Elsevier

Information Sciences

Volume 467, October 2018, Pages 750-765
Information Sciences

Evaluation of the migrated solutions for distributing reference point-based multi-objective optimization algorithms

https://doi.org/10.1016/j.ins.2018.05.015Get rights and content

Highlights

  • Propose an improved method for distributing multiobjective optimization algorithm.

  • A new way to evaluate the migrated members by using crossover.

  • Efficient metric is proposed for evaluation of the performance of the distribution approach.

  • This metric gives the approximate minimum number of function evaluations to reach the solution.

  • Proposals can be able to applied to many-objective decomposition based algorithms.

Abstract

As the number of objectives and/or dimension of a given problem increases, or a real-world optimization problem is modeled in more detail, the optimization algorithm requires more computation time if the computational resources are fixed. Therefore, some more tools are needed to be developed for deployment of these resources. The parallelization is one of these tools based on distribution of the overall problem to different computational units. In this study, a distributed computing approach for multi-objective evolutionary optimization algorithms is proposed by application of a migration policy which is based on sharing the information for inter-processor collaboration. This idea is also intensified with the crossover operator at the evolutionary algorithms where the migrated solutions are applied to the crossover operator so that the performance of the overall approach increases. Besides, a new metric is defined for evaluation of the performance of the proposed distribution methodology. The performance of the proposed approaches is evaluated via well-known two- and three-objective well-known test problems.

Introduction

Many engineering problems are defined to find the best possible configuration or setting so that the overall system can be able to give the desired criteria which are defined by the decision maker -engineer-. Since the aim is to find the best possible configuration of a physical system -or a non-physical system like the economic process- the problems are called as an optimization problem. The purpose of the optimization problem is to dig out a set of unknowns at the problem concerning the objectives and constraints. Hence, optimization algorithms with various properties are suggested to solve these different engineering problems such as the structural design problems like optimal sliding door problem, tension spring problem, welded beam problem, pressure vessel problem [1].

These optimization problems are solved and compared by using evolutionary algorithm -EA- (multi-objective genetic algorithm [2], multi(many)objective optimization algorithm [3]), nature-inspired (gravitational search algorithm [4], firefly algorithm [5], bat algorithm [6], league championship algorithm [6], imperialist competitive algorithm [6], charged system search algorithm [7], moth-flame optimization algorithm [8]), classical-based (Nelder–Mead algorithm [9], hill climbing [4], natural frequency optimization [9]), and hybrid optimization algorithms [10]. Even these algorithms are relatively successful at obtaining approximate solutions. Still, a relatively substantial computational time is needed to obtain the best possible solution where the engineering problem contains the high degree of objective and decision space. That increases the required computation time. Therefore, the problem of increasing the execution speed of these algorithms became one of the engineering problems. In other words, this problem aims to reduce the computational time. In general, the total computational time for converging to the Pareto-optimal solutions of multi-objective optimization problems mostly depends on the decision (search) dimension, objective space dimension, and mathematical representation of the real-world optimization problem, which is modeled as a complex mathematical expression with many unknowns. Hence relatively large population is needed to get an acceptable solution set, which desirably covers the objective space solution-Pareto-front. However, evaluating a large population desires more computational time or computational power for the same computational time, in other words, the necessity to reach the faster computation time rises [11], [12]. Fortunately, with the aid of distributed computational methodologies, the execution time can be reduced dramatically with or without change at the main code of the optimization algorithms. Furthermore, Alba and Tomassini [13] present a survey for distribution, implementation, and application of EAs at the distributed computing environment. The authors also give a section for future achievements that support the aim of this study.

Distributed multi-objective evolutionary algorithm (MOEA) studies have concentrated on the following two manners/methods and their collocation usage [14]: (i) function evaluations of population members are distributed on multiple processors, and (ii) algorithms are evaluated on multiple processors and information is shared among them to constitute a parallel search.

