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A probabilistic approach is considered for robust optimization, where a convex objective function is minimized subject to a parameter dependent convex constraint. A novel sequential randomized algorithm is proposed for solving this optimization employing the stochastic ellipsoid method. It is shown that the upper bounds of the numbers of random samples and updates of the algorithm are much less than those of the stochastic bisection method utilizing the stochastic ellipsoid method at each iteration. This feature actually leads to a computational advantage, which is demonstrated through a numerical example.

A class of robust feasibility problems is considered, which is to find a design variable satisfying a parameter- dependent constraint for all parameter values. A randomized algorithm for solving the problem with a general nonconvex constraint is proposed, where random samples of candidates of the design variable and uncertain parameters are used. The algorithm stops in a finite number of iterations. Then, it gives a design variable satisfying the constraint for almost all parameter values with a prescribed confidence or says that the problem is infeasible in a probabilistic sense.