On Gradient-Based and Swarm-Based Algorithms for Set-Oriented Bicriteria Optimization

This paper is about the numerical solution of multiobjective optimization problems in continuous spaces. The problem is to define a search direction and a dynamical adaptation scheme for sets of vectors that serve as approximation sets. Two algorithmic concepts are compared: These are stochastic optimization algorithms based on cooperative particle swarms, and a deterministic optimization algorithm based on set-oriented gradients of the hypervolume indicator. Both concepts are instantiated as algorithms, which are deliberately kept simple in order to not obfuscate their discussion. It is shown that these algorithms are capable of approximating Pareto fronts iteratively. The numerical studies of the paper are restricted to relatively simple and low dimensional problems. For these problems a visualization of the convergence dynamics was implemented that shows how the approximation set converges to a diverse cover of the Pareto front and efficient set. The demonstration of the algorithms is implemented in Java Script and can therefore run from a website in any conventional browser. Besides using it to reproduce the findings of the paper, it is also suitable as an educational tool in order to demonstrate the idea of set-based convergence in Pareto optimization using stochastic and deterministic search.