Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems

Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems

Abstract

This study presents a hybrid harmony search algorithm (HHSA) to solve engineering optimization problems with continuous design variables. Although the harmony search algorithm (HSA) has proven its ability of finding near global regions within a reasonable time, it is comparatively inefficient in performing local search. In this study sequential quadratic programming (SQP) is employed to speed up local search and improve precision of the HSA solutions. Moreover, an empirical study is performed in order to determine the impact of various parameters of the HSA on convergence behavior. Various benchmark engineering optimization problems are used to illustrate the effectiveness and robustness of the proposed algorithm. Numerical results reveal that the proposed hybrid algorithm, in most cases is more effective than the HSA and other meta-heuristic or deterministic methods.

Introduction

Evolutionary algorithms (EAs) are efficient at exploring entire search space; however, they are relatively poor at finding the precise optimum solution in the region to which the algorithm converges. Many researchers [1–3] have shown that EAs perform well for global searching due to their capability of quickly exploring and finding promising regions in the search space, but they take a relatively long time to converge to a local optimum.

On the other hand, gradient based algorithms are very effective deterministic methods in finding a stationary point near the initial starting point [1]. There exist many efficient gradient descent methods for finding local minima of a function, e.g. the steepest descent method, the Newtonmethod and the quasi Newton methods. In general, gradi-ent based algorithms converge faster and they can obtain solutions with higher accuracy compared to stochastic approaches in fulfilling the local search task. They have been used widely in a large class of problems, especially as a vial component of sophisticated algorithms. However,
these approaches often rely heavily on the initial starting point, the topology of the feasible region and the surface associated with the objective functions. A good starting point is vital for these methods to be executed successfully.

To obtain a more robust optimization technique, it is common to combine different search strategies trying to compensate deficiencies of the individual algorithms. Dur- ing the last few years, new techniques have been developed in order to improve the lack of accuracy of the EAs, using local optimization algorithms. These techniques are based on combination of local optimization procedures, which are good at finding local optima (local exploiter), and glo- bal search methods (global explorer). These are commonly known as hybrid algorithms and have been successfully used to solve a wide variety of problems [1,4] and experi-mental studies. The results show that hybrid methods search more efficiently and often find better solutions [5– 8]. A more detailed and comparative study of these hybrid methods is given at references [9,10].

To improve the efficiency of the HSA, in our previous study [11], we used dynamic parameter adjusting in impro-visation step (defined in Section 2.1.3) and discussed its impacts. The results of our work showed great improve-ment in convergence rate and quality of the HSA solutions. In this study we improve the efficiency of the HSA by incorporation of local search methods. This can be consid-ered as an innovative type of hybridization of the HSA that has not yet been explored in the literature. In particular, we investigate the possibility of the use of SQP as a local opti- mizer. Moreover, different combination strategies in hybridization which are important in computational effi- ciency of the algorithm are discussed.

In this study, first, we present a brief overview of the HSA. Since there are not any precise recommendations for tuning the HSA parameters in the literature, an empir-ical study to determine the impact of different parameters of the algorithm on the solution quality and convergence behavior is performed. Finally the proposed hybrid method is described for engineering optimization problems with continuous design variables. Various standard bench- mark engineering optimization examples including func-tion minimization problems and structural optimization problems from the literature are also presented to show the effectiveness of the HHSA.

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