Combinatorial Optimization, Cross-Entropy, Ants and Rare Events

We show how to solve network combinatorial optimization problems using a randomized algorithm based on the cross-entropy method. The proposed algorithm employs an auxiliary random mechanism, like a Markov chain, which converts the original deterministic network into an associated stochastic one, called the associated stochastic network (ASN). Depending on a particular problem, we introduce the randomness in ASN by making either the nodes or the edges of the network random. Each iteration of the randomized algorithm based on the ASN involves the following two phases: 1. Generation of trajectories using the random mechanism and calculation of the associated path (objective functions) and some related quantities, such as rare-event probabilities. 2. Updating the parameters associated with the random mechanism, like the probability matrix P of the Markov chain, on the basis of the data collected at first phase. We show that asymptotically the matrix P converges to a degenerated one P* d in the sense that at each row of the MC P* d only a single element equals unity, while the remaining elements in each row are zeros. Moreover, the unity elements of each row uniquely define the optimal solution. We also show numericaly that for a finite sample the algorithm converges with very high probability to a very small subset of the optimal values. We finally show that the proposed method can also be used for noisy networks, namely where the deterministic edge distances in the network are replaced by random variables with unknown expected values. Supporting numerical results are given as well. Our numerical studies suggest that the proposed algorithm typically has polynomial complexity in the size of the network.