Adaptive Metrics for Norm-Minimization-Based Outer Approximation in Convex Vector Optimization

Abstract

We develop an adaptive-metric framework for norm-minimization-based outer approximation algorithms in bounded convex vector optimization. The key idea is to let the scalarization metric vary across iterations while measuring approximation error in a fixed Euclidean norm. This enables the algorithm to exploit problem geometry dynamically. Our approach rests on two theoretical foundations. First, we prove that the improved Euclidean convergence rate $O(k^{2/(1-q)})$ -- previously known only for the standard $\ell_2$-norm -- extends to all fixed inner-product norms. Second, we establish a dispersion theorem showing that the cut-normals generated by the algorithm naturally spread across all directions when the upper image has a strictly convex boundary with bounded curvature. This geometric condition guarantees that the adaptive metric remains well-conditioned throughout execution. Building on these results, we derive explicit convergence bounds that quantify how metric conditioning influences the Hausdorff error estimates. Numerical experiments on three test problems validate the theoretical convergence rate; on the problems whose Pareto fronts have sufficient curvature, the adaptive metric additionally reduces the iteration count relative to the fixed Euclidean norm. Our results provide a rigorous foundation for adaptive metric selection in convex vector optimization.