Elsevier

Information Sciences

Volume 450, June 2018, Pages 246-266
Information Sciences

A Hybrid Harmony search and Simulated Annealing algorithm for continuous optimization

https://doi.org/10.1016/j.ins.2018.03.042Get rights and content

Abstract

Harmony search is a powerful metaheuristic algorithm with excellent exploitation capabilities but suffers a very serious limitation of premature convergence if one or more initially generated solutions/harmonies are in the vicinity of local optimal. In order to remove this limitation this paper proposes a novel algorithm based on hybridization of Harmony search and Simulated Annealing called HS-SA to inherit their advantages in a complementary way. Taking the inspiration from Simulated Annealing the proposed HS-SA algorithm accepts even the inferior harmonies with a probability determined by parameter called Temperature. The Temperature parameter is initially kept high to favor exploration of search space and is linearly decreased to gradually shift focus to exploitation of promising search areas. The performance of HS-SA is tested on IEEE CEC 2014 benchmark functions and real life problem from computer vision called Camera Calibration problem. The numerical results demonstrate the superiority of the proposed algorithm.

Introduction

Optimization is defined as the process of selecting the best option from a set of available alternatives. Every process has the potential to be optimized and many challenging problems in business, economics, science and engineering can be formulated as an optimization process. For example, the objective of formulated optimization problems can be the maximization of profit and/or quality, or minimization of time, cost and risk.

Many real life optimization problems are complex and thus difficult to solve in an exact manner within reasonable amount of time. The classical optimization methods have the limitation of being highly sensitive to the initial guess and may frequently converge to a local optimum. Metaheuristic algorithms eradicate some of the afore-mentioned difficulties and are quickly replacing the classical methods in solving complex non linear optimization problems. Metaheuristic algorithms typically intend to find a reasonably good solution close to optimal in reasonable amount of computational time. During the last few decades, several metaheuristic algorithms have been proposed including Genetic Algorithms, Particle Swarm Optimization, Evolutionary Programming, Genetic Programming, Differential Evolution, Ant Colony Optimization, Evolutionary Strategies (ES). Developed by Geem et al. in 2001 the Harmony Search (HS) algorithm is the musicians inspired metaheuristic algorithm [1] that has found applications in diverse fields.

The efficiency of a metaheuristic algorithms depend on the extent of balance between diversification and intensification during the course of the search. Intensification also referred as exploitation is the ability of an algorithm to exploit the search space in the vicinity of the current good solution while diversification also called exploration is the process of exploring the new regions of search space thus allows dissemination of the new information. Proper balance between these two contradicting characteristics is a must to enhance the performance of an algorithm.

Based on the idea of balanced intensification and diversification this article describes a new variant of HS that synergistically incorporates some features of another very powerful optimization algorithm called Simulated Annealing with a view of improving the accuracy and robustness of Harmony Search. Hybridization of intelligent systems is a promising research field of modern artificial intelligence concerned with the development of next generation intelligent systems. Hybrid intelligent systems are becoming popular due to their capabilities in handling many complex real world problems involving uncertainty, imprecision and vagueness [2].

Taking the inspiration from Simulated Annealing the HS-SA algorithm accepts even the suboptimal solutions than the ones stored in its Harmony memory with some probability determined by parameter Temperature. The probability of accepting the suboptimal solutions is gradually decreased by decreasing Temperature parameter during execution so as to enhance the exploration capabilities of the algorithm in the earlier generations and exploitation towards the later generations.

To evaluate the performance of HS-SA comprehensively, it is tested on all the thirty IEEE CEC 2014 benchmark functions [3] and a real life problem from computer vision called Camera Calibration- a highly nonlinear twelve dimensional optimization problem. The numerical results obtained demonstrate the superiority of the proposed algorithm in terms of accuracy and robustness.

Section snippets

Harmony Search and Simulated Annealing

In this section an introduction to Harmony Search and Simulated Annealing is provided.

Proposed Hybrid Harmony Search and Simulated Annealing (HS-SA) algorithm

Harmony Search is a powerful metaheuristic algorithm with excellent exploitation capabilities, however it has a very serious limitation of getting stuck in local optimal usually referred as premature convergence if the initially selected harmonies are in the vicinity of local optimal. In order to remove this limitation HS algorithm is hybridized by proposing HS-SA algorithm so as to increase exploration particularly in the beginning of execution to escape local optima.

The success of a meta

Numerical experiments on CEC 2014 benchmark suite

In this section, the performance of the proposed HS-SA algorithm is evaluated on IEEE CEC 2014 Benchmark functions [3]. The HS-SA algorithm has been compared with standard HS and Simulated Annealing. The experimentation has been carried out on all the IEEE CEC 2014 Benchmark functions using 10 and 30 dimensions. As per the instructions of test suite every problem is tested with 51 independent runs.

The parameter setting adopted for standard HS has been taken from [19] and is shown in Table 1

Comparison with other meta heuristic algorithms

In this section proposed HS-SA algorithm is compared with several state-of-the-art meta heuristic algorithms including Adaptive particle swarm optimization (APSO) [28], Social Spider Optimization algorithm (SSO) [29], Differential Evolution (DE) [30], Differential Evolution with a successful parent selecting framework (DE-SPS) [30], Symbiotic organisms search (SOS) [31], Gravitational Search Algorithm (GSA) [32], Cuckoo Search Algorithm (CS) [33], Evolution Strategy with Covariance Matrix

Camera calibration

The problem of camera calibration has been studied extensively in photogrammetry and computer vision community because of its important applications such as vehicle guidance, robotic navigation and 3D-reconstruction. Camera calibration problem deals with finding the geometrical relationship between the 3D scene and its 2D images taken by the camera(s). It defines exactly how the scene has been projected by the camera to result in the given image(s). Camera calibration involves:

  • 1.

    Determination of

Experimentation on camera calibration

This section presents extensive experiments with HS-SA algorithm so as to evaluates its performance on camera calibration problem. The simulations has been carried out using varying number of control points. The accuracy is evaluated by calculating camera errors and pixel errors. Pixel error is defined as the root mean square Euclidean distance between the observed pixel positions and the corresponding re projected pixel positions calculated using estimated parameters values, whereas camera

Conclusion

This manuscript introduces a hybrid variant of Harmony Search algorithm for continuous optimization problems with the aim to exhibited the desired behavior of exploring the search space at the earlier iterations and exploiting good solutions towards the later iterations. This is achieved by initially setting the parameter Temperature to a high value so as to favor inferior moves. The Temperature parameter is linearly reduced to gradually shift the focus from exploration of search space to

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