Pareto Set Representation Learning with Application to Multi-criteria Order Optimization

Multi-objective optimization seeks to arrive at a diverse set of Pareto-optimal solutions facilitating a posteriori decision-making. However, this becomes challenging for high-dimensional problems with limited compute, imposing a compromise between convergence and diversity of the final solutions. To address this curse of dimensionality, we introduce the concept of Pareto set representation learning, reducing the problem to its smallest possible dimensions while accurately capturing the Pareto-optima. A denoising autoencoder is invoked to discover a compressed latent representation of a sparsely populated Pareto set by leveraging its unique bottleneck architecture. This representation then serves as a means to create compact inverse models, mapping points from the Pareto front in objective space to the (dimensionally reduced) Pareto set in decision space. The method is empirically tested on benchmark problems and an industrial multi-site order planning problem showcasing its effectiveness in reducing the dimensionality of the Pareto set ($\mathbf{~} \mathbf{9 9. 6 \%}$) while achieving significant gains ($\gt \mathbf{2 0 0 \%}$) in Pareto approximation capacity. With such compact yet accurate inverse models, decision makers can readily generate high-dimensional solutions corresponding to any preferred, unexplored subregions of the objective space.