A study of two penalty-parameterless constraint handling techniques in the framework of MOEA/D

Abstract

Penalty functions are frequently employed for handling constraints in constrained optimization problems (COPs). In penalty function methods, penalty coefficients balance objective and penalty functions. However, finding appropriate penalty coefficients to strike the right balance is often very hard. They are problems dependent. Stochastic ranking (SR) and constraint-domination principle (CDP) are two promising penalty functions based constraint handling techniques that avoid penalty coefficients. In this paper, the extended/modified versions of SR and CDP are implemented for the first time in the multiobjective evolutionary algorithm based on decomposition (MOEA/D) framework. This led to two new algorithms, CMOEA/D-DE-SR and CMOEA/D-DE-CDP. The performance of these new algorithms is tested on CTP-series and CF-series test instances in terms of the HV-metric, IGD-metric, and SC-metric. The experimental results are compared with NSGA-II, IDEA, and the three best performers of CEC 2009 MOEA competition, which showed better and competitive performance of the proposed algorithms on most test instances of the two test suits. The sensitivity of the performance of proposed algorithms to parameters is also investigated. The experimental results reveal that CDP works better than SR in the MOEA/D framework.

Introduction

This paper considers the following constrained multiobjective optimization problem (CMOP):

$$\begin{array}{cc}\mathrm{Minimize}\hfill & F(x)=(f_1(x),f_2(x),\dots ,f_m(x){)}^T;\hfill \\ \mathrm{Subject}\mathrm{to}\hfill & g_j(x)\ge 0,j=1,\dots ,p;\hfill \\ \hfill & l_k\le x_k\le u_k,k=1,\dots ,n;\hfill \end{array}$$

where $x=(x_1,\dots ,x_n{)}^T\in {\mathcal{R}}^n$ is an n dimensional vector of decision variables, F is the objective vector function that consists of m real-valued objective functions, and g_i(x) ≥ 0 are inequality constraints. The objective and constraint functions, f_i's and g_j's, could be linear or non linear real-valued functions. l_k and u_k are the lower and upper bounds (called bound constraints) of x_k, k = 1, …, n, respectively, which define the search region $\mathcal{S}=\{x=(x_1,\dots ,x_n{)}^T|l_k\le x_k\le u_k,k=1,\dots ,n\}$.

A solution which satisfies all constraints in (1) is called a feasible solution. More specifically, if for a solution x = (x₁, …, x_n)^T, g_j(x) ≥ 0, j = 1, …, p and l_k ≤ x_k ≤ u_k, k = 1, …, n, then it is called a feasible solution. The set of all feasible solutions is called the feasible region. Mathematically, we can write:

$$\mathcal{F}=\{x\in \mathcal{S}\subset {\mathcal{R}}^n|g_j(x)\ge 0,j=1,\dots ,p\}.$$

However, If a solution is not feasible, we call it infeasible. The set of all infeasible solutions is called the infeasible region.

The feasible attainable objective set (AOS) can be defined as $\{F(x)|x\in \mathcal{F}\}$.

More often, the objectives in (1) contradict one another. Thus, a single solution in the feasible search region could not minimize all the objectives simultaneously. Instead, a set of optimal compromising/tradeoff solutions that satisfy all constraints (i.e., feasible solutions) is desired. The best tradeoffs among the objectives can be defined in terms of Pareto-optimality [1], [2], [3].

A solution x is said to Pareto-dominate or simply dominate another solution y, mathematically denoted as x ⪯ y, if f_i(x) ≤ f_i(y), ∀i = 1, …, m and f_j(x) < f_j(y) for at least one j ∈ {1, …, m}.¹ This definition of domination is sometimes referred to as a weak dominance relation. A solution $x^{*}\in \mathcal{F}$ is Pareto-optimal to (1) if there is no solution $x\in \mathcal{F}$ such that F(x) ⪯ F(x^*). F(x^*) is then called a Pareto-optimal (objective) vector. In other words, any enhancement in a Pareto-optimal solution in one objective must lead to degradation to at least one other objective. The set of all Pareto-optimal solutions is called the Pareto set (PS) in the decision space and Pareto front (PF) in the objective space [1].

In multiobjective evolutionary algorithms (MOEAs), better fitness values are mostly assigned to those individuals that are near the PF and in a less crowded region in the objective space. The two fundamental schemes for fitness assignment in MOEAs are Pareto-based ranking and decomposition [4].

In Pareto-based ranking, individuals are compared based on Pareto dominance. As a result, a scalar value (rank) is assigned to each individual in the population. Then, traditional selection operators can be used. The representative Pareto-based MOEAs include MOGA [5], NPGA [6], PAES [7], SPEA-II [8], and NSGA-II [9].

In decomposition based fitness assignment schemes, all individuals in the population are compared and ranked with respect to a weighted scalar optimization function. This function can be acquired by linearly or nonlinearly aggregating objectives of a CMOP. Contrary to Pareto-based ranking, here the fitness value of each individual is independent of other individuals in the population.

Different weighted scalar functions can be obtained by using different weight vectors, each one represents a particular search direction toward the PF. Therefore, in order to approximate the entire PF, one has to alter the weight vectors of a weighted scalar function during the search. The exemplary decomposition-based MOEAs are IMMOGLS [10], UGA [11], cMOGA [12], MOGLS [13], MOSPS [14], and MOEA/D [3].

The most popular approach to deal with constraints is to use penalty functions. In penalty function methods, appropriate penalty coefficients are required to balance objective and penalty functions. However, it is often very hard to find such appropriate coefficients [15]. They are problems dependent.

SR [15] and CDP [9] are two promising penalty function based constraint handling techniques that try to balance objective and penalty functions explicitly and directly, and thus avoid penalty coefficients. Since they do not employ any penalty coefficient in the adopted penalty functions, a priori knowledge of the constrained problem at hand is also not needed. SR is inspired from the need of balancing objective and penalty functions directly and explicitly in constrained optimization [15]. In SR, a small percentage of infeasible solutions is compared based on objective function values, and the rests are compared based on constraint violation. While in CDP, they are compared based on constraint violation only. Moreover, CDP favors feasible solutions over infeasible solutions.

In this paper, we first modify and then implement these two techniques in the framework of MOEA/D-DE [16], an improved version of MOEA/D.

The rest of this paper is organized as follows. Section 2 presents the two commonly used weight-based decomposition approaches. Section 3 briefly introduces MOEA/D and adapts the algorithmic framework of MOEA/D-DE for CMOPs. Section 4 illustrates constraint handling techniques SR and CDP. Section 5 discusses the experimental settings. Section 6 introduces the metrics that will be used for the performance assessment of the algorithms proposed in this paper. Section 7 shows and discusses the experimental results on CTP-series [2], [17] and CF-series [18] test instances. Section 8 compares our experimental results with NSGA-II [9], IDEA [19], and the three best performers [20], [21], [22] of CEC 2009 MOEA competition. Section 9 comments on the sensitivity of the performance of the suggested algorithms to the parameters involved. Finally, Section 10 concludes this paper with a summary of the work carried out.