The first approach is known/named as the Master–Slave Model (MSM) [15] (Fig. 1). This method is suitable to apply on cores of CPUs and GPUs. It is usually preferred in industrial applications due to its relative ease for application by slightly changing the sequential code in order to form of a distributed implementation with the aid of well-established and redundant libraries. The method relies two main components: master (generally one unit) and slave (more than one unit). The master processor controls the flow of data and distribution of the fragments of the algorithm to slave units (processors). Generally, due to efficiency and performance considerations, the master itself distributes the algorithm by dividing into many pieces which correspond to a part of the algorithm (or the same algorithm) on the slave units (generally for efficiency master also works as a slave after the distribution overs). Three main approaches are usually implemented in the MSM: (i) the objective function evaluation is executed on parallel computational units, (ii) the main algorithm is segmented into different parallel modules and each parallelize sub-programs (i.e. the sequence of search agent initialization, objective function evaluation, search agent update, etc.) is independently evaluated at each parallel computational unit; and (iii) instead of the algorithm, the problem is divided into sub-problems; and each sub-problem is evaluated on parallel computational units without any communication among them. These MSM models are useful if the number of available computational units is considerably high, and the theoretical speedup for MSM is directly proportional to the number of the computational units; hence for a network consisting of K computational units, it is expected that the algorithm performs K times faster than the sequential code, theoretically. On the other hand, an undesired overhead occurs due to the communication between master and slaves, which causes a delay in the process. Since the master device is also a computational unit (even this unit is employed as an additional slave unit, it isn’t always possible to distribute the problem to this unit especially at the GPU implementations), hence there are K+1 processors to be used for parallelization. Even if it is possible to reduce the computation time for an engineering problem from months to hours by utilizing Master–Slave Model; the computational units cannot be utilized efficiently. As an example let us consider the study conducted by Perc et al. [16]. In the study, the authors are proposed a CUDA-based distribution approach on graphics acceleration card for the Monte Carlo simulation method (Master–Slave Model) for solving the publics good game as an example (which may be considered as a harder problem compared to the prisoner’s dilemma [17]). The authors implemented the problem on both CPU and GPU devices; and they achived almost 500 times speed up even though there were 2560 computational units (graphical acceleration card has 2560 streaming multiprocessor - namely CUDA cores).

The second method is called as The Island Model (IM) [18] (Fig. 2). The method is based on execution of the algorithms run independently at different computational units. It is also possible to run same or different optimization algorithms on different computational units. Also, as a part of the method, distributed devices can regularly share information among each other to improve the efficiency of the distribution -increase the speedup-. This information is the some of the solution candidates who are transferred after a definite number of generations; it is called migration. This method is more suitable for the clusters of computation units, which are heterogeneous and asunder from each other that causes communication between computational units is limited or expensive (in time) to achieve. Although, MSM is straightforward and can be able to get almost linear speed-up; the second method is promising and has the potential to surpass linear speed-up [19], [20] with the introduction of new methodologies.

As a part of IM, a couple of methodologies [15], [18] are proposed recently for the problem of distribution of an evolutionary multi-objective algorithms (EMO) [19], [21], even so the problem has still a remarkable case of using these distribution methods. Two mainly-different IM-based distributed computing approaches for multi-objective optimization algorithms were suggested previously [18]. The first approach depends on dividing the objective space, and distributing them into computational units with a cone-domination principle. Then interested part of the Pareto optimal front was found by evaluating EA algorithm on each unit [18], [19], A set of non-dominated solutions from one computational unit to the other units send as an information exchange to improve the efficiency. Even this idea is relatively simple; one significant criticism is the limitation to solve the problems with convex Pareto fronts. Hence, it is not expected from this approach to determine intermediate solutions on the objective space for the problems with non-convex fronts. The second approach [21] divides the objective space by using the pre-defined artificial constraints. The numerical range of the non-dominated front affects determining the location of the constraints adaptively. Therefore, this approach can be able to apply to the problems with a variety of shaped efficient front. However, as an algorithmic point of view, it is an objectionable, and it needed a brute force strategy for distributing the task.

