Engineering Optimization ISSN: 0305-215X (Print) 1029-0273 (Online) Journal homepage: https://www.tandfonline.com/loi/geno20 RePAMO: Recursive Perturbation Approach for Multimodal Optimization Bhaskar Dasgupta , Kotha Divya , Vivek Kumar Mehta & Kalyanmoy Deb To cite this article: Bhaskar Dasgupta , Kotha Divya , Vivek Kumar Mehta & Kalyanmoy Deb (2013) RePAMO: Recursive Perturbation Approach for Multimodal Optimization, Engineering Optimization, 45:9, 1073-1090, DOI: 10.1080/0305215X.2012.725050 To link to this article: https://doi.org/10.1080/0305215X.2012.725050 Published online: 28 Nov 2012. Submit your article to this journal Article views: 150 Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=geno20 Engineering Optimization, 2013 Vol. 45, No. 9, 1073–1090, http://dx.doi.org/10.1080/0305215X.2012.725050 RePAMO: Recursive Perturbation Approach for Multimodal Optimization Bhaskar Dasguptaa*, Kotha Divyab, Vivek Kumar Mehtaa and Kalyanmoy Deba aDepartment of Mechanical Engineering, IIT, Kanpur—208 016, India; bISRO Satellite Centre, Bangalore, India (Received 5 October 2011; final version received 2 August 2012) In this article, a strategy is presented to exploit classical algorithms for multimodal optimization problems, which recursively applies any suitable local optimization method, in the present case Nelder and Mead’s simplex search method, in the search domain. The proposed method follows a systematic way to restart the algorithm. The idea of climbing the hills and sliding down to the neighbouring valleys is utilized. The implementation of the algorithm finds local minima as well as maxima. The concept of perturbing the minimum/maximum in several directions and restarting the algorithm for maxima/minima is introduced. The method performs favourably in comparison to other global optimization methods. The results of this algorithm, named RePAMO, are compared with the GA–clearing and ASMAGO techniques in terms of the number of function evaluations. Based on the results, it has been found that the RePAMO outperforms GA clearing and ASMAGO by a significant margin. Keywords: multimodal optimization; recursive optimization; alternate minimization and maximization; RePAMO algorithm 1. Introduction Suggesting multiple solutions to optimization problems that are multimodal in nature has the benefit of providing flexibility of choice while making decisions. It also leads to a better under- standing of the problem which is very useful in a wide range of applications such as decision making, designing tasks, motion planning, scheduling, etc. Optimization problems often involve human judgements that are not quantifiable. In such problems, an optimization algorithm needs to propose possible alternative solutions that can later be judged by a human decision-maker. There have been two approaches in the development of global optimization algorithms: deter- ministic and stochastic. Deterministic approaches can guarantee absolute success, but only by making restrictive assumptions on the objective function. Stochastic approaches, on the other hand, use evaluation of the objective function values at randomly generated points by making milder assumptions. The convergence is, however, not absolute. But, the probability of their success approaches unity as the sample size tends to infinity. The present work attempts to imbibe the determinism of classical approaches while retaining the diversity of a population-based approach. *Corresponding author. Email: [email protected] © 2013 Taylor & Francis 1074 B. Dasgupta et al. Among deterministic (classical) methods, different approaches have been proposed by researchers, like deflation techniques for the calculation of further solutions of a nonlinear system by Brown and Gearhart (1971) and, more recently, in interval analysis based methods (Hansen 1993, Hansen and Walster 2004). Stochastic approaches promise the global optimum, but they are usually heuristic in nature and expensive in application. In some approaches, gradient-based methods are coupled with certain auxiliary functions to move successfully from one local minimum to a better one. The algorithms developed based on this idea are the tunnelling method of Levy and Montalvo (1985), the bridging method of Liu and Teo (1999), and the filled function method of Liu (2001), which bypass the previously generated local minima. The success of these algorithms relies heavily on the effective construction of suitable auxiliary functions. Yiu et al. (2004) proposed a hybrid descent method, consisting of simulated annealing and a gradient-based method, to retain the robustness of the stochastic optimization method and the speed of the local minimizing algorithm to find the global solution. Shashikala (1992) proposed