Topics in black-box combinatorial optimization

This thesis explores the performance of a class of general optimization heuristics, known as black-box optimization algorithms, on a range of problems possessing a high degree of combinatorial structure. We first explore the performance of one of the most elementary of these algorithms, known as hillclimbing or randomized greedy, on the minimum bisection problem. We prove that a simple hillclimbing algorithm solves a range of problem instances drawn from Jerrum and Sorkin's $G\sb{4n,p,r,}$, random graph model in polynomial time with high probability. In experiments, we show that hillclimbing is perhaps much more effective than this theoretical result suggests. We prove a weak lower bound on the number of local minima in the search space for the problem instances under investigation, and conduct an experimental exploration of these local minima. We then investigate the performance of hillclimbing relative to a class of global optimization algorithms known as genetic algorithms (GAs). We consider a range of combinatorial optimization problems for which GAs have been claimed to work well in the literature: these include the maximum cut and jobshop scheduling problems, a problem known as the multiprocessor document allocation problem, and a genetic programming task involving the discovery of an 11-multiplexer program. We show that simple hillclimbing algorithms can outperform GAs on problem instances for which the GAs were specifically designed. By modifying the structure of the search space for the jobshop problem, however, we are able in turn to make a genetic algorithm more effective than hillclimbing. Finally, we propose an intuitively appealing, mathematical abstraction of genetic algorithms known as the equilibrium genetic algorithm (EGA). We prove the ability of this algorithm to find the optimum of a simple, non-linear problem, which we call MAX, in polynomial time with high probability. We also show experimentally that the EGA is competitive with a conventional GA on the jobshop scheduling problem. This work was supported by a NASA Graduate Research Fellowship, NSF Grant CCR-9505448, and a University of California Regents Fellowship. Part of this work was performed at the Ecole Normale Superieure, rue d'Ulm, Paris.