Investigating Smart Sampling as a population initialization method for Differential Evolution in continuous problems
Introduction
The task of global optimization has arisen in several areas of real-world problems, such as protein structure prediction [8], logistics or circuit design (traveling salesman problem) [6], chemical engineering [35], and airspace design [10]. This task involves the minimization or maximization of a known objective function or an unknown black-box function. In general, these functions are highly complex and may be time-consuming, taking several days, weeks or months to achieve an adequate result, which may not be the global optimum. To solve this type of task, several global optimization metaheuristics have been developed.
Metaheuristics [13], [46], [21] are optimization techniques used to search for high-quality solutions of a problem of which one is expected to be the global optimum. One of the main characteristics is that metaheuristics need neither gradient information to guide the search nor specific knowledge of a problem (heuristic), which makes them useful to solve a wide range of problems, including black-box ones. Several strategies have been investigated to improve the exploratory efficiency of metaheuristics to reach the global optimum of a problem. For instance, strategies have been developed to reduce premature convergence and to increase the chance of escaping from local optima. Basically, those strategies, when applied to populational metaheuristics, involve the maintenance of diversity in the set of solutions. Another procedure that has been investigated to improve a metaheuristic’s performance is the step related to the population initialization [15], [27], [34].
Initial population generation involves an exploration phase. This phase allows the algorithm to select locations to be explored and others to be discarded. Traditional metaheuristics move toward the best solutions. Thus, a bad initialization that generates solutions close to each other (clusters of solutions) could leave large areas with no solutions to explore. On the other hand, if the population is very disperse, a large number of iterations may be required to reach a local optimum. Moreover, if two solutions far from each other are combined, there is a high chance that the offspring will be closer to the best solution found, which could leave a large unexplored gap. Thus, an initialization method capable of providing a better exploration of the search-space and presenting only high-quality solutions should improve the performance of a metaheuristic.
Metaheuristics themselves are naturally guided towards promising regions [36] (see Fig. 1). There is an exploration phase and then the population moves towards the best solution found in order to exploit that region. On the other hand, a desired movement could be the one presented in Fig. 2. The exploration phase may take longer and the population can be split into more than one region. After that, the regions can be exploited independently.
This paper presents an approach to explore the search-space and to find promising regions. The objective is to aid global optimization algorithms by indicating the initial search-space areas with higher possibility of finding the global optimum. The approach is iteratively applied to explore the search-space inside promising regions which become smaller at each iteration, similarly to the strategy proposed in [17], excluding areas considered unfavorable. The new approach, called Smart Sampling (SS), seeks to preserve diversity in more than one promising region, providing a better exploration of the search-space.
First of all, SS generates some solutions, evaluates them using the objective function, and splits them into good and bad based on a threshold applied to the function value of each solution. Then, SS employs a machine learning algorithm to map characteristics of these good solutions, allowing it to check if a new solution is good without being evaluated. If the new solution is identified as a good one, then it can be evaluated by the objective function. This approach works well on problems with small or high numbers of variables. In the last step, another machine learning algorithm separates the different promising regions to be exploited by any metaheuristic, which will refine the high-quality solutions found during the SS process.
Therefore, SS is employed to increase the efficiency of global optimization algorithms, and can be essential to obtain satisfactory results in situations in which the execution of a large number of experiments is not viable. Furthermore, and most important, several researchers have studied ways to improve well-known metaheuristics by using heuristics, or local-search methods, or creating hybrids. Here, we propose a technique that is neither an operator nor a strategy to be included in a search technique, and has been developed for use as a preprocessing phase. It can be directly used to improve the performance of any populational continuous global optimization technique.
SS is tested in conjunction with Differential Evolution (DE), a well-known metaheuristic, in several bound-constrained optimization problems with different properties. The quality of SSDE’s solutions is better than DE’s with fewer evaluations. The paper also presents a performance comparison with other three approaches (ODE, QODE and UQODE) that improve DE to explore the search-space in an attempt to find promising regions and escape from local optima. The results have shown that SSDE provides considerably better performance than the other three approaches.
The paper is organized as follows: Section 2 contains some related works on population initialization and machine learning techniques used to improve metaheuristics; in Section 3, the proposed SS algorithm is presented in details; the Differential Evolution algorithm is briefly described in Sections 4 Differential Evolution, 5 Evaluating SS presents a preliminary study on SSDE’s behavior using some well-known benchmark functions. Some charts show the distribution of high-quality solutions during SS procedure and convergence curves through the optimization process. The experiments comparing SSDE, DE and the other three DE improvements are presented and discussed in Section 5. Finally, Section 6 concludes the paper and presents future works.
Section snippets
Related works
The use of machine learning algorithms to improve metaheuristics is not new. Jourdan et al. [12] showed how classification and clustering techniques are applied to hybridize metaheuristics reducing the computational time and simplifying the objective function by an approximation technique or improve the quality of the search by adding background knowledge to the operators. Ramsey and Grefenstette [34] initialized a genetic algorithm using case-based reasoning to allow the system to bias the
Smart Sampling
The basic flowchart of SS is presented in Fig. 3. First, SS samples the search-space to identify the first large regions which must be explored. The higher the dimensionality of the problem, the larger the first sample. The main idea of SS is to perform a resampling only in areas considered promising regions, avoiding wasting evaluations in non-promising areas. If a simple random resampling is executed inside promising regions, final iterations will lead to local optima. This behavior should be
Differential Evolution
Differential Evolution was introduced by Storn and Price in 1995 [37]. It is a floating-point encoding populational metaheuristic, similar to classical evolutionary algorithms, successfully used to solve several benchmarks and real-world problems [22], [26], [43], [42].
Population P of D dimensions is randomly initialized (using a uniform distribution) inside the problem’s bounds and evaluated using the fitness function for the problem. Next, until a stop criterion has been met, the algorithm
Evaluating SS
The functioning of SS in continuous 2D test functions (Ackley, Alpine, Griewank, Parabola, Rastrigin, Rosenbrock and Tripod, see mathematical definition in Table 1) is illustrated, respectively, in Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13. For this experiment, SS was configured as follows: window_size = 0.01; lower_lim = −0.5 and upper_lim = 1.5. Those lim values are used in Algorithm 2.
The graphs in the figures show the evolution/reduction in promising regions through SS iterations
Computational experiments
This paper presents the effects of SS on global optimization problems using a well-known global optimization algorithm, i.e., the classical Differential Evolution (DE) [38]. The results using SS (our approach) in conjunction with DE – called SSDE – are compared to the results presented by ODE [33], QODE [31] and UQODE [29].
Conclusions and future works
This paper has presented a technique to find promising regions of the search-space of continuous functions. The approach, named Smart Sampling (SS), uses a machine-learning technique to identify promising and non-promising solutions to guide the resampling procedure to smaller areas where higher-quality solutions can be found. This iterative process ends when a stop criterion has been achieved, for instance, when a promising region is too small. At this point, another machine-learning technique
Acknowledgments
The authors would like to acknowledge CAPES (a Brazilian Research Agency) for the financial support given to this research.
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