A Metropolis - type Optimization Algorithm on theIn

Let S(v) be a function deened on the vertices v of the innnite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (X n) deened by P (X n+1 = wjX n = v) = 1 3 e b(S(w)?S(v)) 1 + e b(S(w)?S(v)) for each neighbor w of v, where b is a parameter (\1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is \tree-indexed simple random walk", that is the increments (e) = S(w)?S(v) along parent-child edges e = (v; w) are independent and P (= 1) = p; P (= ?1) = 1 ? p. This algorithm has a \speed" r(p; b) = lim n n ?1 ES(X n). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, the subject is in part analogous to recent work of Lyons-Pemantle-Peres (1995,1996) on the speed on random walk on Galton-Watson trees. A key feature of the model is existence of a critical point p crit below which the problem is infeasible; we study behavior of algorithms as p # p crit. Preliminary version. See homepage for updates.