Some criteria for error bounds in set optimization

We obtain sufficient and/or necessary conditions for global/local error bounds for the distances to some sets appeared in set optimization studied with both the set approach and vector approach (sublevel sets, constraint sets, sets of {\it all } Pareto efficient/ Henig proper efficient/super efficient solutions, sets of solutions {\it corresponding to one} Pareto efficient/Henig proper efficient/super efficient value) and sufficient conditions for metric subregulatity of a set-valued map at efficient solutions. All criteria except one are described in terms of the Mordukhovich coderivatives and coderivative of convex analysis. Our techniques are based on scalarization by mean of the Hiriart-Urruty signed distance function, on exploiting criteria in terms of subdifferentials for error bounds of a lower semicontinuous function and estimates for subdifferentials of marginal functions. We also consider the single-valued case and provide illustrating examples.


Institute of Mathematics, Hanoi, Vietnam, Preprint, August 2012, No. 12-02



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