A genetic algorithm based augmented Lagrangian method for constrained optimization

Among the penalty based approaches for constrained optimization, augmented Lagrangian (AL) methods are better in at least three ways: (i) they have theoretical convergence properties, (ii) they distort the original objective function minimally, thereby providing a better function landscape for search, and (iii) they can result in computing optimal Lagrange multiplier for each constraint as a by-product. Instead of keeping a constant penalty parameter throughout the optimization process, these algorithms update the parameters (called multipliers) adaptively so that the corresponding penalized function dynamically changes its optimum from the unconstrained minimum point to the constrained minimum point with iterations. However, the flip side of these algorithms is that the overall algorithm requires a serial application of a number of unconstrained optimization tasks, a process that is usually time-consuming and tend to be computationally expensive. In this paper, we devise a genetic algorithm based parameter update strategy to a particular AL method. The proposed strategy updates critical parameters in an adaptive manner based on population statistics. Occasionally, a classical optimization method is used to improve the GA-obtained solution, thereby providing the resulting hybrid procedure its theoretical convergence property. The GAAL method is applied to a number of constrained test problems taken from the evolutionary algorithms (EAs) literature. The number of function evaluations required by GAAL in most problems is found to be smaller than that needed by a number of existing evolutionary based constraint handling methods. GAAL method is found to be accurate, computationally fast, and reliable over multiple runs. Besides solving the problems, the proposed GAAL method is also able to find the optimal Lagrange multiplier associated with each constraint for the test problems as an added benefit—a matter that is important for a sensitivity analysis of the obtained optimized solution, but has not yet been paid adequate attention in the past evolutionary constrained optimization studies.