Formulas of first-ordered and second-ordered generalization differentials for convex robust systems with applications

•Formulas for calculating first and second-ordered generalization differentials for uncertain convex systems are provided.•Second-ordered optimality conditions for strict solutions of a convex robust optimization problem are established.•An associated algorithm that quickly converges to a solution for the class of quadratic robust optimization problems is designed. In the paper, we start by establishing first and second-ordered analysis for a convex inequality system that contains uncertainty data, including calculating normal and tangent cones, second-ordered tangent sets for the solution set to this system, and first and second-ordered epi-subderivatives for the indicator function of its solution set. Then, we provide second-ordered necessary and sufficient optimality conditions for strict solutions of convex robust optimization problems. Moreover, an associated algorithm converging quickly to a solution for the class of quadratic robust optimization problems is proposed. The theoretical results are newly obtained under weak qualification conditions, and numerical examples show the advantage of the given method.