Simulated Annealing with Asymptotic Convergence for Nonlinear Constrained Global Optimization

Abstract

In this paper, we present constrained simulated annealing (CSA), a global minimization algorithm that converges to constrained global minima with probability one, for solving nonlinear discrete non-convex constrained minimization problems. The algorithm is based on the necessary and sufficient condition for constrained local minima in the theory of discrete Lagrange multipliers we developed earlier. The condition states that the set of discrete saddle points is the same as the set of constrained local minima when all constraint functions are non-negative. To find the discrete saddle point with the minimum objective value, we model the search by a finite inhomogeneous Markov chain that carries out (in an annealing fashion) both probabilistic descents of the discrete Lagrangian function in the original-variable space and probabilistic ascents in the Lagrange-multiplier space. We then prove the asymptotic convergence of the algorithm to constrained global minima with probability one. Finally, we extend CSA to solve nonlinear constrained problems with continuous variables and those with mixed (both discrete and continuous) variables. Our results on a set of nonlinear benchmarks are much better than those reported by others. By achieving asymptotic convergence, CSA is one of the major developments in nonlinear constrained global optimization today.