# Helper and Equivalent Objectives: An Efficient Approach to Constrained Optimisation

## TL;DR

This paper introduces a novel multi-objective optimization approach using helper and equivalent objectives for constrained problems, demonstrating improved efficiency and performance over existing algorithms, especially on challenging wide gap problems.

## Contribution

It proposes a new method that transforms constrained optimization into a multi-objective problem with helper and equivalent objectives, and provides theoretical and empirical validation of its effectiveness.

## Key findings

- Shortens the crossing time of wide gap problems.
- Achieves top performance on IEEE CEC2017 benchmarks.
- Theoretically reduces exponential crossing time in hard problems.

## Abstract

Numerous multi-objective evolutionary algorithms have been designed for constrained optimisation over past two decades. The idea behind these algorithms is to transform constrained optimisation problems into multi-objective optimisation problems without any constraint, and then solve them. In this paper, we propose a new multi-objective method for constrained optimisation, which works by converting a constrained optimisation problem into a problem with helper and equivalent objectives. An equivalent objective means that its optimal solution set is the same as that to the constrained problem but a helper objective does not. Then this multi-objective optimisation problem is decomposed into a group of sub-problems using the weighted sum approach. Weights are dynamically adjusted so that each subproblem eventually tends to a problem with an equivalent objective. We theoretically analyse the computation time of the helper and equivalent objective method on a hard problem called ``wide gap''. In a ``wide gap'' problem, an algorithm needs exponential time to cross between two fitness levels (a wide gap). We prove that using helper and equivalent objectives can shorten the time of crossing the ``wide gap''. We conduct a case study for validating our method. An algorithm with helper and equivalent objectives is implemented. Experimental results show that its overall performance is ranked first when compared with other eight state-of-art evolutionary algorithms on IEEE CEC2017 benchmarks in constrained optimisation.