Characterizations of robust solution set of convex programs with uncertain data
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In this paper, we study convex programming problems with data uncertainty in both the objective function and the constraints. Under the framework of robust optimization, we employ a robust regularity condition, which is much weaker than the ones in the open literature, to establish various properties and characterizations of the set of all robust optimal solutions of the problems. These are expressed in term of subgradients, Lagrange multipliers and epigraphs of conjugate functions. We also present illustrative examples to show the significances of our theoretical results.
KeywordsConvex programs under uncertainty Robust solution sets Constraint qualifications Subdifferential Lagrangian multipliers Epigraphs of conjugate functions
The authors express their deep gratitude to the anonymous referees and the associate editor for their valuable comments and suggestions that helped to improve this article. Xiao-Bing Li would like to thank Prof. Song Wang for his hospitality during his stay from February 2015 to February 2016 at the School of Mathematics and Statistics of the Curtin University in Perth. This research was partially supported by the Basic and Advanced Research Project of Chongqing (cstc2015jcyjBX0131, cstc2015jcyjA30009), the Program of Chongqing Innovation Team Project in University under Grant (CXTDX201601022) and the Program for Core Young Teacher of the Municipal Higher Education of Chongqing (47).