Jih-Yiing Lin and Ying-Ping Chen, Member, IEEE
Abstract—The synergy between exploration and exploitation has been a prominent issue in optimization. The rise of memetic algorithms, a category of optimization techniques which fea-ture the explicit exploration-exploitation coordination, much accentuates this issue. While memetic algorithms have achieved remarkable success in a wide range of real-world applications, the key to successful exploration-exploitation synergies still re-mains obscure as conclusions drawn from empirical results or theoretical derivations are usually quite algorithm specific and/or problem dependent. This paper aims to provide a theoretical model that can depict the collaboration between global search and local search in memetic computation on a broad class of objective functions. In the proposed model, the interaction between global search and local search creates a set of local search zones, in which the global optimal points reside, within the search space.
Based on such a concept, the quasi-basin class (QBC) which categorizes problems according to the distribution of their local search zones is adopted. The subthreshold seeker, taken as a representative archetype of memetic algorithms, is analyzed on various QBCs to develop a general model for memetic algorithms.
As the proposed model not only well describes the expected time for a simple memetic algorithm to find the optimal point on different QBCs but also consists with the observations made in previous studies in the literature, the proposed model may reveal important insights to the design of memetic algorithms in general.
Index Terms—Global search, local search, memetic algorithms, quasi-basin class, subthreshold seeker.
I. Introduction
O
PTIMIZATION, finding the optimal element among a set of feasible ones, is a type of problem commonly encountered in many fields. Many real-world and theoret-ical problems can be formulated as optimization problems and solved by applying or developing various optimization techniques. Early optimization techniques, such as Newton’s method, simplex method, conjugate gradient algorithm, and the like, have been well developed on problems with certain mathematical characteristics. However, as many real-world optimization problems are black-box problems of which a priori problem knowledge is not available, the use of meta-heuristics started to prevail. Meta-meta-heuristics are generally population-based algorithms which explore the search spaceManuscript received April 30, 2010; revised September 29, 2010 and March 11, 2011; April 19, 2011. Date of current version September 30, 2011. This work was supported in part by the National Science Council of Taiwan, under Grant NSC 99-2221-E-009-123-MY2.
The authors are with the Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: jylin@nclab.tw;
ypchen@nclab.tw).
Digital Object Identifier 10.1109/TEVC.2011.2150754
stochastically according to some heuristics. As they are not problem-specific, they have a good chance to perform well on black-box optimization problems. Evolutionary algorithms, particle swarm optimizations, ant colony algorithms, and the like, are some of the renowned meta-heuristics which have been widely adopted.
The generality of meta-heuristics which provides the wide applicability also limits the efficiency of meta-heuristics.
When complicated problems are encountered, without taking advantages of problem-specific information given a priori or retrieved during optimization, meta-heuristics can merely deliver mediocre performance. As problem-specific heuristics can generally take advantages of problem-specific information, techniques that hybrid general meta-heuristics and problem-specific heuristics have been developed to provide more effi-cient optimization techniques for more complicated problems.
These techniques which employ general meta-heuristics as global search and problem-specific heuristic as local search are commonly referred to as memetic algorithms (MAs).
With an appropriate coordination, memetic algorithms cannot only exhibit a good explorative ability as a population-based global search algorithm does but also deliver a good exploitive performance as a local search algorithm does. As a result, memetic algorithms perform better than pure population-based global search algorithms or stand-alone local search algorithms. As the research interests and activities of memetic algorithms thrive, memetic computing has been evolved from hybridization of global search and local search to hybridization with adaptation and has the potential to be applied to com-putational intelligence [1]. Manifold of successful memetic algorithms in various application domains, ranging from NP-hard combinatorial problems to non-linear programming prob-lems, have been reported [2]. Besides the various application domains mentioned in [2], recent memetic algorithm applica-tions in Cartesian robot control [3], e-learning systems [4], image segmentation [5], feature selection [6], mission man-agement [7], and portfolio selection [8] also demonstrate the efficacy of memetic algorithms in different application domains.
