Free Energy Rates for a Class of Very Noisy Optimization Problems

We study a class of stochastic optimization problems for which the cardinality of the set of feasible solutions (called also configurations) m and the size of every feasible solution N satisfy logm = o(N). Assuming the data to be random, e.g., weights of a graph, edges in spanning tree problem, elements of matrices in assignment problem, etc. fluctuate due to measurements, we adopt the maximum entropy framework by weighting solutions with a Boltzmann distribution where the inverse computational temperature β controls the cost resolution of the solution space. Large fluctuations of the costs due to high input randomness correspond to a low cost resolution β. For such a high noise level in the instances implying low β, we estimate the free energy in the asymptotic limit. This quantity plays a significant role in many applications, including algorithm analysis, robust optimization and so on. In particular, we prove that the free energy exhibits a phase transition in the second order term.