Optimization by rare-event simulation

Abstract

In this dissertation we present several papers about two optimization methods that are derived from rare-event simulation techniques, i.e., the splitting method and the cross-entropy (CE) method. The splitting method is a well-known method for rare-event simulation, where sample paths of a Markov process are split into multiple copies during the simulation, so as to make the occurrence of a rare event more frequent. Motivated by the splitting algorithm we introduce a novel global optimization method for continuous optimization (SCO) that is both very fast and accurate. We also introduce a new multi-objective optimization (MOO) methodology based the SCO algorithm. The method, called MOS, generalizes the elite set selection of the traditional splitting framework, and uses both local and global sampling to sample in the decision space. Both SCO and MOS were compared with state-of-the art algorithms using a prevailing set of benchmark problems. Numerical experiments demonstrate that the new splitting methods are competitive with famous methods in the fields. The thesis also contains two new applications of the CE method on Laguerre fitting problems. The Laguerre fitting problems are about recovering the weighted generator points from a given Laguerre tessellation. When the description of the Laguerre tessellation is exact, the solutions are not unique and different weighted generator points can create the same tessellation. To recover pertinent generator points we formulate the problem as an optimization problem and apply the modified CE method to solve it. However, the representation of a Laguerre tessellation is typically not exact when working with real data, such as tomographic image data. In this case, we formulate an optimization problem where we minimize the discrepancy between the image data and our approximated tessellations. We then solve the optimization problem using the CE method. Last, we introduce a new R package that implements the general CE methodology for optimization and well as some useful modifications. The usage and efficacy of CEoptim is demonstrated through a variety of optimization examples, including model fitting, combinatorial optimization, and maximum likelihood estimation.