, Luis Rodríguez-Marín. 2007

252 citations recorded by the source corpus.

Abstract The aim of this work is to obtain scalar representations of set-valued optimization problems without any convexity assumption. Using a criterion of solution introduced by Kuroiwa [D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, pp. 221–228], which is based on ordered relations between sets, we characterize this type of solutions by means of nonlinear scalarization. The scalarizing function is a generalization of the Gerstewitz's nonconvex separation function. As applications of our results we give two existence theorems for set-valued optimization problems.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1bfc5af2063075a5cddf0bd90a791e49c1a57c7b

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Citations in the corpus

Lagrangian Duality in Set-Valued Optimization (2007, unknown): “[9–13].”
Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization (2016, background): “Hernández and Rodríguez-Marín [12] introduce an extension of the functional zC,k (see Equation (1)) in order to characterize the inclusion B ⊆ A+C (compare also [13–15] and the references given therein).”
Well-posedness and stability of solutions for set optimization problems (2017, background): “7 of [4], we know that El ( F,K ) ⊆ Wl (F,K).”