# Comprehensive Survey of the Hybrid Evolutionary Algorithms

Wali Khan Mashwani (Department of Mathematics, Kohat University of Science & Technology (KUST), Kohat, Khyber Pukhtunkhwa, Pakistan)
DOI: 10.4018/jaec.2013040101

## Abstract

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) and an improved non-dominating sorting multiobjective genetic algorithm (NSGA-II) is two well known multiobjective evolutionary algorithms (MOEAs) in the field of evolutionary computation. This paper mainly reviews their hybrid versions and some other algorithms which are developed for solving multiobjective optimization problems (MOPs. The mathematical formulation of a MOP and some basic definitions for tackling MOPs, including Pareto optimality, Pareto optimal set (PS), Pareto front (PF) are provided in Section 1. Section 2 presents a brief introduction to hybrid MOEAs. The authors present literature review in subsections. Subsection 2.1 provides memetic multiobjective evolutionary algorithms. Subsection 2.2 presents the hybrid versions of well-known Pareto dominance based MOEAs. Subsection 2.4 summarizes some enhanced Versions of MOEA/D paradigm. Subsection 2.5 reviews some multimethod search approaches dealing optimization problems.
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## 1. Introduction

A multiobjective optimization problem (MOP) can be stated as follows:1minimize F(x) = (f1(x), . . ., fm(x))T(1) subject to x ∈ ΩWhere Ω is the decision variable space, x = (x1, x2, . . ., xn)T is a decision variable vector and xi, i =1. . . . n are called decision variables, F(x): Ω → Rm consist of m real valued objective functions and Rm is called the objective space.

If Ω is closed and connected region in Rn and all the objectives are continuous of x, a problem (1) is said to be a continuous MOP.

Very often, the objectives of the problem (1) are in conflict with one another or are incommensurable. There doesn’t exist a single solution in the search space Ω that can minimize all the objectives functions simultaneously. Instead, one has to find the best tradeoffs among the objectives. These tradeoffs can be better de-fined in terms of Pareto optimality. The Pareto optimality concept was first introduced by eminent economists Pareto and Edgeworth (Edgeworth, 1881). A formal definition of the Pareto optimality is given as follows (Coelle Coello, Lamont, & Veldhuizen, 2002); (Deb, 2002); (Deb, 2001); (Miettinien, 1999):

• Definition: Let u = (u1, u2, . . ., um)T and v = (v1, v2, . . ., vm)T be any two given vectors in Rm. Then u is said to dominate v, denoted as uv, if and only if the following two conditions are satisfied.

• 1.

uivi for everyi ∈ {1, 2, . . ., m}

• 2.

u j < v j for at least one index j ∈ {1, 2, . . ., m}.

• Remarks: For any two given vectors, u and v, there are two possibilities:

• 1.

Either u dominates v or v dominates u

• 2.

Neither u dominates v nor does v dominate u.

• Definition: A solution x ∈ Ω is said to be a Pareto Optimal to the problem (1) if there is no other solution x ∈ Ω such that F(x) dominates F(x). F(x) is then called Pareto optimal (objective) vector.

• Remarks: Any improvement in a Pareto optimal point in one objective must lead to deterioration in at least one other objective .

• Definition: The set of all the Pareto optimal solutions is called Pareto set (PS): PS = {x ∈ Ω, F(y) ≺ F(x)}

• Definition: The image of the Pareto optimal set (PS) in the objective space is called Pareto front (PF), PF = {F(x)|xPS }.

• Weight Sum Approach: the weighted sum of the m objectivists is defined as gws(x, λ) = λ1f1(x) + λ2f2(x) + . . . + λm fm(x), where ∑ mj=1 λ j = 1 and λ j ≥ 0.

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