Among these memetic algorithms, in addition to the se-lection of the global search component and the local search operator, the synergy between global search and local search has always been one of the key design issues. The design of most memetic algorithms follows the seminal studies on memetic algorithms proposed in [9] and [10]. In these studies, the authors observed that memetic algorithms favor infrequent
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LIN AND CHEN: ANALYSIS ON THE COLLABORATION BETWEEN GLOBAL SEARCH AND LOCAL SEARCH IN MEMETIC COMPUTATION 609
starts and long running time of local search. They also proposed several renowned strategies for selecting solution candidates on which the local search operator is applied: the fitness based selection and the diversity based selection. How-ever, with the aids of these guidelines, designing a memetic algorithm for a specific problem still requires considerable time as the optimal design is not only algorithm specific but also problem dependent. To cope with this issue, the concept of systematically adjusting the parameters of local search is proposed [11]. Although this technique is robust, it does not guarantee the best performance. Another line of research is regarding the concept of memes [12]–[14]. In these studies, the local search algorithms, encoded as memes, can adapt to the underlying problem and thus improve the efficiency as the memetic algorithm progresses. This framework is robust as well as efficient with the expense of the learning cost of memes.
In spite of the light shed on the design issue of memetic algorithms by the aforementioned studies, the question of how one can achieve the optimal design of memetic algorithms on a specific problem remains. The key to achieve this ultimate goal apparently include a full awareness of the physics behind the algorithm and the problem. As theoretical studies can help to understand the internal mechanism of algorithms, they can provide important insights to the design issue. Compared to the progress of theoretical studies on evolutionary computation, which is still in its infancy [15]–[23], theoretical studies of memetic algorithms are even scarce. Recent studies [24], [25]
investigated the behavior of simple memetic algorithms on sev-eral classes of functions. The proposed theoretical models on the demonstrative classes of functions reaffirmed that param-eterizing memetic evolutionary algorithms can be extremely difficult. As these theoretical models are developed according to different classes of functions, they are capable of depicting the algorithmic behavior from their respective perspectives on the adopted classes of functions instead of providing a unified principle for the design of memetic algorithms.
The concept of basins of attraction [26] provides another perspective and gives an opportunity to conduct general anal-ysis on memetic algorithms. In [27] and [28], the search space is viewed as a union of basins of attraction, and the optimal allowable local search length of simple memetic algorithms is theoretically estimated. A similar concept, quasi-basins defined by the subthreshold seeker, was adopted to prove the searchability of general functions [29] and to investigate the subthreshold seeking behavior [30].
In this paper, we aim to establish a theoretical model that can depict the collaboration between global search and local search in memetic computation on a wide range of problems.
To achieve this, we propose the concept of local search zones which are the regions that local search exploits. In this per-spective, these local search zones are defined by the landscape of the problem as well as the collaboration between global search and local search. As local search zones are generally not easy to assess, we adopt quasi-basins to estimate local search zones and define the quasi-basin class (QBC) which categorizes problems by their quasi-basin distributions as the basis on which memetic algorithms are investigated. Then, we
analyze the performance of the subthreshold seeker, which is regarded as a representative archetype of memetic algorithms, to develop a theoretical model for the global-local search collaboration in memetic computation. The derived theoretical model can describe how the distribution of local search zones and the efficiency of the global search algorithm and the local search algorithm are related to the expected time for a memetic algorithm to find the optimal solution. Because this model, empirically verified, is consistent with the observations made in many previous studies in the literature, it may be considered valid for representing various memetic algorithms on a wide range of problems and may give important insights to the future design of advanced memetic algorithms.
The rest of this paper is arranged in the following manner.
Section II gives a survey on the current progress of analysis on memetic algorithms and elaborates the need of a general the-oretical model which can describe the collaboration between global search and local search in memetic computation on a broad range of problems. Section III expounds the fundamental concepts on the analysis of memetic algorithms and provides the definitions of our framework to form the basis for further derivation. As a memetic algorithm comprises global search and local search, we first analyze the global search component of the subthreshold seeker and discuss how this analysis is related to the behavior of common global search algorithms in Section IV. Based on the analysis of global search and the concept of QBC, we derive and empirically verify the formula that describes the behavior of the subthreshold seeker working with local search operators of different efficiency on various QBCs in Section V. After the empirically verifying the proposed model, we expound how our model can repre-sent the general behavior of memetic algorithms and discuss possible extensions and future work of the proposed model in Section VI. Finally, we recap the significance of our model and conclude this paper in Section VII.
