A high-speed interval type 2 fuzzy system approach for dynamic parameter adaptation in metaheuristics

Abstract

Fuzzy dynamic adaptation of parameters in meta-heuristic algorithms has recently been shown to provide an improvement in efficiency with respect to meta-heuristic algorithms with static parameters. However, executing a fuzzy inference in each iteration represents an increase in the computational cost, and this is even more critical in the case of using Type-2 Fuzzy Logic systems. On the other hand, fuzzy dynamic adaptation with Type-2 Fuzzy Logic has shown better performance when compared with respect to Type-1 Fuzzy Logic in diverse areas of application; therefore, the goal of this paper is aimed at reducing the computational cost of Type-2 Fuzzy Logic processing for dynamic adaptation of parameters in meta-heuristic algorithms. To reduce the computational cost of processing the Interval Type-2 Fuzzy system for dynamic adaptation of metaheuristic parameters, the use of an approximation to the Continuous Karnik–Mendel method (CEKM) is proposed. The proposed approach provides an analytical approximation to the CEKM method, in this way reducing the computational cost of evaluating the Interval Type-2 Fuzzy System. The performance of the proposed approach was tested with five benchmark functions and with one benchmark control problem. The proposed approach was tested with two different meta-heuristic algorithms, the Differential Evolution algorithm (DE) and the Harmony Search algorithm (HS), in both cases achieving a reduction in the computational cost, while maintaining the performance with respect to the Type-2 Dynamic adaptation of parameters with the conventional type reduction methods.

Introduction

The solution to complex optimization problems represents a challenge for conventional methods, this is because these problems are usually not linear and with a high number of dimensions, and as a consequence meta-heuristic algorithms are a good alternative for this kind of problems. Meta-heuristic methods provide an interesting approach to solve optimization problems, and many of them are inspired by organism behavior, natural phenomena, physics laws, etc. Some representative examples of these methods can be found in Ali and Hassanien (2016), Blum and Roli (2008) and Caserta and Voß (2009).

Meta-heuristic algorithms are iterative methods, and their behavior is determined by their corresponding parameters and equations. However, their performance can be improved by dynamic adaptation of parameters. Based on recent works on meta-heuristic algorithms (Caraveo et al., 2016, Bernal et al., 2017, Amador-Angulo and Castillo, 2016, Peraza et al., 2016, Castillo et al., 2016), it has been shown that is possible to use Fuzzy Inference Systems for realizing an efficient dynamic adaptation of parameters and improving in this way the corresponding optimization performance. In a general way we can say that meta-heuristic methods intelligently combine different concepts to properly explore and exploit the search space and are currently applied to real-life problems such as: engineering problems, robotics, medicine, combinatorial optimization, just to name a few. Some advantages that meta-heuristics offer are: general purpose algorithms, great success in practice, easily implementable, easily parallelizable and among the disadvantages that we can mention are: approximate algorithms, not exact, are non-deterministic (probabilistic), there is not always an established theoretical basis. For this paper, two meta-heuristics are used for experimentation, the first is the harmony search algorithm created in 2001 and the second is the differential evolution algorithm created in 1994. Both algorithms have recently been applied in different types of problems. The alternative to meta-heuristics are the derivative-based (gradient-based algorithms), and these methods use the gradient of the objective function to perform the optimization. These methods are commonly applied in control optimization, and some examples of these methods are: steepest descent or conjugate gradient. However, the disadvantage of these methods is the higher possibility of finding local minima, which is a common issue in complex problems and this is the reason why derivative-free methods are receiving increasing attention in control optimization.

The main contribution of the paper is the implementation of a faster approach for type reduction in order to accelerate the decision making for Interval Type-2 Fuzzy Inference Systems (IT2 FIS) for Dynamic parameter adaptation in meta-heuristics. In this regard, an analysis of the improvement of reducing the computational cost of IT2 FIS in the performance of optimization problems computed with meta-heuristics with dynamic parameters is presented.

The organization of the present work is as follows: Section 2 presents a background of Interval Type 2 Fuzzy Logic, including the explanation of implication under uncertainty in the IT2 FL model and the Type-Reduction algorithms, Section 3 presents a background on Meta-heuristic algorithms, in particular the description of the HS and DE algorithms, Section 4 describes the dynamic parameter adaptation and the proposed approach for computational cost reduction, Section 5 reports the experimental results focused on verifying the improvement provided by the proposed approach, and finally, Section 6 contains the conclusions of the performed experiments.