Abstract
A randomized algorithm is proposed for solving a robust optimization which is to minimize a convex objective function subject to a parameter dependent convex constraint. The probabilistic analytic center cutting plane method is employed for solving a series of robust feasibility problems defined by an optimality cut which corresponds to a provisional optimal value, where the parameter is randomly sampled in each iteration and the cut is sequentially updated for optimization. The algorithm always stops in a finite number of iterations, and finds a suboptimal solution and a suboptimal value. The suboptimal solution satisfies the constraint with a given probability and with a given probabilistic confidence, while the suboptimal value ensures that the feasibility set whose objective function is less than this value is too small to contain a ball with a given radius. The upper bounds of the numbers of random samples and updates of the algorithm are of polynomial order of the problem size.
