Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization

TL;DR

This paper establishes that the convergence rate of outer approximation algorithms in convex vector optimization is independent of the $ ext{ell}_p$ norm used, matching the Euclidean case, and introduces a novel proof technique.

Contribution

The authors prove a $p$-independent convergence rate for $ ext{ell}_p$ norm-based algorithms using a Euclidean intermediary approach, extending known Euclidean results.

Findings

Convergence rate $ ext{O}(k^{2/(1-q)})$ holds for all $p eq 1, ext{infinity}$.

Numerical experiments confirm the theoretical $p$-independent rate.

A new proof technique exploits the ambient inner product structure to bypass $ ext{ell}_p$ smoothness limitations.

Abstract

We analyze convergence rates of norm-minimization-based outer approximation algorithms for convex vector optimization when the scalarization uses an $\ell_p$ norm with $p \in (1,\infty)$. While the Euclidean case ($p=2$) achieves the optimal rate $O(k^{2/(1-q)})$, the behavior under general $\ell_p$ norms has remained open. A direct approach via the modulus of smoothness yields only the weaker exponent $\min(p,2)$, which degrades for $1 < p < 2$. We prove that the Hausdorff approximation error satisfies $\delta_H(P_k, A) = O(k^{2/(1-q)})$ for \emph{every} $p \in (1,\infty)$, where $q$ is the number of objectives and $k$ is the iteration count. The proof introduces a Euclidean intermediary technique that exploits the ambient inner product structure of $\R^q$ to obtain a quadratic bound on the hyperplane distance, bypassing the $\ell_p$ smoothness limitation; norm equivalence then converts this to any $\ell_p$ metric at the cost of only a dimension-dependent constant, not a loss of exponent. Numerical experiments confirm the $p$-independent rate predicted by the theory.