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Optimality Functions and Lopsided Convergence

  • Johannes O. Royset
  • Roger J-B Wets
Article

Abstract

Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.

Keywords

epi-Convergence Lopsided convergence Consistent approximations Optimality functions Optimality conditions 

Mathematics Subject Classification

90C46 49J53 

Notes

Acknowledgments

This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under Grant Numbers 00101-80683, W911NF-10-1-0246 and W911NF-12-1-0273.

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Copyright information

© Springer Science+Business Media New York (outside the US) 2015

Authors and Affiliations

  1. 1.Naval Postgraduate SchoolMontereyUSA
  2. 2.University of California, DavisDavisUSA

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