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Advanced Fitness Landscape Analysis and the Performance of Memetic Algorithms

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Advanced Fitness Landscape Analysis and the Performance of Memetic Algorithms Peter Merz [email protected] University of Kaiserslautern, Department of Computer Science P.O. Box 3049, D-67653 Kaiserslautern, Germany Abstract Memetic algorithms (MAs) have demonstrated very effective in combinatorial opti- mization. This paper offers explanations as to why this is so by investigating the per- formance of MAs in terms of efficiency and effectiveness. A special class of MAs is used to discuss efficiency and effectiveness for local search and evolutionary meta- search. It is shown that the efficiency of MAs can be increased drastically with the use of domain knowledge. However, effectiveness highly depends on the structure of the problem. As is well-known, identifying this structure is made easier with the notion of fitness landscapes: the local properties of the fitness landscape strongly influence the effectiveness of the local search while the global properties strongly influence the effectiveness of the evolutionary meta-search. This paper also introduces new techniques for analyzing the fitness landscapes of com- binatorial problems; these techniques focus on the investigation of random walks in the fitness landscape starting at locally optimal solutions as well as on the escape from the basins of attractions of current local optima. It is shown for NK-landscapes and landscapes of the unconstrained binary quadratic programming problem (BQP) that a random walk to another local optimum can be used to explain the efficiency of recom- bination in comparison to mutation. Moreover, the paper shows that other aspects like the size of the basins of attractions of local optima are important for the efficiency of MAs and a local search escape analysis is proposed. These simple analysis techniques have several advantages over previously proposed statistical measures and provide valuable insight into the behaviour of MAs on different kinds of landscapes. Keywords: Memetic Algorithms, Fitness Landscapes, Combinatorial Optimization 1 Introduction Memetic algorithms (MAs) (Moscato, 1989; Moscato, 1999) have been shown to be very effective for many combinatorial optimization problems, including the traveling sales- man problem (TSP) (Merz and Freisleben, 2001), the quadratic assignment problem (QAP) (Merz and Freisleben, 2000a), the binary quadratic programming problem (BQP) (Merz and Katayama, 2002), graph bipartitioning (GBP) (Merz and Freisleben, 2000b), NK-landscapes (Merz, 2002), and minimum sum of squares clustering (MSSC) (Merz and Zell, 2002). In addition to memetic algorithms, many other modern heuristics have been proposed for combinatorial optimization problems. Most work has been con- cerned with showing that the proposed heuristic is effective and superior to others, but it is not shown why. To date, there is not much known about many meta-heuristics that have been developed in the operations research literature, and about the problems c 2004 by the Massachusetts Institute of Technology Evolutionary Computation 12(3): 303-325 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/1063656041774956 by guest on 25 September 2021 P. Merz on which these algorithms are tested. Fortunately, for evolutionary algorithms, there are well developed theoretical foundations. However, the theoretical analysis is lim- ited to relatively simple algorithms and problems. Moreover, fitness landscape analy- sis has been proposed for analyzing problems with respect to a family of algorithms. Nevertheless, the link between theory and landscape properties has not yet been well developed. In an attempt to find reasons why memetic algorithms perform well, both memetic algorithms and combinatorial optimization problems are investigated experimentally by utilizing fitness landscape analysis techniques to find properties of a problem (in- stance) a memetic algorithm can exploit. These techniques allow us to independently investigate the characteristics of the problem that have a high influence on the perfor- mance of the MAs two basic components, namely the local search and the evolution- ary meta-search. This paper introduces two new fitness landscape analysis techniques which are important for memetic algorithms. These new techniques are relevant for comparing recombination-based MAs with mutation based variants or with iterated local search (ILS) (Lourenco et al., 2003). The studies concentrate on a special class of memetic algorithms: hybrid evolutionary algorithms incorporating local search. In these algorithms, all individuals in the population represent local optima with respect to the local search used, and the evolutionary meta-search is performed by applying either a