Characterizations of ɛ-Approximate Solutions for Robust Convex Semidefinite Programming Problems

In this article, we explore the characterizations of -approximate solutions for convex semidefinite programming problems that involve uncertain data. It first reviews essential findings regarding the optimality condition and duality of robust convex semidefinite programming problems. Then, we establish the optimality and duality conditions concerning the problem by assuming specific constraint qualifications. The study investigates -Kuhn-Tucker vectors and their relationships with the optimal solutions, maximizers of the corresponding Lagrangian dual problem, saddle points of the Lagrangian, and Kuhn-Tucker vectors. Finally, the article establishes the characterization of -approximate solution sets for the problem by studying the connection among three sets: the set of Lagrange multipliers corresponding to -approximate solutions, the set of -Kuhn-Tucker vectors, and the set of approximate solutions for their Lagrangian dual problems. The characterization is illustrated with several examples.