Optimizing expected path lengths with ant colony optimization using fitness proportional update

Abstract

We study the behavior of a Max-Min Ant System (MMAS) on the stochastic single-destination shortest path (SDSP) problem. Two previous papers already analyzed this setting for two slightly different MMAS algorithms, where the pheromone update fitness-independently rewards edges of the best-so-far solution.The first paper showed that, when the best-so-far solution is not reevaluated and the stochastic nature of the edge weights is due to noise, the MMAS will find a tree of edges successfully and efficiently identify a shortest path tree with minimal noise-free weights. The second paper used reevaluation of the best-so-far solution and showed that the MMAS finds paths which beat any other path in direct comparisons, if existent. For both results, for some random variables, this corresponds to a tree with minimal expected weights.In this work we analyze a variant of MMAS that works with fitness-proportional update on stochastic-weight graphs with arbitrary random edge weights from [0, 1]. For δ such that any suboptimal path is worse by at least δ than an optimal path, then, with suitable parameters, the graph will be optimized after O n 3 ln (n/δ) δ 3 iterations (in expectation). In order to prove the above result, the multiplicative and the variable drift theorem are adapted to continuous search spaces.

Improved time complexity analysis of the Simple Genetic Algorithm

“…Therefore, the standard assumption of the variable drift theorem (see Rowe and Sudholt, 2012 for the most recent version) does not hold. We use the following variant (correcting a minor mistake in a formulation by Feldmann and Kötzing, 2013) instead. Its proof is given in Appendix B.…”

Robustness of populations in stochastic environments

“…In essence, it was shown that the (1+1) EA can deal with small noise levels, but not medium noise levels. Recently, there was a sequence of paper discussing ant colony optimization for path finding problems in the presence of uncertainty [ST12,DHK12,FK13], see also [GP96,Gut03] for early work in this area.…”

“…For example, one can add a value drawn from a centered normal distribution (or add a value drawn from any other chosen distribution; we call such noise additive posterior noise). Posterior noise is essentially the model used in [GP96,ST12,DHK12,FK13].…”

Runtime analysis of ant colony optimization on dynamic shortest path problems

“…We have chosen the single-destination shortest path problem (SDSP) as object of our analysis as this is probably the combinatorial optimization problem that ACO has been understood best on. There are even runtime analyses of ACO on stochastic optimization problems (Sudholt and Thyssen, 2012b;Doerr et al, 2012;Feldmann and Kötzing, 2013), which, together with dynamic problems, can be subsumed under the term "optimization under uncertainty". However, methods for the analysis of stochastic optimization problems are not directly applicable to dynamic problems.…”