Sequential randomized algorithms: A probabilistic cutting plane technique based on maximum volume ellipsoid center

Abstract

A sequential randomized algorithm is developed for robust optimization which is to minimize a linear objective function subject to a parameter dependent convex constraint for all uncertain parameter values. The algorithm is realized as a probabilistic cutting plane technique based on maximum volume ellipsoid center, where candidates of the optimal value and of the optimal solution are sequentially updated by a series of cutting planes generated by random samples of the uncertain parameters. This algorithm stops in a finite number of iterations which is of polynomial order of the problem size, and provides a feasible solution of a chance constraint optimization which corresponds to a probabilistic relaxation of the robust optimization with a prescribed probability. Then, it is shown that the algorithm can find a suboptimal value whose lower bound is the optimal value of the chance constrained optimization with the prescribed probability and whose upper bound is determined by the optimal value of the robust optimization.