Dynamically Dimensioned Search Embedded with Piecewise Opposition-Based Learning for Global Optimization

Hindawi Scientific Programming Volume 2019, Article ID 2401818, 20 pages https://doi.org/10.1155/2019/2401818 Research Article Dynamically Dimensioned Search Embedded with Piecewise Opposition-Based Learning for Global Optimization Jianzhong Xu,1 Fu Yan ,1 Kumchol Yun,1,2 Sakaya Ronald,3 Fengshu Li,1 and Jun Guan4 1 School of Economics and Management, Harbin Engineering University, Harbin 150001, China Faculty of Mechanics, Kim Il Sung University, Pyongyang 950003, Democratic People’s Republic of Korea 3 College of Civil and Building Engineering, Kyambogo University, Kampala, Uganda 4 College of Economics and Management, Northeast Forestry University, Harbin 150040, China 2 Correspondence should be addressed to Fu Yan; Received 2 January 2019; Revised 4 April 2019; Accepted 12 May 2019; Published 26 May 2019 Academic Editor: Basilio B. Fraguela Copyright © 2019 Jianzhong Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dynamically dimensioned search (DDS) is a well-known optimization algorithm in the field of single solution-based heuristic global search algorithms. Its successful application in the calibration of watershed environmental parameters has attracted researcher’s extensive attention. The dynamically dimensioned search algorithm is a kind of algorithm that converges to the global optimum under the best condition or the good local optimum in the worst case. In other words, the performance of DDS is easily affected by the optimization conditions. Therefore, this algorithm has also suffered from low robustness and limited scalability. In this work, an improved version of DDS called DDS-POBL is proposed. In the DDS-POBL, two effective methods are applied to improve the performance of the DDS algorithm. Piecewise opposition-based learning is introduced to guide DDS search in the right direction, and the golden section method is used to search for more promising areas. Numerical experiments are performed on a set of 23 classic test functions, and the results represent significant improvements in the optimization performance of DDSPOBL compared to DDS. Several experimental results using different parameter values demonstrate the high solution quality, strong robustness, and scalability of the proposed DDS-POBL algorithm. A comparative performance analysis between the DDSPOBL and other powerful algorithms has been carried out by statistical methods by using the significance of the results. The results show that DDS-POBL works better than PSO, CoDA, MHDA, NaFA, and CMA-ES and gives very competitive results when compared to INMDA and EEGWO. Moreover, the parameter calibration application of the Xinanjiang model shows the effectiveness of the DDS-POBL in the real optimization problem. 1. Introduction The rapid development of productivity of human society has brought a great demand for optimization algorithms. Obtaining a good solution to the complex optimization problems in the real world becomes the specialized task for the optimization algorithms. Traditional optimization algorithms such as Newton’s method and the gradient method, which are based on mathematical theory, can hardly solve these complex optimization problems due to the extreme computation burdens. Therefore, highly efficient optimization algorithms have become the focus of research in recent years. The metaheuristic algorithm inspired from various phenomena of nature is one of the prevailing highly efficient algorithms. The biggest characteristic of these algorithms is to continuously evaluate candidate solutions through multiple iterations and try to improve upon these solutions. These metaheuristic algorithms are usually classified into two main categories [1]: single solution-based heuristic global search algorithms and population-based heuristic algorithms. Some of the famous single solutionbased heuristic global search algorithms are simulated annealing (SA) [2], threshold accepting method (TA) [3], microcanonical Annealing (MA) [4], tabu search (TS) [5], guided local search (GLS) [6], and dynamically dimensioned search (DDS) [7, 8]. Population-based ones include evolutionary algorithms (EA) [9], genetic algorithms (GA) [10], particle swarm optimization (PSO) [11], dragonfly algorithm 2 (DA) [12, 13], and shuffled complex evolution (SCE) algorithms [14]. According to the No Free Lunch (NFL) theorem [15], it is hard for researchers to propose a metaheuristic algorithm that is best suited for solving all optimization problems. That is to say, a particular algorithm may show very promising solutions only on certain problems but not on others. From this view, both the single solution-based heuristic global search algorithms and population-based heuristic algorithms have their respective strengths and weaknesses. The main trouble they all encounter is that the rates of convergence are very low, thus bringing them both a high computing burden and low results accuracy and limiting their applications in the real world. This study will focus on single solution-based heuristic global search algorithms especially the DDS algorithm and try to rectify its slow convergence speed and low solutions accuracy. The dynamically dimensioned search (DDS) algorithm, introduced by Tolson and Shoemaker [7], provides a relatively new potential for the family of single solution-based heuristic global search algorithms. At the initial stages of iteration, the algorithm is mainly based on the global search and is converted to local search at the later stages of iteration. This special search mechanism of the DDS algorithm is achieved by dynamically and probabilistically reducing the number of dimensions in the neighborhood [7]. Different versions of DDS have been proposed and successfully applied to practical engineering optimization problems such as the hybrid discrete dynamically dimensioned search (HD-DDS) which was used to solve discrete, single-objective, constrained water distribution system (WDS) design problems [8], the modified dynamically dimensioned search (MDDS) which was presented to optimize the parameter for distributed hydrological model [16], the DDS algorithm which was used to automate the calibration process of an unsteady river flow model [17], the Pareto archived dynamically dimensioned search (PA-DDS) which was applied for multi-objective optimization [18], and the combining filter method and dynamically dimensioned search which was designed for constrained global optimization problems [19]. Although the DDS algorithm partly overcomes the common drawback of single solution-based search algorithms to some extent, it does not still provide an ideal solution to address the poor and slow convergence of the global optimum in the best case or an acceptable local optimum in the worst case completely. The drawbacks of the DDS algorithm, by which the g (...truncated)