Global Optimization, the Gaussian Ensemble, and Universal Ensemble Equivalence.Mathematics and Statistics Department Faculty Publication Series
AbstractGiven a constrained minimization problem, under what conditions does there exist a related, unconstrainedproblem having the same minimum points? This basic question in global optimizationmotivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, itreduces to the fundamental question of the equivalence and nonequivalence of ensembles, which isanalyzed using the theory of large deviations and the theory of convex functions.In a 2000 paper appearing in the Journal of Statistical Physics, we gave necessary and sufficientconditions for ensemble equivalence and nonequivalence in terms of support and concavityproperties of the microcanonical entropy. In later research we significantly extended those resultsby introducing a class of Gaussian ensembles, which are obtained from the canonical ensemble byadding an exponential factor involving a quadratic function of the Hamiltonian. The present paperis an overview of our work on this topic. Our most important discovery is that even when the microcanonicaland canonical ensembles are not equivalent, one can often find a Gaussian ensemblethat satisfies a strong form of equivalence with the microcanonical ensemble known as universalequivalence. When translated back into optimization theory, this implies that an unconstrained minimizationproblem involving a Lagrange multiplier and a quadratic penalty function has the sameminimum points as the original constrained problem.The results on ensemble equivalence discussed in this paper are illustrated in the context of theCurie-Weiss-Potts lattice-spin model.
Citation InformationM Costeniuc, RS Ellis, H Touchette and B Turkington. "Global Optimization, the Gaussian Ensemble, and Universal Ensemble Equivalence." (2006)
Available at: http://works.bepress.com/richard_ellis/11/