Decomposition gradient descent method for bi-objective optimisation

Population-based decomposition methods decompose a multi-objective optimisation problem (MOP) into a set of single-objective subproblems (SOPs) and then solve them collaboratively to produce a set of Pareto optimal solutions. Most of these methods use heuristics such as genetic algorithms as their search engines. As a result, these methods are not very efficient. This paper investigates how to do a gradient search in multi-objective decomposition methods. We use the NBI-style Tchebycheff method to decompose a MOP since it is not sensitive to the scales of objectives. However, since the objectives of the resultant SOPs are non-differentiable, they cannot be directly optimised by the classical gradient methods. We propose a new gradient descent method, decomposition gradient descent (DGD), to optimise them. We study its convergence property and conduct numerical experiments to show its efficiency.