SURFACE: Preference Distribution Matching on Pareto Fronts via Scalarization

Constructing a finite Pareto representation requires deciding where a limited number of solutions should be placed along a continuous Pareto front (PF). A desired preference distribution provides a direct way to specify this allocation by assigning more solutions to some PF regions and fewer to others. Scalarization generates Pareto solutions by varying objective weights and solving single-objective problems. However, a distribution of scalarization weights generally does not translate into the same distribution of Pareto solutions along the PF, because the weight-to-solution map is nonlinear and unknown in advance. We introduce SURFACE (Scalarization Updates for Redistributing Finite Approximations with Chart Estimation), which adaptively updates scalarization weights to match a desired PF distribution. SURFACE estimates how each region of the weight grid corresponds to an area of the PF from the current scalarized solutions, then redistributes the weights accordingly. The method supports parallel, warm-started, inexact optimization with standard single-objective solvers. For smooth positive preference distributions, we prove that the error decreases geometrically to an $\mathcal{O}(N^{-2/(M-1)})$ finite-sample level, with the same rate for the mismatch between the desired and achieved PF allocation, where $N$ is the number of adaptive weights and $M$ is the number of objectives. Experiments across diverse benchmarks and comparisons with nine baselines consistently demonstrate strong performance.