Naoya Akedo

Behavior of Multiobjective Evolutionary Algorithms on Many-Objective Knapsack Problems

We examine the behavior of three classes of evolutionary multiobjective optimization (EMO) algorithms on many-objective knapsack problems. They are Pareto dominance-based, scalarizing function-based, and hypervolume-based algorithms. NSGA-II, MOEA/D, SMS-EMOA, and HypE are examined using knapsack problems with 2-10 objectives. Our test problems are generated by randomly specifying coefficients (i.e., profits) in objectives. We also generate other test problems by combining two objectives to create a dependent or correlated objective. Experimental results on randomly generated many-objective knapsack problems are consistent with well-known performance deterioration of Pareto dominance-based algorithms. That is, NSGA-II is outperformed by the other algorithms. However, it is also shown that NSGA-II outperforms the other algorithms when objectives are highly correlated. MOEA/D shows totally different search behavior depending on the choice of a scalarizing function and its parameter value. Some MOEA/D variants work very well only on two-objective problems while others work well on many-objective problems with 4-10 objectives. We also obtain other interesting observations such as the performance improvement by similar parent recombination and the necessity of diversity improvement for many-objective knapsack problems.

Behavior of EMO algorithms on many-objective optimization problems with correlated objectives

Recently it has been pointed out in many studies that evolutionary multi-objective optimization (EMO) algorithms with Pareto dominance-based fitness evaluation do not work well on many-objective problems with four or more objectives. In this paper, we examine the behavior of well-known and frequently-used EMO algorithms such as NSGA-II, SPEA2 and MOEA/D on many-objective problems with correlated or dependent objectives. First we show that good results on many-objective 0/1 knapsack problems with randomly generated objectives are not obtained by Pareto dominance-based EMO algorithms (i.e., NSGA-II and SPEA2). Next we show that the search ability of NSGA-II and SPEA2 is not degraded by the increase in the number of objectives when they are highly correlated or dependent. In this case, the performance of MOEA/D is deteriorated. As a result, NSGA-II and SPEA2 outperform MOEA/D with respect to the convergence of solutions toward the Pareto front for some many-objective problems. Finally we show that the addition of highly correlated or dependent objectives can improve the performance of EMO algorithms on two-objective problems in some cases.

A Study on the Specification of a Scalarizing Function in MOEA/D for Many-Objective Knapsack Problems

In recent studies on evolutionary multiobjective optimization, MOEA/D has been frequently used due to its simplicity, high computational efficiency, and high search ability. A multiobjective problem in MOEA/D is decomposed into a number of single-objective problems, which are defined by a single scalarizing function with evenly specified weight vectors. The number of the single-objective problems is the same as the number of weight vectors. The population size is also the same as the number of weight vectors. Multiobjective search for a variety of Pareto optimal solutions is realized by single-objective optimization of a scalarizing function in various directions. In this paper, we examine the dependency of the performance of MOEA/D on the specification of a scalarizing function. MOEA/D is applied to knapsack problems with 2-10 objectives. As a scalarizing function, we examine the weighted sum, the weighted Tchebycheff, and the PBI penalty-based boundary intersection function with a wide range of penalty parameter values. Experimental results show that the weighted Tchebycheff and the PBI function with an appropriate penalty parameter value outperformed the weighted sum and the PBI function with no penalty parameter in computational experiments on two-objective problems. However, better results were obtained from the weighted sum and the PBI function with no penalty parameter for many-objective problems with 6-10 objectives. We discuss the reason for these observations using the contour line of each scalarizing function. We also suggest potential usefulness of the PBI function with a negative penalty parameter value for many-objective problems.

A many-objective test problem for visually examining diversity maintenance behavior in a decision space

Recently distance minimization problems in a two-dimensional decision space have been utilized as many-objective test problems to visually examine the behavior of evolutionary multi-objective optimization (EMO) algorithms. Such a test problem is usually defined by a single polygon where the distance from a solution to each vertex is minimized in the decision space. We can easily generate different test problems from different polygons. We can also easily generate test problems with multiple equivalent Pareto optimal regions using multiple polygons of the same shape and the same size. Whereas these test problems have a number of advantages, they have no clear relevance to real-world situations since they are artificially generated unrealistic test problems. In this paper, we generate a distance minimization problem from a real-world map. Our test problem has four objectives, which are to minimize the distances to the nearest elementary school, junior high school, railway station, and convenience store. Using our test problem, we examine the behavior of well-known and frequently-used EMO algorithms in terms of their diversity maintenance ability in the two-dimensional decision space.

Relation between Neighborhood Size and MOEA/D Performance on Many-Objective Problems

MOEA/D is a simple but powerful scalarizing function-based EMO algorithm. Its high search ability has been demonstrated for a wide variety of multiobjective problems. MOEA/D can be viewed as a cellular algorithm. Each cell has a different weight vector and a single solution. A certain number of the nearest cells are defined for each cell as its neighbors based on the Euclidean distance between weight vectors. A new solution is generated for each cell from current solutions in its neighboring cells. The generated solution is compared with the current solutions in the neighboring cells for solution replacement. In this paper, we examine the relation between the neighborhood size and the performance of MOEA/D. In order to examine the effect of local mating and local replacement separately, we use a variant of MOEA/D with two different neighborhoods: One is for local mating and the other is for local replacement. The performance of MOEA/D with various combinations of two neighborhoods is examined using the hypervolume in the objective space and a diversity measure in the decision space for many-objective problems. Experimental results show that MOEA/D with a large replacement neighborhood has high search ability in the objective space. However, it is also shown that small replacement and mating neighborhoods are beneficial for diversity maintenance in the decision space. It is also shown that the appropriate specification of two neighborhoods strongly depends on the problem.

