Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>] 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.</p>

Robust duality for generalized convex nonsmooth vector programs with uncertain data in constraints

Robust optimization has come out to be a potent approach to study mathematical problems with data uncertainty. We use robust optimization to study a nonsmooth nonconvex mathematical program over cones with data uncertainty containing generalized convex functions. We study sufficient optimality conditions for the problem. Then we construct its robust dual problem and provide appropriate duality theorems which show the relation between uncertainty problems and their corresponding robust dual problems.

Characterization of Robust Solution Quadratically Constrained Quadratic Optimization Problem Subjected to Data Uncertainty

Robust Optimization (RO) arises in two stages of optimization, first level for maximizing over the uncertain data and second level for minimizing over the feasible set. It is the most suitable mathematical optimization procedure to solve real-life problem models. In the present work, we characterize robust solutions for both homogeneous and non-homogeneous quadratically constrained quadratic optimization problem where constraint function and cost function are uncertain. Moreover, we discuss about optimistic dual and strong robust duality of the considered uncertain quadratic optimization problem. Finally, we complete this work with an example to illustrate our solution method. Mathematics Subject Classification: (2010) 90C20 - 90C26 - 90C46-90C47 Keywords: Robust Optimization, Data Uncertainty, Quadratic Optimization Strong Duality, Robust Solution, DPJ-Convex.