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

TL;DR

This paper introduces an adaptive-metric framework for convex vector optimization algorithms, allowing the metric to vary dynamically to exploit problem geometry and improve convergence rates.

Contribution

It extends Euclidean convergence rates to all fixed inner-product norms and establishes geometric conditions ensuring well-conditioned adaptive metrics.

Findings

Adaptive metrics reduce iteration count by 31-33% on curved Pareto front problems.

The convergence rate $O(k^{2/(1-q)})$ applies to all fixed inner-product norms.

Theoretical foundations link metric conditioning to Hausdorff error bounds.

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 validate the theoretical rates and demonstrate that adaptive metrics achieve 31--33\% fewer iterations than the fixed Euclidean norm on problems with curved Pareto fronts. Our results provide a rigorous foundation for adaptive metric selection in convex vector optimization.