Equispaced Pareto front construction for constrained bi-objective optimization

Abstract

We consider constrained bi-objective optimization problems. One of the extant issues in this area is that of uniform sampling of the Pareto front. We utilize equispacing constraints on the vector of objective values, as discussed in a previous paper dealing with the unconstrained problem.We present a formulation and a dual formulation based on arc-length homotopy continuation and illustrate this method (using standard nonlinear programming tools) on some problems from the literature. We contrast the performance of our method with the results of three other algorithms, showing several orders of magnitude speed-up with respect to evolutionary algorithms, while simultaneously providing perfectly sampled fronts by construction. We then consider a large-scale application: the variational approach to mesh generation for partial differential equations in complex domains. Balancing multiple criteria leads to significantly improved mesh design.