Evolutionary algorithm with multiobjective optimization technique for solving nonlinear equation systems
Introduction
Nonlinear equation systems (NESs) appear in many areas such as robotics [1], physics [2], engineered materials [3], chemical processing [4], economics [5], and so on [6], [7]. In general, a NES can be defined as follows [8]:where denotes a D-dimensional decision vector, is the decision space, the ith equation is , and m is the number of equations. S can be defined as follows:where and denote the lower and upper bounds of , respectively.
If , then is called an optimal solution (or a root) of a NES. Generally speaking, a NES may contain multiple optimal solutions, and all these optimal solutions are equally important. Fig. 1 shows that a NES (i.e., F03) has two nonlinear equations and 15 optimal solutions. The main task of solving NESs is to locate these optimal solutions simultaneously in a single run.
To solve NESs, many classical methods have been proposed based on traditional optimization [9], [10], [11], [12], such as steepest descent method, Newton method, conjugate gradient method, quasi-Newton method and so on. However, these methods have some disadvantages, mainly involving the initial point, differentiability, and convergence. Specifically, the initial point plays an important role in these methods. If the initial point is not selected properly, the result may be poor. These classical methods usually require gradient information. What’s more, only one optimal solution can be found by these methods in a single run. However, evolutionary algorithms (EAs) can overcome the drawbacks of the traditional methods. They may locate multiple optimal solutions in a single run.
In general, the methods of solving NESs by EAs consist of two steps. In the first step, a transformation technique is designed to transform a NES into an optimization problem. In the next step, the transformed optimization problem is solved by an optimization algorithm.
Currently, the main transformation techniques fall into three categories: 1) the transformation technique based on single-objective optimization [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]; 2) the transformation technique based on constrained optimization [23]; 3) the transformation technique based on multiobjective optimization [24], [25], [26], [27].
A single-objective optimization problem is built through the first transformation technique as follows [16]:orCombined with this transformation technique, many algorithms are proposed to deal with NESs. For example, Liao et al. [21] proposed the memetic niching-based evolutionary algorithm to solve NESs and M. Ariyaratne et al. [28] developed a modified firefly algorithm for NESs.
In addition, the second transformation technique transforms a NES into a constrained optimization problem as follows [23]:where .
The third transformation technique transforms a NES into a multiobjective optimization problem as follows [24]:This transformation technique is a simple yet direct transformation method, which considers each equation in a NES as an objective function. However, the performance of this method decreases with the increase of the number of equations. The multiobjective optimization problem that has m [24] objectives may cause the “curse of dimensionality” if m is a big number. To solve this problem, Song et al. [25] presented an algorithm called MONES. In MONES, a NES is transformed into a biobjective optimization problem. This transformed problem can be divided into two parts: the system function and the location function. This biobjective optimization problem is given bywhere denotes the first decision variable.
However, since only one decision variable is used to design the location function, MONES may lose some optimal solutions. Based on this consideration, Gong et al. [27] proposed an algorithm called A-WeB where a weighted biobjective optimization is defined by the following equations:where denotes the jth decision variable, . is the weight vector.
In this paper, we present a new evolutionary algorithm with multiobjective optimization technique for NESs, called MOPEA. In MOPEA, a NES is transformed into a single-objective optimization problem as shown in Eq. (4). Moreover, a diversity indicator is introduced to keep the diversity of the population. The main contributions of this paper can be described as follows:
- (1)
To balance the diversity and the convergence, the multiobjective optimization technique is applied to solve NESs. Specifically, NSGA-II is employed to obtain a lot of candidate solutions, and an approximate fitness landscape can be obtained by all candidate solutions in this process.
- (2)
A K-means clustering-based selection strategy is used to divide all candidate solutions into several subregions. The selection strategy may screen out the promising solutions and accelerate the convergence of the algorithm. Subsequently, the local search is designed to locate the optimal solutions quickly.
The rest of this paper is organized as follows. The basic backgrounds about the multiobjective optimization and clustering are introduced in Section 2. Section 3 discusses the motivation of this work and describes the proposed algorithm in detail. The experimental studies and parameter analysis are presented in Section 4. Finally, Section 5 draws the conclusion of this paper.
