An analysis of Harmony Search for solving Sudoku puzzles

Harmony search (HS) algorithm is a heuristic optimization algorithm which is newly developed in recent years. In this paper, according to the shortcomings of the existing harmony search algorithm, an improved adaptive harmony search algorithm (IAHS) is proposed. In the IAHS algorithm, the adaptive parameters HMCR and PAR and BW are used to adjust the global and local search, so as to improve the robustness and speed of convergence of the algorithm. IAHS algorithm is tested by five standard benchmark functions and contrasted with HS、HIS and GHS algorithm. Experimental results demonstrated that the proposed IAHS algorithm has the favorable abilities of accuracy and escaping local minimums. Introduction Harmony search (HS) algorithm is a new heuristic optimization algorithm which put forward by Geem, etc in 2001 [1]. Similar with Particle swarm optimization (PSO) algorithm, HS algorithm is based on the music improvisation process. In the process, musicians repeatedly adjusting the tone of each instrument in the band, and finally get a good harmony. HS algorithm has the advantages of simple model, strong randomicity, good ergodicity and global search ability. HS and its improved algorithm has been successfully applied to many practical optimization problems [2-5], such as environmental economic load dispatch optimization, traffic path optimization, the optimization of water distribution system and fault location in distribution networks and other issues. Since the advent of harmony search algorithm, a series of achievements are made in various fields such as optimization problem: bus lines, water network design problem, the problem of reservoir scheduling, civil engineering problems. At present, this method has been widely applied in the problem of multi dimensional function optimization, pipeline optimization design, slope stability analysis etc. Research shows that the HS algorithm in solving multidimensional function optimization problems show a genetic algorithm and simulated annealing algorithm to optimize the better. However, the standard HS exists some defects such as setting BW blindly harmony memory diversity gradually dissipating with iterations, falling into local optimum easily, low accuracy and so on[6].Therefore, in order to improve the performance of the HS, this paper propose an improved adaptive harmony search algorithm (IAHS) whose parameters are adjusted adaptively. IAHS algorithm is tested by five standard benchmark functions and contrasted with HS、HIS and GHS algorithm. Experimental results demonstrated that the proposed IAHS algorithm has the favorable abilities of accuracy and escaping local minimums. Standard harmony search algorithm Harmony search (HS) algorithm was inspired by the improvising process of composing a piece of music. In the play, each player generates a tone to constitute a harmony vector. If the harmony is better, write it down, so that the next time to produce better harmony. In the algorithm, the tones of music instrument are analogous to the decision variables (i 1, 2, , n) j xi   of the optimization problem, each harmony are analogous to the solution vector ( , , , ) 1 2 j j j j x x x xn   , aesthetic evaluation are analogous to the objective function   j f x , the musicians want to find the beautiful harmonies by defined by aesthetic evaluation, the researchers want to find the global optimal solution defined by objective function .HS algorithm contains a series of optimization International Conference on Computational Science and Engineering (ICCSE 2015) © 2015. The authors Published by Atlantis Press 54 factors, such as the harmony memory (HM), harmony memory size (HMS), harmony memory consideration rate (HMCR), pitch adjusting rate (PAR), bandwidth (BW) stores the feasible solution vector represents the probability is the probability of disturbing to the Aesthetic evaluation is issued by the decision, as the value of the objective function brief description is given of the above statement Table 1 Analog elements best condition be evaluated by

The harmony search (HS) method is an emerging meta-heuristic optimization algorithm inspired by the natural musical performance process, which has been extensively applied to handle numerous optimization problems during the past decade. However, it usually lacks of an efficient local search capability, and may sometimes suffer from weak convergence. In this paper, a memetic HS method, m-HS, with local search function is proposed and studied. The local search in the m-HS is based on the principle of bee foraging like strategy, and performs only at selected harmony memory members, which can significantly improve the efficiency of the overall search procedure. Compared with the original HS method and particle swarm optimization (PSO), our m-HS has been demonstrated in numerical simulations of 16 typical benchmark functions to yield a superior optimization performance. The m-HS is further successfully employed in the optimal design of a practical wind generator.