Many-objective and many-variable test problems for visual examination of multiobjective search

In the development of evolutionary multiobjective optimization (EMO) algorithms, it is important to implement a good balancing mechanism between the convergence of solutions towards the Pareto front and their diversity over the Pareto front. When an EMO algorithm is applied to a two-objective problem, the balance can be easily visualized by showing all solutions at each generation in the two-dimensional objective space. However, such a visual examination of the multiobjective search is difficult for many-objective problems with four or more objectives. The use of many-objective test problems with two decision variables has been proposed in some studies to visually examine the search behavior of EMO algorithms. Such test problems are defined by a number of points in a two-dimensional decision space where the distance minimization from each point is an objective. Thus the number of objectives is the same as the number of points. The search behavior of EMO algorithms can be visually examined in the two-dimensional decision space. In this paper, we propose the use of many-objective test problems for visual examination of the search behavior in a high-dimensional decision space. More specifically, our m-objective test problem with n variables is generated by specifying m points on a plane in an n-dimensional decision space. We examine the behavior of EMO algorithms through computational experiments on such an m-objective n-variable test problem. Our experimental results show that the number of variables has a large effect on the search behavior of EMO algorithms with respect to the diversity of solutions.

Recombination of similar parents in SMS-EMOA on many-objective 0/1 knapsack problems

In the evolutionary multiobjective optimization (EMO) community, indicator-based evolutionary algorithms (IBEAs) have rapidly increased their popularity in the last few years thanks to their theoretical background and high search ability. Hypervolume has often been used as an indicator to measure the quality of solution sets in IBEAs. It has been reported in the literature that IBEAs work well on a wide range of multiobjective problems including many-objective problems on which traditional Pareto dominance-based EMO algorithms such as NSGA-II and SPEA2 do not always work well. In this paper, we examine the behavior of SMS-EMOA, which is a frequently-used representative IBEA with a hypervolume indicator function, through computational experiments on many-objective 0/1 knapsack problems. We focus on the effect of two mating strategies on the performance of SMS-EMOA: One is to select extreme parents far from other solutions in the objective space, and the other is to recombine similar parents. Experimental results show that the recombination of similar parents improves the performance of SMS-EMOA on many-objective problems whereas the selection of extreme parents is effective only for a two-objective problem. For comparison, we also examine the effect of these mating strategies on the performance of NSGA-II.

How to strike a balance between local search and global search in multiobjective memetic algorithms for multiobjective 0/1 knapsack problems

An important implementation issue in the design of hybrid evolutionary multiobjective optimization algorithms with local search (i.e., multiobjective memetic algorithms) is how to strike a balance between local search and global search. If local search is applied to all individuals at every generation, almost all computation time is spent by local search. As a result, global search ability of memetic algorithms is not well utilized. We can use three ideas for decreasing the computation load of local search. One idea is to apply local search to only a small number of individuals. This idea can be implemented by introducing a local search probability, which is used to choose only a small number of initial solutions for local search from the current population. Another idea is a periodical (i.e., intermittent) use of local search. This idea can be implemented by introducing a local search interval (e.g., every 10 generations), which is used to specify when local search is applied. The other idea is an early termination of local search. Local search for each initial solution is terminated after a small number of neighbors are examined. This idea can be implemented by introducing a local search length, which is the number of examined neighbors in a series of iterated local search from a single initial solution. In this paper, we discuss the use of these three ideas to strike a local-global search balance. Through computational experiments on a two-objective 500-item knapsack problem, we compare various settings of local search such as short local search from all individuals at every generation, long local search from only a few individuals at every generation, and periodical long local search from all individuals. Global search in this paper means genetic search by crossover and mutation in multiobjective memetic algorithms.

Many-objective test problems with multiple Pareto optimal regions in a decision space

In evolutionary multi-objective optimization (EMO) algorithms, diversity maintenance has been mainly discussed in the objective space in order to search for uniformly distributed non-dominated solutions along the entire Pareto front. In this paper, we propose three types of many-objective test problems with multiple Pareto optimal regions in the decision space. One type has multiple equivalent Pareto optimal regions. Another type has different but somewhat similar Pareto optimal regions. The other type has Pareto and local Pareto optimal regions. Our many-objective test problems are generated by placing multiple polygons of the same or similar shapes in a decision space. The ith objective is the minimization of the distance from a solution to the nearest ith vertex over all polygons. Thus the number of objectives is the same as the number of vertices of the polygons. The number of equivalent or similar Pareto regions in the decision space is the same as the number of the polygons.

Two-objective solution set optimization to maximize hypervolume and decision space diversity in multiobjective optimization

Diversity maintenance in the decision space is a recent hot topic in the field of evolutionary multiobjective optimization (EMO). In this paper, we propose the use of a decision space diversity measure as an objective function in a two-objective formulation of solution set optimization where the hypervolume measure is used as the other objective. In the proposed approach, a given multiobjective problem with an arbitrary number of objectives is handled as a two-objective solution set optimization problem. A solution of our two-objective problem is a set of non-dominated solutions of the original multiobjective problem. An EMO algorithm is used to search for a number of solution sets along the tradeoff surface between the diversity maximization in the decision space and the hypervolume maximization in the objective space. In this paper, first we numerically examine the diversity measure of Solow & Polasky (1994), which was used in recent studies of Ulrich et al. (2010, 2011), through computational experiments on many-objective distance minimization problems in a two-dimensional decision space. Then we formulate a two-objective solution set optimization problem to maximize the decision space diversity and the objective space hypervolume. Finally we demonstrate that a number of non-dominated solution sets can be obtained along the diversity-hypervolume tradeoff surface. Through computational experiments, we also examine the difference between the following two settings for diversity calculation: All solutions in a solution set are used in one setting while only non-dominated solutions are used in the other setting.