A Genetic Algorithm Based Augmented Lagrangian Method for Accurate , Fast and Reliable 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 a self-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 constrained 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 all test problems as an added benefit – a matter that is important for a sensitivity analysis of the obtained solution, but has not yet been paid adequate attention in the past evolutionary constrained optimization studies.

[1]  Irena Lasiecka,et al.  Sensitivity analysis of hyperbolic optimal control problems , 2012, Comput. Optim. Appl..

[2]  José Mario Martínez,et al.  Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization , 2012, Comput. Optim. Appl..

[3]  Samuel Amstutz,et al.  Augmented Lagrangian for cone constrained topology optimization , 2011, Comput. Optim. Appl..

[4]  Kalyanmoy Deb,et al.  A Genetic Algorithm Based Augmented Lagrangian Method for Computationally Fast Constrained Optimization , 2010, SEMCCO.

[5]  Kalyanmoy Deb,et al.  A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach , 2010, IEEE Congress on Evolutionary Computation.

[6]  Kalyanmoy Deb,et al.  Investigating EA solutions for approximate KKT conditions in smooth problems , 2010, GECCO '10.

[7]  Laura Palagi,et al.  A truncated Newton method in an augmented Lagrangian framework for nonlinear programming , 2010, Comput. Optim. Appl..

[8]  Angel Eduardo Muñoz Zavala,et al.  Continuous Constrained Optimization with Dynamic Tolerance Using the COPSO Algorithm , 2009 .

[9]  Janez Brest,et al.  Constrained Real-Parameter Optimization with ε -Self-Adaptive Differential Evolution , 2009 .

[10]  Tetsuyuki Takahama,et al.  Constrained Optimization by the ε Constrained Differential Evolution with Gradient-Based Mutation and Feasible Elites , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[11]  Jing J. Liang,et al.  Problem Deflnitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization , 2006 .

[12]  A. Ravindran,et al.  Engineering Optimization: Methods and Applications , 2006 .

[13]  Kalyanmoy Deb,et al.  Optimization for Engineering Design: Algorithms and Examples , 2004 .

[14]  Yuren Zhou,et al.  Multi-objective and MGG evolutionary algorithm for constrained optimization , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[15]  KalyanmoyDebandSamirAgrawal KanpurGeneticAlgorithmsLaboratory,et al.  A Niched-Penalty Approach for Constraint Handling in Genetic Algorithms , 2002 .

[16]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[17]  Stefan Volkwein,et al.  Application of the Augmented Lagrangian-SQP Method to Optimal Control Problems for the Stationary Burgers Equation , 2000, Comput. Optim. Appl..

[18]  C. Coello.  TREATING CONSTRAINTS AS OBJECTIVES FOR SINGLE-OBJECTIVE EVOLUTIONARY OPTIMIZATION , 2000 .

[19]  Zbigniew Michalewicz,et al.  Evolutionary Algorithms for Constrained Parameter Optimization Problems , 1996, Evolutionary Computation.

[20]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

[21]  Christopher R. Houck,et al.  On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA's , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[22]  Abdollah Homaifar,et al.  Constrained Optimization Via Genetic Algorithms , 1994, Simul..

[23]  Singiresu S Rao,et al.  Genetic algorithmic approach for multiobjective optimization of structures , 1993 .

[24]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1992, Artificial Intelligence.