Two-stage stochastic standard quadratic optimization

Two-stage stochastic decision problems are characterized by the property that in stage one part of the decision has to be made before relevant data are observed and the rest of the decision can be made in stage two after data are available. It is generally assumed that while the data missing in stage one are not known, their probability distribution is known. In this paper, we consider a special form of such problems where the objective function is quadratic in the first and second stage decisions and the goal is to minimize the expected quadratic cost function. We assume that the decisions must lie in a standard simplex, but do not require convexity of the uncertain objective. It is well known that such quadratic optimization programs are NP-complete and thus belong to the most complex optimization problems. A viable method to attack such problems is to find bounds for the original complicated problem by solving easier approximate problems. We describe such approximations and show that by exploiting their properties, good but not necessarily optimal solutions can be found. We also present possible applications, such as selecting invited speakers for conferences with the aim to generate a community with high coherence. Finally, systematic numerical results are provided.