Hybridizing sine cosine algorithm with differential evolution for global optimization and object tracking

Highlights

• A novel Hybrid SCA-DE algorithm is introduced for global optimization and object tracking.

• The proposed hybrid algorithm has better capability to escape from local optima with faster convergence.

• The performance of the Hybrid SCA-DE algorithm was better than with other state-of-the-art metaheuristic algorithms.

• The hybrid SCA-DE algorithm is applied for visual tracking as a real thought- provoking case study to demonstrate and verify the performance of this algorithm in practice.

Abstract

A new optimization algorithm called Hybrid Sine-Cosine Algorithm with Differential Evolution algorithm (Hybrid SCA-DE) is proposed in this paper for solving optimization problems and object tracking. The proposed hybrid algorithm has better capability to escape from local optima with faster convergence than the standard SCA and DE. The effectiveness of this algorithm is evaluated using 23 benchmark functions, which are divided into three groups: unimodal, multimodal, and fixed dimension multimodal functions. Statistical parameters have been employed to observe the efficiency of the Hybrid SCA-DE qualitatively and results prove that the proposed algorithm is very competitive compared to the state-of-the-art metaheuristic algorithms. The proposed algorithm is applied for object tracking as a real thought-provoking case study. To demonstrate the tracking ability of a Hybrid SCA-DE-based tracker, a comparative study of tracking accuracy and speed of the Hybrid SCA-DE-based tracker with four other trackers, namely, Particle Filter, Scale-invariant feature transform, Particle swarm optimization and Bat algorithm are presented. Comparative results show that the Hybrid SCA-DE-based tracker can robustly track an arbitrary target in various challenging conditions than the other trackers.

Introduction

Global optimization problems are continually unavoidable in current engineering and science fields. Mathematically, an optimization problem can be expressed as

$${}_{x\in {ℛ}^n}{}^{minimize}f{}_i\left(x\right),\left(i=\mathrm{1,2},\dots ,M\right)$$

$$\begin{array}{cc}\mathrm{s}\mathrm{.}\mathrm{t}& h_j\left(x\right)=0,\left(j=\mathrm{1,2},\dots ,J\right)\end{array}$$

$$g_k\left(x\right)\le 0,\left(k=\mathrm{1,2},\dots ,K\right)$$

Here $f_i\left(x\right),g_k\left(x\right)$ and $h_j\left(x\right)$ are design vector functions and

$$x={\left(x_1,x_2,\dots ,x_n\right)}^T$$

where x_i of x are termed decision or design variables, which are real discrete, continuous or combination of the two.

The $f_i\left(x\right)$ is termed cost or objective function, and for M = 1 case, there is a single objective. The area covered by the design variables is termed the search space or design space ${ℛ}^n$, while the area formed by the cost function results is termed as the response or solution space. The inequalities for g_k and equalities for h_j are termed as constraints.

Conventional mathematical and analytical methods are not suitable for solving some difficult optimization problems with complex characteristics such as highly non-linearity, multimodality, non-differentiability, with steep and flat search regions [1]. Therefore, nature-inspired heuristic algorithms are designed to tackle the difficult optimization problems. Genetic Algorithm (GA) may be the first and popular algorithm inspired by natural genetic variation and natural selection [2]. Particle swarm algorithm (PSO) was inspired by the social behavior of bird flocking or fish school [3], [4]. Artificial bee colony (ABC) which simulate the foraging behavior of bee swarm [5], [6]. Ant colony algorithm (ACO) [7] was another optimization algorithm inspired by the foraging behavior of ant colonies. Differential evolution (DE) [8], [9] is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. It currently stands out as a very attractive evolutionary algorithm for optimization in continuous search spaces, mainly for of its simplicity, a small number of parameters to tune and notable performance. Teaching–Learning-Based Optimization (TLBO) [10] algorithm works on the effect of the influence of a teacher on learners. Mine blast algorithm (MBA) [11] is based on the mine bomb explosion concept. Exchange market algorithm (EMA) [12] is inspired by the procedure of trading the shares on stock market. Group counseling optimization (GCO) [13] emulate the behavior of human beings in life problem solving through counseling within a group. Grasshopper Optimisation Algorithm (GOA) [14] mimics the behavior of grasshopper swarms in nature for solving optimization problems. These algorithms have been applied to various research areas and have gained a lot of success [15], [16], [17], [18], [19], [20], [21], [22]. However, up to now, there is no algorithm that performs well in all the fields. Some algorithms perform much better for some particular problems, while worse for other problems. Until now, how to design a new heuristic optimization algorithm for the optimization problem is still an open problem [23].

Different exploration and exploitation strategies are used in metaheuristic algorithms for high dimensionality optimization problems. Hybrid metaheuristic algorithm is the latest research trend for solving high dimensionality problems to overcome the poor exploration ability of one algorithm and poor exploitation ability of the other algorithm. There is a significant number of hybrid meta-heuristics in the literature such as: Hybrid CSO [24], PS–ABC [25], Hybrid spiral-dynamic bacteria-chemotaxis (HSDBC) algorithm [26], GACE [27], CA-MMTS [28], IGAL-ABC [29] and Hybrid TVAC-GSA-PSO [30] etc.

In this paper, a combination of a Sine-Cosine Algorithm (SCA) and a Differential Evolution (DE) method was presented for solving optimization problems and object tracking. The research is more concentrated to investigate the application of optimization algorithms to object tracking, due to the rapid growth of modern optimization algorithms [31], [32], [33], [34], [35], [36]. The assumptions made about the shape of distribution or noise of the system are not required for these optimization methods, which forms the major advantage for object tracking. Therefore it enables these methods as a potential method for precise solutions even in challenging uncertain environments. So, the proposed algorithm is applied for object tracking as a real thought-provoking case study. To demonstrate the tracking ability of Hybrid SCA-DE based tracker, the tracking performances of Hybrid SCA-DE, Particle Filter (PF) [37], Scale-invariant feature transform (SIFT) [38], Particle swarm optimization (PSO) [33] and Bat algorithm (BA) [32] are studied comparatively.

The organization of this paper is as follows. The basics of SCA and DE are briefly introduced in Section 2 and 3. The proposed algorithm is presented in Section 4. Section 5 deals with the evaluation of proposed algorithm using Twenty-three well-known benchmark functions. Application of SCA-DE for object tracking is introduced in Section 6 and is compared with PF, SIFT, PSO and BA is presented. Finally, the conclusion is given in Section 7.