Global Optimization for Constrained Nonlinear Programming

TLDR: Constrained simulated annealing (CSA) is developed, a global optimization algorithm that asymptotically converges to constrained global minima (CGM) with probability one, for solving discrete constrained nonlinear programming problems (NLPs).

Abstract: In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary and sufficient condition for constrained local minima (CLM) in the theory of discrete constrained optimization using Lagrange multipliers developed in our group. The theory proves the equivalence between the set of discrete saddle points and the set of CLM, leading to the first-order necessary and sufficient condition for CLM. To find a CGM, CSA searches for a discrete saddle point with the minimum objective value by carrying out both probabilistic descents in the original-variable space of a discrete augmented Lagrangian function and probabilistic ascents in the Lagrange-multiplier space. We prove that CSA converges asymptotically to a CGM with probability one. We also extend CSA to solve continuous and mixed-integer constrained NLPs. By achieving asymptotic convergence, CSA represents one of the major developments in nonlinear constrained global optimization today, which complements simulated annealing (SA) in unconstrained global optimization. Based on CSA, we have studied various strategies of CSA and their trade-offs for solving continuous, discrete, and mixed-integer NLPs. The strategies evaluated include adaptive neighborhoods, distributions to control sampling, acceptance probabilities, and cooling schedules. An optimization software package based on CSA and its various strategies has been implemented. Finally, we apply CSA to solve a collection of engineering application benchmarks and design filters for subband image coding. Much better results have been reported in comparison with other existing methods.