II. Background
Designing a memetic algorithm requires not only selecting a global search mechanism as well as a local search operators but also establishing a subtle coordination to exhibit the van-tage of both ends. Hart [9] in his seminal study for designing efficient memetic algorithms investigated the following four questions on continuous optimization problems.
1) How often should local search be applied?
2) On which solutions should local search be used?
3) How long should local search be run?
4) How efficient does local search need to be?
In his framework, he noted that the memetic algorithms that employ elitism will be most efficient with large population sizes and infrequent local search. He also proposed two strategies, fitness based selection and diversity based selection, for selecting solution candidates to apply local search. He concluded that these two strategies help much. Land [10]
extended Hart’s study to combinatorial domains. In his study, he adopted steady state genetic algorithms as global search and proposed a local search potential based strategy in se-lecting local search candidates. The local search potential
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strategy turned out to be not very useful. Yet, he observed that his steady state memetic algorithm favored smaller rates and longer runtime for local search, consistent with Hart’s study.
Although limited to specific problems, the studies of Hart and Land gave some insights to the first three questions and have inspired the successive memetic algorithms in a wide variety of applications. The concepts of selecting the best or some qualified individuals for local search which resemble the fitness based selection have been adopted in [31]–[33].
The steady state memetic algorithm with adaptive local search has been applied in [34]–[36], while other studies exhibit the vantage of utilizing the diversity information in their design of memetic algorithms [37], [38]. Investigations into the balance between global search and local search for some applications available in the literature also accord with Hart’s and Land’s observations [39], [40]. Despite that the accordance of these results reveals some essential design principles of efficient memetic algorithms, designing a memetic algorithm still requires a considerable amount of effort due to the lack of detailed knowledge on how the key mechanism of memetic al-gorithms, the synergy between global search and local search, working on the underlying problem. An interesting technique of adapting local search intensity in a simulated annealing way was proposed to cope with the MA parameterizing issue [11].
More robust than the fixed local search intensity setting, this method still requires a range setting and does not guarantee the best performance.
In addition to the parameterizing issue caused by using memetic algorithms to handle different problems, the effi-ciency of a local search operator is particularly problem dependent. [41] provided a landscape analysis for memetic al-gorithms. Following this, the concept of memes [12]–[14] was proposed. In these frameworks, the local search component is designed to adapt to the underlying problem as the optimiza-tion progresses. These meme evolving or learning memetic algorithms are robust regardless of the underlying problem and efficient. Recent studies [42], [43] have also proposed several metrics to assess the improvement of applying a local search algorithm on a problem.
Furthermore, theoretical analysis has always been a preva-lent way to provide clues to the design of algorithms. For continuous problems, convergence analysis is widely adopted in performance assessment for evolutionary computation [16], [19], [22]. For discrete problems, the (1+1) evolutionary algorithm (EA) has been widely adopted in theoretical anal-ysis on evolutionary algorithms [15], [17], [18], [20], [21], [23]. The (1+1)-EA is a rather simple algorithm with one individual and an evolutionary operator flipping each bit of the individual with a uniform probability. Following these studies, the theoretical analysis of memetic algorithms starts from the (1+1)-MA and goes to the (μ+λ)-MA [24], [25]. On three discrete functions, Sudholt investigated the behavior of the (1+1)-MA and the (μ+λ)-MA, and these studies reaffirmed the parameterizing of memetic algorithms is extremely hard.
Theoretical models developed in this way are capable of providing different perspectives, according to the adopted classes of functions, to analyze a memetic algorithm.