Recently, the authors proposed a novel method [22] by borrowing two concepts from the existing literature: one of them is the reference point idea from many-objective optimization, and the island model approaches from distributed computing. This new reference point based approach [23] is applied to divide the objective space. Each unit has a sub-set of reference points which are pre-assigned to find a limited part of the Pareto front. It was shown that, with the aid of the proposed approach [22], it is generically possible to solve problems having multi-shape of the Pareto frontier. Although, the number of function evaluations per computational unit is smaller than calculations for the single computational unit, however in the total count for objective function evaluations at multi-processor is more than the single processor. It means that the number of parallel devices is smaller than the decreasing in function evaluations. If a single processor is evaluated N objective functions, it is desired that each computational units have to assess at least N/P tasks per computational units for each of the P units, or reaching to closest results. For solving this issue, in [22], with the introduction of delay approach, the total number of evaluated objective functions is reduced with the sacrifice of parallelization, because the approach is based on assessing the algorithm for a given number of generation on a single computational unit, and then the population is divided into P units. Therefore, the approach can be viewed more sequential then parallel.

In this paper, the performance of the proposed reference point distribution based parallelization approach is improved by means of integration of the migration idea. The impact of migration is increased by directly using migration with crossover operator. This new migration idea is beneficiary in terms of transfering information of different portions of the objective space. The proposed methods are applied to two and three objective benchmark problems to show the performance of them.

The organization of the paper is as follows: a brief description of the existing island model approaches for distributed computing approaches for efficient distributed computation of Pareto fronts is presented as a sub-section of this Introduction section. Section II gives a detailed description of our proposed method together with some fundamental information regarding the R-NSGA-II algorithm; the novel migration idea is also explained in this section. The implementation is divided into two sections; The section III for two objective problems, and The section IV is for three objective problems) based on the number of objectives of the problems. The last section outlines the conclusion and potential future works for this study.

The general mathematical formulation of an unconstrained multi-objective optimization problem is given as follows: min(f1(x),f2(x),,fm(x))where x={x1,x2,,xd} is the d dimensional decision variable x ∈ Ω, f is the objective function F: Ω → Rm, and m is the number of objectives.

Even though the distribution for Multi-objective Evolutionary Algorithms (MOEAs) are applied successfully to many real-world problems (vehicle routing problem [24], flow-shop problem [25], [26], ammonia synthesis plant [27], broadcast operation in MANETs on DFCN protocol [14], greenhouse corp planing [28], antenna position problem [29], logistics network design [30], energy-aware scheduling [31]), the number of novel schemes for distribute of the MOEAs and problems is relatively low. These existing schemes can be grouped as: (i) the cone-domination approach, which is based on transformation of the objective space, (ii) the constrained-based approach, which depends on definition of constraints for different portions of the objective space, and (iii) the clustered population approach, with the idea of construction of sub-populations of search agents as regards their layout in the objective space. These approaches are briefly explained below.

Two main ideas are presented in this subsection. The first idea is introduced by Sato et al. [32], 33] and it is implemented by Cheshmehgaz et al. [34] with a novel (named as VIP dominance) migration idea. The idea depends on transforming the objective space and convert the rectangular definitions of the solution candidates at objective space to polar coordinate form. At each iteration, the minimum of the objective function values is recorded and assigned as the new origin. Next, based on this new origin, the objective space is transformed, and solutions are converted from rectangular form to polar coordinate. Finally, it is divided into sub-populations based on the angles of polar coordinates.

The second one (proposed by Deb et al. in [18]) is based on the changing the domination area of the fundamental domination definition.

