Uncertain convex programs: randomized solutions and confidence levels
Many engineering problems can be cast as optimization problems subject to con- vex constraints that are parameterized by an uncertainty or 'instance' parameter. A recently emerged successful paradigm for attacking these problems is robust optimiza- tion, where one seeks a solution which simultaneously satisfies all possible constraint instances. In practice, however, the robust approach is effective only for problem families with rather simple dependence on the instance parameter (such as affine or polynomial), and leads in general to conservative answers, since the solution is usually computed by transforming the original semi-infinite problem into a standard one, by means of relaxation techniques. In this paper, we take an alternative 'randomized' or 'scenario' approach: by randomly sampling the uncertainty parameter, we substitute the original infinite constraint set with a finite set of N constraints. We show that the resulting randomized solution fails to satisfy only a small por- tion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.