Multi-objectivization, fitness landscape transformation and search performance: A case of study on the hp model for protein structure prediction

https://doi.org/10.1016/j.ejor.2014.06.009Get rights and content

Highlights

  • Multi-objectivization is investigated from a fitness landscape analysis perspective.

  • Multi-objectivization introduces incomparability, increasing landscape neutrality.

  • Multi-objectivization forms neutral connections, merging different neutral networks.

  • The new defined neutral paths play an important role in escaping from local optima.

  • Multi-objectivization significantly improved search performance.

Abstract

Multi-objectivization represents a current and promising research direction which has led to the development of more competitive search mechanisms. This concept involves the restatement of a single-objective problem in an alternative multi-objective form, which can facilitate the process of finding a solution to the original problem. Recently, this transformation was applied with success to the HP model, a simplified yet challenging representation of the protein structure prediction problem. The use of alternative multi-objective formulations, based on the decomposition of the original objective function of the problem, has significantly increased the performance of search algorithms. The present study goes further on this topic. With the primary aim of understanding and quantifying the potential effects of multi-objectivization, a detailed analysis is first conducted to evaluate the extent to which this problem transformation impacts on an important characteristic of the fitness landscape, neutrality. To the authors’ knowledge, the effects of multi-objectivization have not been previously investigated by explicitly sampling and evaluating the neutrality of the fitness landscape. Although focused on the HP model, most of the findings of such an analysis can be extrapolated to other problem domains, contributing thus to the general understanding of multi-objectivization. Finally, this study presents a comparative analysis where the advantages of multi-objectivization are evaluated in terms of the performance of a basic evolutionary algorithm. Both the two- and three-dimensional variants of the HP model (based on the square and cubic lattices, respectively) are considered.

Introduction

The term multi-objectivization was originally coined by Knowles et al. to refer to the process of reformulating a single-objective optimization problem in terms of two or more objective functions, i.e., as a multi-objective problem (Knowles, Watson, & Corne, 2001). This transformation can be either based on the addition of new supplementary objectives (Brockhoff et al., 2007, Jensen, 2004), or it can be based on the decomposition of the original objective function of the problem (Handl et al., 2008b, Knowles et al., 2001). In either case, multi-objectivization may result in fundamental changes to the fitness landscape of the problem. Since the performance of search algorithms is dictated by their interaction with the underlying fitness landscape (Watson, 2010), multi-objectivization can thus significantly impact on the ability of these algorithms to solve a given optimization task.

It is commonly assumed that the higher the number of objective functions, the more difficult a problem is; and this is usually the case (Ishibuchi et al., 2008, Knowles and Corne, 2007). A single-objective to multi-objective transformation, however, has led to the development of more competitive search mechanisms. A considerable number of successful applications of multi-objectivization have been reported in the literature. For a recent review on applications of multi-objectivization, the reader can be referred to Segura, Coello Coello, Miranda, and León (2013). Multi-objectivization has been largely studied in the context of well-known combinatorial problems such as the traveling salesman problem (Jähne et al., 2009, Jensen, 2004, Knowles et al., 2001, Lochtefeld and Ciarallo, 2014), the job-shop scheduling problem (Jensen, 2004, Lochtefeld and Ciarallo, 2011), the bin packing problem (Segredo et al., 2013, Segura, Segredo and León, 2011), the vehicle routing problem (Watanabe & Sakakibara, 2007), and the shortest path and minimum spanning tree problems (Neumann & Wegener, 2008). Also, multi-objectivization has found interesting applications in the fields of mobile communications (Segura, Segredo, González et al., 2011, Segura et al., 2013), computational mechanics (Greiner, Emperador, Winter, & Galván, 2007), power system planning (Trivedi, Sharma, & Srinivasan, 2012), structural topology optimization (Sharma, Deb, & Kishore, 2014), computer aided manufacturing (Churchill, Husbands, & Philippides, 2013), robotics (Mouret, 2011) and computer vision (Vite-Silva, Cruz-Cortés, Toscano-Pulido, & de la Fraga, 2007). Finally, multi-objectivization has also been proposed to deal with bioinformatic problems, such as those related to gene regulatory networks (Thomas & Jin, 2013) and protein structure prediction (Becerra et al., 2010, Cutello et al., 2005, Cutello et al., 2006, Cutello et al., 2008, Day et al., 2002, Handl et al., 2008a, Olson and Shehu, 2013, Soares Brasil et al., 2011, Sudha et al., 2013).

