Dynamic multi-objective differential evolution for solving constrained optimization problem

Dynamic constrained multi-objective differential evolution (DCMODE) is designed for solving constrained optimization problem (COP). Main feature presented in this paper is to construct dynamic multi-objective optimization problem (DMOP) from COP. The two evolved objectives are original function objective and violation objective. Constraints are controlled by dynamic environments, where the relaxed constraints boundaries are gradually tightened to original boundaries. After this dynamic process, DMOP solutions are close to COP solution. This new algorithm is tested on benchmark problems of special session at CEC2006 with 100% success rates of all problems. Compared with several state-of-the-art DE variants referred in this paper, our algorithm outperforms or performs similarly to them. The satisfactory results suggest that it is efficient and generic when handling inequality/equality constraints.

I. Introduction

In science and engineering, it is easy to meet COP trying to minimize or maximize the function objective value on the premise that solution in the search space satisfies requirements. These constraints could be inequality or/and equality ones. Assuming, without loss of generality, a minimization COP is: $$\eqalignno{min\qquad &y=f(\overrightarrow{x})\cr st:\qquad\ &g_{i}(\overrightarrow{x})\leq 0, i=1,2, \ldots, q\cr &h_{i}(\overrightarrow{x})=0, i=q+1, q+2, \ldots, m\cr { where}\quad &\overrightarrow{x}=(x_{1}, x_{2}, \ldots, x_{n})\in {\bf X}&\hbox{(1)}\cr &{\bf X}=\{(x_{1}, \cdots, x_{n})\vert l_{i}\leq x_{i}\leq u_{i}, i=1, \cdots, n\}\cr &\overrightarrow{l}=(l_{1}, l_{2}, \ldots, l_{n})\cr &\overrightarrow{u}=(u_{1}, u_{2}, \ldots, u_{n})}$$ where is decision vector, X denotes decision space, and are the lower bound and the upper bound of decision space. is an objective function, and are inequality constraints and equality constraints, respectively.