an ingenious heuristic to find all the extrema of a given function. The problem of determining the critical points of a function on a manifold is transformed to the problem of finding all the equilibrium points of an appropriate vector field. The solution to the vector field is evaluated through a numerical approach wherein this algorithm (Branin 1972, Shashikala 1992, Shashikala et al. 1992, Sudarsan and Sathiya Keerthi 1998) used the constrained stabilization approach and Adam’s formulas for integration. Another recent method, called A Simple Multistart Algorithm for Global Optimization (ASMAGO) was proposed by Hickernell and Yuan (1997). This algorithm starts with a quasir- andom sample of N points representing the feasible set. The algorithm applies p iterations of an inexpensive local search to concentrate on the sample set and retains q points with the smallest function values. The remaining (N − q) points are replaced by new quasirandom points and then the concentration step is repeated. Any point that is retained for s iterations of concentration steps is used to start an efficient local search, provided that its function value is not significantly larger than the smallest function value obtained thus far. The algorithm terminates when no new local minimum is found after several ‘major iterations’. The strength of this algorithm lies in choosing quasirandom rather than random sample points, thus targetting better coverage of the feasible region. But the algorithm fails to locate the global optima in the case of highly oscillating functions. Various approaches in the history of evolutionary algorithms, which facilitate the determination of multiple solutions, can be divided into three subgroups even though all the techniques fall under the major group called niching techniques: (i) the sharing function approach, (ii) sequential niching and (iii) crowding. In multimodal GAs, instead of assigning the usual fitness to all individuals in a population, the population is divided into species and each individual in a specie is forced to share the fitness of all the other individuals in its specie. The incorporation of forced sharing of resources discourages crowding of the population at a particular optimal point and causes the formation of stable sub- populations (species) at different optima (niches). Furthermore, the number of individuals devoted to each niche is proportional to the expected niche payoff. A practical scheme that directly uses the sharing metaphor to induce niching and speciation was detailed in Goldberg and Richardson (1987), Deb (1989) and Deb and Goldberg (1989). In this scheme, a sharing function is defined to determine the neighbourhood and degree of sharing for each string in the population. It has been reported that the algorithm fails to establish the niches in massively multimodal problems (Goldberg et al. 1992) and niches with required peaks of relative fitness worse than the global optimum. Scaling the fitness function has been proposed as a remedy to overcome the problem. Although other fitness scaling methods, such as linear scaling, exponential scaling, sigma truncation scaling, etc., exist, power law scaling has been Engineering Optimization 1075 adjudged the best in a study by Kreinovich et al. (1993). Modifications based on niche radius were proposed by Yu (2005) where the niche radius is encoded in the chromosome itself which was the major problem in setting this parameter in the previous case. In the above algorithms, calculation of the sharing fitness value requires computation of the order O(N2), hence various algorithms have been proposed with the aim of reducing the com- plexity. Some such algorithms are Monte Carlo sampling (Goldberg and Richardson 1987), k-mean clustering analysis (Yin and Germay 1993), and k-mean clustering analysis with collec- tive sharing (Pictet et al. 1996). Miller and Shaw (1996) used a technique called dynamic niche sharing, which dynamically identifies the niches based on a greedy algorithm. Oei et al. (1991) chose tournament selection as compared to other selection methods to reduce the computational complexity. Gan and Warwick (1999) proposed a niching scheme called dynamic niche clustering-1(DNC- I) that used a clustering algorithm for the identification of peaks followed by a sharing function. DNC-I was originally proposed for one-dimensional space, and later extended to DNC-II (Gan and Warwick 2000), in which the authors introduced a new triangular sharing function and the use of hill–valley function topology. It has been reported that DNC-II requires an increased number of fitness evaluation calculations, which significantly add to the computational time as compared to the other simplistic approaches. Hanagandi