Ref-erence [44] illustrated that different problems favor different population sizes, while [45] and [46], which investigated the effect of recombination operators, provided counter per-spectives. The issue of such an analysis technique is that the derived theoretical behavior is naturally confined and largely determined by the adopted objective functions. As an undesirable result, the different conclusions obtained from various theoretical models cannot form a unified guideline to the design of algorithms.
Another line of analysis involves the concept of basins of attraction. The basin of attraction of a local optimum is the set of points in the search space such that a local search process starts from any member within a basin will eventually find the local optimum in that basin [26]. In this line of research, the search space is a union of basins of attraction. References [27] and [28] adopted this concept to estimate the optimal local search length. In these papers, basins of attraction in the search space are categorized into two types, in which target solutions can or cannot be reached. The optimal local search length is estimated via acquiring the probability of hitting the former basins.
A closely related concept, quasi-basin defined by subthresh-old seeker, was introduced by [29] in investigating searchable functions in which the No Free Lunch theorem does not hold.
The submedian seeker which starts local search when hitting a point with a submedian value and turns to do random search when hitting a point with a supermedian value was considered.
By applying the submedian seeker to functions with a certain degree of similarity, that the functions exhibiting self-similarity are searchable was proved. Whiteley and Rowe further proposed the subthreshold seeker, a generalized 1-D submedian seeker, and investigated its seeking behavior [30].
In their work, the subthreshold seeking behavior, the ratio of the sampled subthreshold points to superthreshold points, was used as a performance index. Their theoretical analysis detailed the conditions under which the subthreshold seeker could outperform random search and showed that a higher bit-precision could improve the performance.
Finally, in this paper, we aim to provide a general model for the collaboration between global search and local search in memetic algorithms on a broad class of problems. The proposed model will describe how the expected performance of a memetic algorithm is related to the efficiency of the local search operator, the landscape of the problem, and the collaboration between global search and local search. In order to achieve this goal, we propose the concept of local search zones. Local search zones are the regions which local search prefers and the global optimal point resides in. As generally local search zones are not easy to assess, we adopt the idea of quasi-basins to estimate local search zones and define the QBC to categorize problems according to their quasi-basin distributions. The subthreshold seeker, taken as a rep-resentative archetype of memetic algorithms, is analyzed over different QBCs as a general theoretical model for memetic algorithms. Thus, the proposed model can depict the essence of the collaboration between global search and local search in memetic algorithms on various problems and may shed light on the design of memetic algorithms.
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III. Quasi-Basin Classes and Subthreshold Seeker In this section, we introduce the concept of local search zones and give definitions to the fundamental terminologies of our framework. The concept of the local search zones is described based on the formal definitions of the search process of an algorithm on a problem and the search space viewed by a search process. Then, based on the concept of local search zones, we introduce the QBC and the generalized subthreshold seeker on which the theoretical analysis is based.
A. Local Search Zones
The task to handle an optimization problem is to optimize a given objective function f :X → Y. For convenience, we specify our optimization goal as to find a point x∗ ∈ X with the minimum value y∗∈ Y. We assume that both X and Y are finite sets. Such an assumption makes a practical sense because optimization problems are generally numerically solved on digital computers. In this paper, for simplifying the derivation, we also assume that every function maps different x ∈ X to different y ∈ Y. In order to formally describe a search process of an algorithm on a function, we adopt part of the terminologies defined in [47] as the following definitions.
Definition 1 (Search Process): Given two finite setsX and Y:
1) A trace of length m is a sequence Tm := ((xi, yi))m1 = ((x1, y1), (x2, y2), . . . , (xm, ym))∈ (X × Y)m with dis-tinct xi. “x ∈ Tm” denotes that x = xi for some i∈ {1, 2, . . . , m}. Let T0be the empty sequence andT be the set containing all the traces of a length smaller than or equal to .
2) Let AT, where T ∈ T|X |−1, be a random variable over X satisfying that Prob{AT = x} = 0 for all x ∈ T . An algorithm A is a collection of such random variables,
2) Let AT, where T ∈ T|X |−1, be a random variable over X satisfying that Prob{AT = x} = 0 for all x ∈ T . An algorithm A is a collection of such random variables,