A relaxation algorithm with a probabilistic guarantee for robust deviation optimization

Three measures of robustness (absolute robustness, deviation robustness and relative robustness), whose choice depends on the goals of the decision maker, have been proposed for uncertain optimization problems. Absolute robustness has been thoroughly studied, whereas the others have been studied to less of a degree.We focus on deviation robustness for uncertain convex quadratic programming problems with ellipsoidal uncertainties and propose a relaxation technique based on random sampling for robust deviation optimization problems. We theoretically and experimentally show that solving the relaxation problem gives a tighter lower bound than solving a simple sampled relaxation problem. Furthermore, we measure the robustness of the solution in a probabilistic setting. The number of random samples is estimated for obtaining an approximate solution with a probabilistic guarantee, and the approximation error is evaluated a-priori and a-posteriori. Our relaxation algorithm with a probabilistic guarantee utilizes a-posteriori assessment to evaluate the accuracy of the approximate solutions.