The Harmony Search (HS) method is an emerging meta-heuristic optimization algorithm, which has been extensively applied to handle numerous optimization problems during the past decade. However, it usually lacks of an efficient local search capability, and may sometimes suffer from weak convergence. In this paper, a memetic HS method, m-HS, with local search function is proposed and studied. The local search in the m-HS is inspired by the principle of bee foraging, and performs only at selected harmony memory members, which can significantly improve the efficiency of the overall search procedure. Compared with the original HS method, our m-HS has been demonstrated in numerical simulations of 16 typical benchmark functions to yield a superior optimization performance.

Harmony search (HS) algorithm was applied to solving Sudoku puzzle. The HS is an evolutionary algorithm which mimics musicians' behaviors such as random play, memory-based play, and pitch-adjusted play when they perform improvisation. Sudoku puzzles in this study were formulated as an optimization problem with number-uniqueness penalties. HS could successfully solve the optimization problem after 285 function evaluations, taking 9 seconds. Also, sensitivity analysis of HS parameters was performed to obtain a better idea of algorithm parameter values.

The dam is the wall that holds the water in, and the operation of multiple dams is complicated decisionmaking process as an optimization problem (Oliveira & Loucks, 1997). Traditionally researchers have used mathematical optimization techniques with linear programming (LP) or dynamic programming (DP) formulation to find the schedule. However, most of the mathematical models are valid only for simplified dam systems. Accordingly, during the past decade, some meta-heuristic techniques, such as genetic algorithm (GA) and simulated annealing (SA), have gathered great attention among dam researchers (Chen, 2003) (Esat & Hall, 1994) (Wardlaw & Sharif, 1999) (Kim, Heo & Jeong, 2006) (Teegavarapu & Simonovic, 2002). Lately, another metaheuristic algorithm, harmony search (HS), has been developed (Geem, Kim & Loganathan, 2001) (Geem, 2006a) and applied to various artificial intelligent problems, such as music composition (Geem & Choi, 2007) and Sudoku puzzle (Geem, 2007). The HS algorithm has been also applied to various engineering problems such as structural design (Lee & Geem, 2004), water network design (Geem, 2006b), soil stability analysis (Li, Chi & Chu, 2006), satellite heat pipe design (Geem & Hwangbo, 2006), offshore structure design (Ryu, Duggal, Heyl & Geem, 2007), grillage system design (Erdal & Saka, 2006), and hydrologic parameter estimation (Kim, Geem & Kim, 2001). The HS algorithm could be a competent alternative to existing metaheuristics such as GA because the former overcame the drawback (such as building block theory) of the latter (Geem, 2006a). To test the ability of the HS algorithm in multiple dam operation problem, this article introduces a HS model, and applies it to a benchmark system, then compares the results with those of the GA model previously developed.

Metaheuristic profiling to assess performance of hybrid evolutionary optimization algorithms applied to complex wellbore trajectories

To overcome the drawbacks of the harmony search (HS) algorithm and further enhance its effectiveness and efficiency, an improved differential HS (IDHS) is proposed to solve numerical function optimization problems. The proposed IDHS has a novel improvisation scheme that integrates DE/best/1/bin and DE/rand/1/bin from the differential evolution (DE) algorithm to enhance its local search and exploration capabilities and a new pitch adjustment rule that benefits from the best solution in the harmony memory to increase its convergence speed. With dynamically adjusted parameters, the proposed IDHS can balance exploitation and exploration throughout the search process. The numerical results of an experiment with classic testing functions and those of a comparative experiment show that IDHS outperforms eight algorithms in the HS family and three widely used population-based algorithms in different families, including DE, particle swarm optimization, and improved fruit fly optimization algorithm. IDHS demonstrates fast convergence and an especially good capability to handle difficult high-dimensional optimization problems.