Definition 1.1

Domination: Let x1={x1,x2,,xd} and x2={x1,x2,,xd} are two decision vectors defined as x1, x2 ∈ Ω. The solution candidate x1 said to dominate x2 if fi(x1)fi(x2)i=1,2,..,m and at least for one i=j such as fj(x1) < fj(x2)

In the two-dimensional objective space, -as an example-, the conventional domination idea is is to make all solutions fall in the positive quadrant of x objective space, dominant. By considering this definition, a solution x dominates all of the solution candidates lying above of the positive cone (positive quadrant) which spans 90° in angle of the objective space. Thus, the solutions which do not get dominated by other solution candidates in the objective space are, by definition, called Pareto-optimal solutions. The geometrical shape formed by pointing these several objective space solutions which are named as efficient solutions; and this form of objective space is named as the efficient front. The definition of domination can also be generalized to any number of objectives. The description of the domination can be extended for cone which is used for more than 90° at two-dimensional objective space, as an example; where the subset of the efficient front will be the new cone dominated front (Fig. 3(a)).

In the proposed island approach [18], a variety of cones were assigned to each unit, by this way every feasible region on objective space and corresponding solutions become a members of the cone-dominated Pareto-optimal set. The cone-domination principle works only for MO problems with a convex Pareto front; for problems with non-convex Pareto front, only the boundary solutions can be obtained.

Without loss of generality, which works on both convex and non-convex shaped problems; constraint-based approaches are proposed to check/countermeasure the violated sections by defining some artificial constraints. Unlike the cone-domination approach (Fig. 3(b)) each computational unit uses the same domination principle while searching the solutions for each unit remains on the defined sectors on objective space [21]. Moreover, since the same domination principle is shared among all units, it is possible to extend this idea to a relatively higher number of objectives. If the objective space is segmented in a deterministic way, a large number of computational units are needed. Alternatively, in [18], and adoptive placement scheme for constraints to alleviate this problem is proposed.

The first idea is proposed by Hiroyasu et al. [35] such that at a given migration frequency, all of the subpopulations are gathered and sorted as regards the first objective value. This process is repeated until the program is terminated. Since it is needed to be re-evaluated, there is no guarantee that the solutions remain in the search area of the islands. A similar segmentation idea was proposed by us in [22] for the delay approach, which is executed just once during the dispatch of the population among the islands. The second idea is proposed by Streichert et al. in [36]. The primary population is segmented into sub-populations by using the K-means clustering algorithm. First, the algorithm searches for the proper segmentation. Then sub-populations are formed by concerning the closeness to each cluster center.

Section snippets

Distributing the reference points

The proposed idea gets benefits of the concept of the reference point from many-objective optimization, and hence in this study the reference point-based elitist non-dominated sorting genetic algorithm (R-NSGA-II) [23] is selected as the primary optimization algorithm. The first step of the proposed distribution IM-based approach begins with the definition of trade-off reference points set by using Das and Dennis’s systematic approach [37] (Detailed information about point selection can be read

Two objective problems

The performance of the proposed distribution computation approach is demonstrated on benchmark problems. At the beginning of the analysis, first two objective problems are considered. For this purpose, three well-known benchmark problems (ZDT1, ZDT2, and ZDT4) are selected as test-bed. The mathematical description of the selected benchmark problems and their properties can be obtained from [20]. The proposed distributed approach is implemented on the benchmark problems by using R-NSGA-II

Three objective problems

In the previous section, the performance of the proposed method on two-objective benchmark problems was investigated. Since two axes represent the objective space, the segmentation of the objective space and checking the boundary violation offspring are relatively easy (especially when one decision variable represents the first objective). However, in this section, the performance of the proposed method is evaluated on relatively harder three-objective problems. Similarly, statistics based on

Conclusion

In this study, the performance of the proposed distributed approach for parallelization of multi-objective problems is improved up to a super-linear speed-up level by suggesting the collaboration of migration and crossover operators for two- and three-objective problems. With the introduction of the proposed method, instead of replacing the members in the host computational unit, the offsprings are produced with the immigrants, and best of them are determined by the selection operator of EAs.

Acknowledgment

O. Tolga Altinoz has been supported by the grant from TUBITAK under 2214/A Research Scholar Program for the research period at Michigan State University.

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