Chance constrained programs with Gaussian mixture models

In this article, we discuss input modeling and solution techniques for several classes of Chance constrained programs (CCPs). We propose to use a Gaussian Mixture Model (GMM) to fit the data available and to model the randomness. We demonstrate the merits of using a GMM. We consider several scenarios that arise from practical applications and analyze how the problem structures could embrace alternative optimization techniques. More specifically, for several scenarios, we study how to assess the gradient of the chance constraint and incorporate the results into gradient-based nonlinear optimization algorithms, and for a class of CCPs, we propose a spatial branch-and-bound procedure and solve the problems to global optimality. We also conduct numerical experiments to test the efficiency of our approach and propose an example of hedge fund portfolio to illustrate the practical application of the method.