Helper and Equivalent Objectives: An Efficient Approach to Constrained Optimisation

Numerous multi-objective optimisation evolutionary algorithms have been designed for solving constrained optimisation problems in last two decades. Their idea is to transform a constrained optimisation problem into a multi-objective optimisation problem without inequality and equality constraints and then to apply a multi-objective evolutionary algorithm into solving it. This paper investigates the helper and equivalent objective method for constrained optimisation, in which one objective is equivalent to solving the original constrained problem and other objectives play a helper role. Several new contributions are made in this paper. First, the helper and equivalent objective method is analysed in a rigorous way. It is proven that using helper and equivalent objectives may shorten the expected hitting time of a multi-objective algorithm leaving a local optimum with respect to a single objective algorithm. Secondly, in order to reduce the preference of feasible solutions over infeasible ones, a new equivalent objective function is constructed. Then the multi-objective problem consisting of helper and equivalent objectives is decomposed into several single objective problems using the weighted sum approach. Weights are dynamically adjusted so that each single objective eventually tends to an equivalent objective. At the end, a new multi-objective evolutionary algorithm is designed for constrained optimisation. This algorithm is run on benchmarks in IEEE CEC 2017 and 2018 constrained optimisation competitions. Comparative experiment shows that the proposed algorithm is capable of producing dominating results compared with all algorithms participated in the two competitions.