Section snippets
Multiobjective optimization (MOP)
In general, the form of a MOP can be defined as follows [25], [29]:where denotes a decision vector with D decision variables and denotes the decision space. denotes an objective vector and denotes the objective space. M denotes the number of objectives.
For a MOP, two candidate solutions are compared by Pareto dominance. To be specific, let and be two candidate solutions, is said to Pareto dominate iff
Motivations
As mentioned above, this paper presents an algorithm for NESs based on multiobjective optimization technique. The motivations of the proposed algorithm mainly include the following two points:
- (1)
The classical methods for a NES have some weaknesses. For instance, the classical methods depend on the differentiability and are highly sensitive to the initial points. Furthermore, they are easy to get trapped in the local optimal solutions and obtain one optimal solution at each run. Based on these
Benchmark test functions
In this paper, 30 NESs (denoted as F01–F30) with different characteristics are chosen from the literature [17] to test the performance of different methods. Some of them are derived from the real world, such as multiple steady states problem (F08) [38], robot kinematics problem (F17) [39] and so on. Table 1 describes the information of 30 NESs. D is the number of the decision variables. LE and NE are the number of linear and nonlinear equations, respectively. NOP is the number of the known
Conclusion
Many real-world problems are NESs with multiple optimal solutions. It is a challenge to locate multiple optional solutions of NESs simultaneously in a single run. In this paper, MOPEA is presented for solving NESs. A NES is transformed into a single-objective optimization, and a diversity indicator is designed to keep the diversity of population. Then, NSGA-II is employed to obtain a lot of candidate solutions. Afterward, the selection strategy is applied to select the promising solutions from
CRediT author statement
Weifeng Gao: Writing - original draft. Yuting Luo: Writing - original draft. Jingwei Xu Writing - original draft. Shengqi Zhu: Writing - original draft.
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.
References (44)
Forward kinematics of planar parallel manipulators in the clifford algebra of p 2
Mech. Mach. Theory
(2002)- et al.
Extension of multimoora method with interval numbers: an application in materials selection
Appl. Math. Model.
(2016) - et al.
Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering
Swarm Evol. Comput.
(2012) - et al.
On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth- and fifth-order convergence
Appl. Math. Model.
(2016) - et al.
A new approach based on the newton’s method to solve systems of nonlinear equations
J. Comput. Appl. Math.
(2017) - et al.
Conjugate direction particle swarm optimization solving systems of nonlinear equations
Comput. Math. Appl.
(2009) - et al.
Particle swarm algorithm for solving systems of nonlinear equations
Comput. Math. Appl.
(2011) - et al.
A decomposition-based differential evolution with reinitialization for nonlinear equations systems
Knowl. Based Syst.
(2020) - et al.
Fuzzy neighborhood-based differential evolution with orientation for nonlinear equations system
Knowl. Based Syst.
(2019) - et al.
Memetic niching-based evolutionary algorithms for solving nonlinear equation system
Expert Syst. Appl.
(2020)
Solving systems of nonlinear equations using a modified firefly algorithm (modfa)
Swarm Evol. Comput.
Mixed fuzzy inter-cluster separation clustering algorithm
Appl. Math. Model.
On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods
Nonlinear Anal. Real World Appl.
Finding all solutions of nonlinear systems using a hybrid metaheuristic with fuzzy clustering means
Appl. Soft Comput.
A new filled function method for an unconstrained nonlinear equation
J. Comput. Appl. Math.
Calculation of critical points of thermodynamic mixtures with differential evolution algorithms
Ind. Eng. Chem. Res.
Solution of the mathematical model of a nonlinear direct current circuit using particle swarm optimization
Dyna
Joint range and angle estimation using mimo radar with frequency diverse array
IEEE Trans. Signal Process.
An adaptive range-angle-doppler processing approach for fda-mimo radar using three-dimensional localization
IEEE J. Sel. Top. Signal Process.
Solving nonlinear systems of equations by means of quasi-newton methods with a nonmonotone strategy
Opt.Meth. Softw.
A class of methods for solving nonlinear simultaneous equations
Math. Comput.
A newton-like method for nonlinear system of equations
Numer. Algorithms
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