Harmony Search (HS) and Teaching-Learning-Based Optimization (TLBO) as new swarm intelligent optimization algorithms have received much attention in recent years. Both of them have shown outstanding performance for solving NP-Hard optimization problems. However, they also suffer dramatic performance degradation for some complex high-dimensional optimization problems. Through a lot of experiments, we find that the HS and TLBO have strong complementarity each other. The HS has strong global exploration power but low convergence speed. Reversely, the TLBO has much fast convergence speed but it is easily trapped into local search. In this work, we propose a hybrid search algorithm named HSTLBO that merges the two algorithms together for synergistically solving complex optimization problems using a self-adaptive selection strategy. In the HSTLBO, both HS and TLBO are modified with the aim of balancing the global exploration and exploitation abilities, where the HS aims mainly to explore the unknown regions and the TLBO aims to rapidly exploit high-precision solutions in the known regions. Our experimental results demonstrate better performance and faster speed than five state-of-the-art HS variants and show better exploration power than five good TLBO variants with similar run time, which illustrates that our method is promising in solving complex high-dimensional optimization problems. The experiment on portfolio optimization problems also demonstrate that the HSTLBO is effective in solving complex read-world application.

A parallel membrane inspired harmony search for optimization problems: A case study based on a flexible job shop scheduling problem

Identifying Optimal Design of Office Buildings Using Harmony Search Optimization Algorithm

In the last few years, there has been explosive growth in the application of Harmony Search (HS) in solving NP-complete problems in computer science. The success of the HS algorithm in finding relatively good solutions to these problems discriminates it as an affirmative alternative to other conventional optimization techniques. This chapter surveys the existing literature on the application of HS in combinatorial optimization problems. We begin by presenting HS based algorithms for solving problems such as Sudoku puzzle, music composition, orienteering problem, and vehicle routing. We then turn to solve a multicast routing problem with two constraints (i.e. bandwidth and delay constraints). Finally, we show how to apply HS to a clustering problem.

In evolutionary multi-objective optimization, an evolutionary algorithm is invoked to solve an optimization problem involving concurrent optimization of multiple objective functions. Many techniques have been proposed in the literature to solve multi-objective optimization problems including NSGA-II, MOEA/D and MOPSO algorithms. Harmony Search (HS), which is a relatively new heuristic algorithm, has been successfully used in solving multi-objective problems when combined with non-dominated sorting (NSHS) or the breakdown of the multi-objectives into scalar sub-problems (MOHS/D). In this paper, the performance of NSHS and MOHS/D is enhanced by using a previously proposed hybrid framework. In this framework, the diversity of the population is measured every a predetermined number of iterations. Based on the measured diversity, either local search or a diversity enhancement mechanism is invoked. The efficiency of the hybrid framework when adopting HS is investigated using the ZDT, DTLZ and CEC2009 benchmarks. Experimental results confirm the improved performance of the hybrid framework when incorporating HS as the main algorithm.

Sudoku is a logical puzzle that has achieved international popularity. Given this, there have been a number of computer solvers developed for this puzzle. Various methods including genetic algorithms, simulated annealing, particle swarm optimization and harmony search have been evaluated for this purpose. The approach described in this paper combines human intuition and optimization to solve Sudoku problems. The main contribution of this paper is a set of heuristic moves, incorporating human expertise, to solve Sudoku puzzles. The paper investigates the use of genetic programming to optimize a space of programs composed of these heuristics moves, with the aim of evolving a program that can produce a solution to the Sudoku problem instance. Each program is a combination of randomly selected moves. The approach was tested on 1800 Sudoku puzzles of differing difficulty. The approach presented was able to solve all 1800 problems, with a majority of these problems being solved in under a second. For a majority of the puzzles evolution was not needed and random combinations of the moves created during the initial population produced solutions. For the more difficult problems at least one generation of evolution was needed to find a solution. Further analysis revealed that solution programs for the more difficult problems could be found by enumerating random combinations of the move operators, however at a cost of higher runtimes. The performance of the approach presented was found to be comparable to other methods used to solve Sudoku problems and in a number of cases produced better results.

MEMPSODE: A global optimization software based on hybridization of population-based algorithms and local searches

So far, water distribution networks have been hydraulically analyzed by using various techniques such as Hardy Cross method, Newton-Raphson method, linear theory, and Todini algorithm. This study proposes a novel technique for the analysis of water distribution networks using a phenomenon-mimicking optimization algorithm, harmony search (HS). The HS algorithm mimics the behaviors of musicians in improvisation process, where they play musical notes based upon randomness, experience, or variation of experience. In order to use the HS optimization algorithm, the network analysis is formulated as an optimization problem where energy conservation equation is incorporated in an objective function and mass conservation equations are incorporated in constraints. The proposed HS model is successfully applied to two example networks.