Recently, the concept of multi-objectivization was applied with success to the hydrophobic-polar (HP) model, a reduced representation of the protein structure prediction problem (Dill, 1985). This model abstracts the fact that hydrophobicity is a major determinant of the folded state of proteins. Despite its limited biological significance, from the computational point of view this model still represents an interesting and challenging problem in combinatorial optimization (Berger and Leighton, 1998, Crescenzi et al., 1998). Three different multi-objectivization schemes for the HP model have been proposed, all of them based on the decomposition of the original energy (objective) function of the problem (Garza-Fabre et al., 2012a, Garza-Fabre et al., 2012b, Garza-Fabre, Toscano-Pulido et al., 2012). Decomposition introduces plateaus of incomparable solutions, an effect that can be exploited in order to overcome search difficulties such as that of becoming trapped in local optima (Handl et al., 2008b, Knowles et al., 2001). In this way, the use of alternative multi-objective formulations of the HP model has led to an important increase in the performance of search algorithms (Garza-Fabre et al., 2012a, Garza-Fabre et al., 2012b, Garza-Fabre, Toscano-Pulido et al., 2012), motivating further research in this direction.

The present study significantly extends preliminary research regarding the multi-objectivization of the HP model. While previous analyses were concerned only with the benefits of multi-objectivization in terms of search performance (Garza-Fabre et al., 2012a, Garza-Fabre et al., 2012b, Garza-Fabre, Toscano-Pulido et al., 2012), the primary goal of this study is to thoroughly investigate the potential effects that this transformation has on the characteristics of the problem. As pointed out before, multi-objectivization influences the comparability relation among solutions. As a means of illustrating and, to some extent, quantifying this effect, it is first evaluated how the comparability between the different defined fitness classes can be affected. The alteration in the comparability of solutions directly impacts on an essential property of the fitness landscape, neutrality. Hence, a detailed analysis is conducted with the aim of understanding multi-objectivization from a fitness landscape perspective, by focusing on neutrality. Finally, a comparative study is presented where the three multi-objectivization proposals are evaluated with respect to each other, and with respect to the conventional single-objective formulation of the HP model, in terms of the performance of a basic single-solution-based evolutionary algorithm.

The remainder of this document is organized as follows. Section 2 provides background concepts and sets the notation used in this study. The three studied multi-objectivization approaches for the HP model are described in detail in Section 3. Section 4 is devoted to the analysis of the effects of multi-objectivization. The comparative study which focuses on search performance is covered in Section 5. Finally, Section 6 discusses the main findings and presents the conclusions of this study. Appendices at the end of this document contain supplementary information with regard to implementation details of the considered search algorithms, performance measures, test instances, the methodology followed for the statistical significance analyses, and the utilized experimental platform.

Section snippets

Single-objective and multi-objective optimization

Without loss of generality, a single-objective optimization problem can be formally stated as follows:Minimize0.35em0exf(x),subjectto0.35em0exxXF,where x is a solution vector; XF is the feasible set, i.e., the set of all feasible solution vectors in the search space X,XFX; and f:XR is the objective function to be optimized.

Similarly, a multi-objective optimization problem can be formally defined as follows:Minimize0.35em0exf(x)=[f1(x),f2(x),,fk(x)]T,subjectto0.35em0exxXF,where f(x) is the

Multi-objectivization of the HP model

Three different multi-objectivization schemes for the HP model are analyzed in this study: the parity decomposition (Garza-Fabre et al., 2012a), the locality decomposition (Garza-Fabre, Toscano-Pulido et al., 2012) and the H-subsets decomposition (Garza-Fabre et al., 2012b). As their name indicates, these approaches are based on the decomposition of the original energy (objective) function of the HP model. Decomposition, as discussed in Section 2.2, has the potential effect of introducing

Effects of multi-objectivization

This section is devoted to investigating the potential effects that can be achieved by multi-objectivization. Although three different multi-objectivization schemes for the HP model are covered by this research work, only the locality decomposition (defined in Section 3.2), using a value of δ = 7, is considered in this section due to the high computational demands of the conducted analyses. The locality decomposition has been found, in the authors’ previous work (Garza-Fabre et al., 2012b,

Search performance

The aim of this section is to investigate the influence that multi-objectivization can exert on the search behavior of metaheuristic algorithms. To this end, the three studied multi-objectivization schemes, based on the parity (PD), locality (LD) and H-subsets (HD) decompositions, are evaluated and compared with respect to each other and with respect to the conventional single-objective (SO) formulation of the HP model. A basic single-solution-based evolutionary algorithm (EA), the so-called (1

Discussion and final remarks

Multi-objectivization concerns the reformulation of a single-objective problem as a multi-objective one (Knowles et al., 2001). When applied to the particular case of study of this research project, the HP model for protein structure prediction, it has been reported that this transformation provides significant improvements in terms of the performance of search algorithms (Garza-Fabre et al., 2012a, Garza-Fabre et al., 2012b, Garza-Fabre, Toscano-Pulido et al., 2012). Motivated from these

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