mutation or recombination operator to separate the effects of mutation and re- combination. Generally, it is an open question whether local search should be done until local optimality or not. However, opposed to the continuous case (Hart, 1994), it appears to be beneficial to do a complete local search for combinatorial optimization problems. Therefore, this work focuses on that case. In addition to previously proposed landscape analysis techniques, this paper pro- poses a local search escape analysis for finding effective escape strategies for evolution- ary variation operators. Moreover, we show how the effectiveness of mutation relative to recombination can be estimated by using a random walk analysis starting at local optima. The paper is organized as follows. In section 2, memetic algorithms, fitness land- scapes and two combinatorial problems are introduced: NK-landscapes and binary quadratic programming (BQP). The efficiency of local search and evolutionary meta- search is discussed in Section 3. In Section 4, well-known fitness landscape analysis techniques for investigating the effectiveness of memetic algorithms are reviewed. Sec- tion 5 presents two new fitness landscape analysis techniques that are especially im- portant for memetic algorithms: the local search escape analysis and random walks starting at local optima. Section 6 concludes the paper and outlines areas of future research. 2 Fitness Landscapes and Memetic Algorithms Evolutionary algorithms incorporating local search are usually referred to as memetic algorithms. A typical framework of a memetic algorithm differs from other evolution- ary algorithms in the way that after each creation or variation of a solution (individual) a local search is applied. Fig. 1 shows the pseudo code of such a memetic algorithm. In this framework, variation is performed by recombination or (macro-)mutation. Both operators are applied independently from each other in contrast to other evolution- 304 Evolutionary Computation Volume 12, Number 3 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/1063656041774956 by guest on 25 September 2021 Performance of Memetic Algorithms procedure MA; begin initialize population P ; foreach i P do i := Local-Search(i); repeat 2 for i := 1 to #recombinations do select two parents ia; ib P randomly; 2 ic := Recombine(ia; ib); ic := Local-Search(ic); add individual ic to P ; endfor; for i := 1 to #mutations do select parent ia P randomly; 2 ic := Mutate(ia); ic := Local-Search(ic); add individual ic to P ; endfor; P := select(P ); until terminate=true; end; Figure 1: The memetic algorithm ary algorithms. Furthermore, the role of recombination and mutation is different from variation in evolutionary algorithms without local search. For example, both operators should be designed to escape from the basins of attraction of the current local optima, since in this framework, all individuals represent local optima. The framework differs from others in that all individuals are local optima and mutation is completely inde- pendent of recombination. This allows us to study the effects of using recombination or mutation in the evolutionary search separately. Note that local search itself can be considered as a kind of mutation, but since it is done until local optimality, its role differs highly from mutation in other evolutionary algorithms. We consider that local search is very strong at exploitation, for example by using k-opt neighborhoods. Hence, the main purpose of the variation operators is exploration. Other memetic algorithms exist with spatially structured populations (Gorges- Schleuter, 1989), or tree-structured populations (Moscato and Tinetti, 1994). Further- more, there are variants using self-adaptation of the local search strategy (Krasnogor and Smith, 2000; Krasnogor and Smith, 2001). For simplicity, the studies presented here focus on simple memetic algorithms with a panmictic population and no self- adaptation. 2.1 Memetic Algorithm Performance The performance of a given heuristic is determined by its efficiency and its effective- ness. While the effectiveness of an algorithms refers to the quality of a solution found by the algorithm or its reliability in finding a desired solution, efficiency is usually char- acterized by its runtime behavior, i.e., the order of its computation time and its memory requirements depending on the problem size. Both average and worst-case runtime be- havior are of interest. However, these types of analysis are not always applicable; an Evolutionary Computation Volume 12, Number 3 305 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/1063656041774956 by guest on 25 September 2021 P. Merz example is a black-box optimization where there's not enough knowledge about the ob- jective function. Furthermore, many combinatorial optimization problems
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