Research paperBayesian Interactive Search Algorithm: A New Probabilistic Swarm Intelligence Tested on Mathematical and Structural Optimization 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.
Section snippets
Interactive Search Algorithm (ISA)
In the current section, Interactive Search Algorithm (ISA) is concisely described. The ISA method is the gradient-free and population-based search algorithm which has been introduced by Mortazavi et al. [40]. Each agent in the ISA method, based on its tendency factor (τi) uses either tracking or interacting phase to update its location. In the tracking phase, the agent searches the vicinity of the locations spotted three certain agents as the best agent (XG), the weighted agent (XW), and best
Bayesian Interactive Search Algorithm (BISA)
Based on the given definitions in the previous section, in the ISA approach the tendency factor reveals that, the interacting search phase accounts thirty percent of the search capacity of the ISA method. However, studies show that in the more complex problems decreasing the contribution of this phase does not considerably affects the search performance of the method [45,46]. Also, the ISA applies a simple random approach to adjust its search behavior that does not take into account the
Numerical Tests
In this section the performance of the proposed BISA is tested on a suite of unconstrained mathematical functions and constrained structural problems with both discrete and continuous variables. The algorithms are run on the computer equipped with the intel™-i7 CPU and 12 MB of installed RAM. It should be noted that, the main purpose of the current study is to evaluate achieved enhancements for BISA over it parent method (i.e. ISA), but to provide more insight into BISA's performance, in
Discussion on the Proposed Bayesian Mechanism
The proposed Bayesian formula dynamically updates the memory impact factor (ω) based on the collecting evidence during the optimization process. As described before (see section 3.1.2) activation period (k) specifies that the how often Bayesian regulator mechanism should be activated (i.e. it should be activated in each k iterations). So, if k is small, the contribution of the memory concept is updated more frequently. However, if k is high, the contribution of the memory concept is updated
Conclusion
The current study deals with developing a new forecasting Bayesian strategy to enhance the search capability of Interactive Search Algorithm (ISA). The introduced method is named as Bayesian Interactive Search Algorithm (BISA). During the optimization process, the defined Bayesian module of the proposed BISA gathers required information, and based on its activation factor (k), it is periodically engaged and tunes the algorithm's search behavior. To acquire an optimal value for the activation
Authorship contributions
Please indicate the specific contributions made by each author (list the authors’ initials followed by their surnames, e.g., Y.L. Cheung). The name of each author must appear at least once in each of the three categories below.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
All persons who have made substantial contributions to the work reported in the manuscript (e.g., technical help, writing and editing assistance, general support), but who do not meet the criteria for authorship, are named in the Acknowledgements and have given us their written permission to be named. If we have not included an Acknowledgements, then that indicates that we have not received substantial contributions from non-authors.
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