Bayesian Interactive Search Algorithm: A New Probabilistic Swarm Intelligence Tested on Mathematical and Structural Optimization Problems

Highlights

• Introducing a new probabilistic-based metaheuristic search technique

• Employing a hierarchical Bayesian formulation

• Evaluating the new algorithm on solving distinct optimization problems

Abstract

Metaheuristic algorithms are general optimization techniques that demonstrate remarkable performance in solving different classes of optimization problems. However, equipping their stochastic search mechanisms with auxiliary logical strategies can still increase their search capability. Based on this fact, in the current study, the search performance of the Interactive Search Algorithm (ISA), as a metaheuristic search method, is improved by adding a new Bayesian regulator strategy to adjust its search behavior. In this regard, the search patterns of the ISA method are unified and classified according to the memory and learning concepts. Subsequently, during the optimization process, the developed Bayesian module dynamically regulates the ratio of the exploration and exploitation search behaviors by tuning the effect of memory concept. The recent technique is named Bayesian Interactive Search Algorithm (BISA), and its search performance tested on a suite of unconstraint mathematical functions and constrained engineering problems. Acquired outcomes indicate that the proposed BISA considerably speeds up the convergence rate, and improves the stability of the process as well as the accuracy of the solutions, for both engineering and mathematical problems.

Introduction

Generally, the metaheuristic algorithms are population-based and gradient-free search techniques inspired by the natural rules or physical and/or social phenomena [1], [2], [3], [4], [5]. They do not demand any continuous objective function and its gradient information to determine the search direction and step sizes. This feature makes them convenient techniques to solve complex optimization problems in which it is very difficult (or even impossible) to define the continuous and differentiable objective functions. On this subject, in the last decades, these methods are broadly employed to solve different mathematical and engineering optimization problems [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].

Based on the existing studies, although each of these methods has its weaknesses and strengths in solving different problems, they are mostly suffered from the lack of a proper trade-off between exploration and exploitation balance specially in the case of more complicated optimization problems. To mitigate this shortcoming, there are several works addressed in the literature [19], [20], [21], [22], [23], [24], [25], [26]. Some of them hybridize the affirmative local and global search capabilities of two or more methods, while some other add the auxiliary module(s) to the algorithm to tune its search behavior. As the instances, Wen-Jun and Xiao-Feng (2003) combined Deferential Evolution (DE) with Particle Swarm Optimization (PSO) [27], Deep et al. (2009) hybridized Genetic Algorithm (GA) and PSO with Quadratic Approximation (QA) [28,29], Barroso et al. (2017) proposed a hybrid GA-PSO to optimize the laminated composites [30]. Yalaoui et al. (2013) applied a fuzzy programing in solving scheduling problem [31]; Nobile et al. (2018) introduced a fuzzy formulation for adjusting the search behavior of PSO algorithm [32]. To provide a deeper insight into the previous studies, Table 1 chronologically summarizes some relevant work done in the last decade.

Interactive Search Algorithm (ISA) is one these methods which is the recently developed. It employs a search strategy consists of two distinct search patterns called tracking and interacting phases. The proposed ISA applies tendency factor as an internal parameter to adjust the balance between these two phases. This algorithm shows a good performance on solving mathematical and structural problems [41,44]. However, assessing the work mechanism of ISA reveals that it still has two drawbacks. First, to increase the search capacity of this method, its internal setting parameter (i.e. tendency factor) should be determined for the current optimization problem performing a series of costly sensitivity analyses. Second, in the interacting phase of the algorithm a simple random search direction is provided which cannot accomplish the required local search particularly in the case of complex optimization problems.

In the current study, initially the search patterns of the ISA algorithm (i.e. tracking and interacting) are combined together. Then, the components of the unified formulation are divided into memory and learning concepts. Consequently, a Bayesian regulator formulation is developed to tuned the contribution level of the memory concept in the search process. Such that, increasing and decreasing the impact of the memory concept respectively amplifies the exploration and exploitation search behaviors of the algorithm. The recent method is named Bayesian Interactive Search Algorithm (BISA). Bayesian regulating mechanism of the proposed BISA provides two main advantages in comparison with conventional approaches (e.g. fuzzy-based mechanisms). First, it is based on probabilistic rules and its developing/application does not require any prior user knowledge about the problem and/or the how the search algorithm works. Second, it dynamically adjusts the algorithm search behavior based on the governing conditions of the current problem, therefore it gives a self-adaptive property to the algorithm. Consequently, the search performance of the proposed BISA is tested on a suite of different constrained and unconstrained optimization problems with both continuous and discrete variables and the acquired results are reported and discussed.

The rest of this study is organized as follows. In the next section, ISA method is briefly described. In section 3, the proposed BISA and its Bayesian decision mechanism are described in detail. The Section 4 is devoted to test the search performance of proposed BISA on distinct mathematical and engineering optimization problems. In section 5, to give more insight about introduced method, its important features are discussed in more detail. Consequently, a brief conclusion on achievements and observations is given in the last section.