Unconventional initialization methods for differential evolution

Abstract

The crucial role played by the initial population in a population-based heuristic optimization cannot be neglected. It not only affects the search for several iterations but often also has an influence on the final solution. If the initial population itself has some knowledge about the potential regions of the search domain then it is quite likely to accelerate the rate of convergence of the optimization algorithm. In the present study we propose two schemes for generating the initial population of differential evolution (DE) algorithm. These schemes are based on quadratic interpolation (QI) and nonlinear simplex method (NSM) in conjugation with computer generated random numbers. The idea is to construct a population that is biased towards the optimum solution right from the very beginning of the algorithm. The corresponding algorithms named as QIDE (using quadratic interpolation) and NSDE (using non linear simplex method), are tested on a set of 20 traditional benchmark problems with box constraints and 7 shifted (non-traditional) functions taken from literature. Comparison of numerical results with traditional DE and opposition based DE (ODE) show that the proposed schemes considered by us for generating the random numbers significantly improves the performance of DE in terms of convergence rate and average CPU time.

Introduction

DE is comparatively a recent addition to class of population based search heuristics. Nevertheless, it has emerged as one of the techniques most favored by engineers for solving continuous optimization problems. DE [1] has several attractive features. Besides being an exceptionally simple evolutionary strategy, it is significantly faster and robust for solving numerical optimization problems and is more likely to find the function’s true global optimum. Also, it is worth mentioning that DE has a compact structure with a small computer code and has fewer control parameters in comparison to other evolutionary algorithms. Originally Storn and Price proposed a single strategy for DE, which they later extended to ten different strategies [2].

DE has been successfully applied to a wide range of problems including optimization of process synthesis and design problems [3], application of DE in image processing [4], [5], [6], traveling salesman problem [7], multi class support vector machine [8], optimization of directional over-current relay settings [9], multi-objective optimization [10] etc.

Despite having several striking features and successful applications to various fields DE is sometimes criticized for its slow convergence rate for computationally expensive functions. By varying the control parameters the convergence rate of DE may be increased but it should be noted that it does not affect the quality of solution. Generally, in population based search techniques like DE an acceptable trade-off should be maintained between convergence and type of solution, which even if not a global optimal solution should be satisfactory rather than converging to a suboptimal solution which may not even be a local solution. Several attempts have been made in this direction to fortify DE with suitable mechanisms to improve its performance. Most of the studies involve the tuning or controlling of the parameters of algorithm and improving the mutation, crossover and selection mechanism, some interesting modifications that helped in enhancing the performance of DE include introduction of greedy random strategy for selection of mutant vector [11], modifications in mutation and localization in acceptance rule [12], DE with preferential crossover [13], crossover based local search method for DE [14], self adaptive differential evolution algorithm [15], new donor schemes proposed for the mutation operation of DE [16], DE with Cauchy mutation [17]. There are also work have done on parameter analysis [18] and hybridization [19], [20]. All the modified versions have shown that a slight change in the structure of DE can help in improving its performance. However, the role of the initial population, which is the topic of this paper, is widely ignored. The opening sentence of these algorithms is usually “generate an initial population” without indicating how this should be done. There is only few literature is available on this topic [21], [22], [23], [24]. An interesting method for generating the initial population was suggested by Rahnamayan et al. [25], [26] in which the initial population was generated using opposition based rule.

To further continue the research in this direction, in this paper we propose two schemes for generating the initial population of basic DE algorithm. These schemes are based on quadratic interpolation (QI) method and nonlinear simplex method in conjugation with random numbers. The corresponding modified DE versions are named (1) Quadratic interpolation method called QIDE and (2) non linear simplex method called NSDE. Both QI and NSM are well known local search methods. Their use provides additional information about the potential regions of the search domain.

In the present study our aim is to investigate the effect of initial population on payoff between convergence rate and solution quality. Our motivation is to encourage discussions on methods of initial population construction. Performances of the proposed algorithms are compared with Basic DE and differential evolution initialized by opposition based learning (ODE), which is a recently modified version of differential evolution [25], on a set of twenty unconstrained benchmark problems and 7 shifted functions.

Remaining of the paper is organized in following manner; in Section 2, we give a brief description of DE. In Section 3, we have given a brief description of the initialization schemes used in this paper. The proposed algorithms are explained in Section 4. Section 5 deals with experimental settings and parameter selection. Benchmark problems considered in the present study and the results are given in Section 6. The conclusions based on the present study are finally drawn in Section 7.