Optimization Under Uncertainty Using Probability Collectives

In this paper, we review an optimization technique called Probability Collectives (PC), and compare its approach with that of quadratic Response Surface Methods (RSM), on some standard continuous test functions for unconstrained minimization, both in the presence and absence of uncertainty in the function evaluations. Probability Collectives (PC) is an optimization framework where optimization is performed over probability distributions over the variables of interest, rather than the variables themselves. In order to find a solution to the original optimization problem, we sample the final solution of the transformed problem, which is a probability distribution. In other words, PC is a transform technique with the special property that the inverse transform is performed stochastically. This transformation yields algorithms that are relatively insensitive to structural properties of the underlying objective function, like continuity, convexity, and dierentiability; optimization under uncertainty becomes natural and straightforward; finally, the distributions yield sensitivity information about the original problem. Some PC algorithms bear a strong resemblance to RSM, and in this paper, we investigate these similarities, and compare performances of these techniques on some simple test functions.