(301s) An Effective Transformation for Enhancing Stochastic Global Optimizations

Stochastic global optimization techniques have been attracting greater attention and interest because of their ability to provide near global optimal solutions within stipulated times. Among the many, Differential Evolution (DE) (Storn and Price, 1997) is one of the promising stochastic global optimization methods, and has been successfully employed in many applications (Mayer et al., 2005; and Bingul, 2004) because of its simplicity, robustness and a few control variables. However, the performance of DE is affected at times due to the highly non-linear and non-convex nature of the objective functions (Moles et al., 2003). In this work, an effective transformation of the original objective function is proposed in order to facilitate escaping from a local minimum and thus improve the performance of DE.

The main features of the proposed transformation are: it preserves the relative ordering of local and global minima as in TRUST algorithm (Barhen et al., 1997), it transforms the current local minimum to a local maximum so that a global optimization method such as DE escapes from the current local minimum, and it is simple. The transformation involves two steps: transforming the objective function such that the function value becomes zero where the function value is greater than or equal to that of the current local minimum, and building a concave envelope such that the algorithm escapes from the local minimum. The transformation proposed in this work is different from that in the literature (Parsopoulos and Vrahatis, 2004) since it does not require tuning of parameters and also solving a system of differential equations as in TRUST algorithm.

In general, DE works over a set of individuals known as population and explores the globally optimal regions with its mutation and crossover mechanisms. Once the optimal region is identified, DE needs several functional evaluations to obtain the refined solution compared to a local optimization technique. It is also difficult to asses whether the algorithm (DE) explored any optimal (either local or global) region or not. In this study, a measure namely maximum distance criterion (Zielinski et al., 2005) is used to predict the convergence of the algorithm to an optimal region, and also to start the local optimization. This criterion calculates the maximum distance from every individual to the best individual in the current population, and indicates the convergence of DE if it lies below a threshold value. In this study, a local optimization technique (fast convergent quasi-Newton method) is used at the end of DE to enhance its computational efficiency.

The proposed transformation will be studied in different ways along with the local optimization to enhance the performance of DE. Though the transformation is studied for DE in this work, it is equally applicable to other stochastic methods such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Initially, the transformation will be studied over a set of benchmark problems involving 2 to 50 variables and a few to thousands of local minima. The enhanced DE is then tested for challenging phase equilibrium calculations (Teh and Rangaiah, 2003) involving multiple components and popular thermodynamic models. The results of the study will be presented and discussed in detail.

References Barhen, J., Protopopescu, V. and Reister, D. TRUST: A deterministic algorithm for global optimization. Science, 276, pp.1094-1097. 1997. Bingul, Z. A new PID tuning technique using differential evolution for unstable and integrating processes with time delay, ICONIP, Proceedings of ICONIP, Lecture Notes in Computer Science, 3316, pp.254-260. 2004. Mayer, D. G., Kinghorn, B. P. and Archer, A. A. Differential evolution ? an easy and efficient evolutionary algorithm for model optimization. Agricultural Systems, 83, pp.315-328. 2005. Moles, C. G., Mendes, P. and Banga, J. R. Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Research, 13, pp.2467-2474. 2003. Parsopoulos, K. E. and Vrahatis, M. N. On the computation of all global minimizers through particle swarm optimization. IEEE Transactions on Evolutionary Computatioin, 8 (3), pp.211-224. 2004. Storn, R. and Price, K. Differential evolution ? A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11, pp.341-359. 1997. Teh, Y. S., and Rangaiah, G. P. Tabu search for global optimization of continuous functions with application to phase equilibrium calculations, Computers and Chemical Engineering, 27, pp.1665-1679. 2003. Zielinski, K., Peters, D. and Laur, R. Stopping Criteria for Single-Objective Optimization, Proceedings of the Third International Conference on Computational Intelligence, Robotics and Autonomous Systems, Singapore. 2005.