Proves that ℓ_p norm minimization yields p-independent Hausdorff convergence rate O(k^{2/(1-q)}) in convex vector optimization via Euclidean intermediary and norm equivalence.
Adaptive Metrics for Norm-Minimization-Based Outer Approximation in Convex Vector Optimization
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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.
fields
math.OC 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Introduces parallel subproblem evaluation and batch addition of up to K cuts per iteration for a convex vector optimization algorithm, proves the batch variant preserves the O(k^{2/(1-q)}) convergence rate, and reports 62-80% fewer iterations with variable wall-clock gains.
citing papers explorer
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Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization
Proves that ℓ_p norm minimization yields p-independent Hausdorff convergence rate O(k^{2/(1-q)}) in convex vector optimization via Euclidean intermediary and norm equivalence.
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On Parallel and Batch-Cutting Strategies for Norm-Minimization-Based Convex Vector Optimization
Introduces parallel subproblem evaluation and batch addition of up to K cuts per iteration for a convex vector optimization algorithm, proves the batch variant preserves the O(k^{2/(1-q)}) convergence rate, and reports 62-80% fewer iterations with variable wall-clock gains.