Open AccessGlobal optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators
Author(s) -
Jutamas Kerdkaew,
Rabian Wangkeeree,
Rabian Wangkeeree
Publication year - 2022
Publication title -
numerical algebra, control and optimization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.303
H-Index - 20
eISSN - 2155-3289
pISSN - 2155-3297
DOI - 10.3934/naco.2021053
Subject(s) - karush–kuhn–tucker conditions , duality (order theory) , mathematics , mathematical optimization , optimization problem , differentiable function , type (biology) , robust optimization , strong duality , function (biology) , dual (grammatical number) , pure mathematics , art , ecology , literature , evolutionary biology , biology
In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [ 1 , 2 ] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.