On the Complexity of p-Order Cone Programs

TL;DR

This paper advances the understanding of the computational complexity of feasibility problems over p-order cones, providing refined bounds, solution estimates, and insights into problem structure that improve upon previous semidefinite programming approaches.

Contribution

It introduces new complexity bounds and solution estimates for p-order cone problems, enhancing both theoretical understanding and practical solving strategies.

Findings

Refined complexity bounds for p-order cone feasibility problems

Explicit solution bounds when feasible

Analysis of infeasibility measures and problem structure

Abstract

This manuscript explores novel complexity results for the feasibility problem over $p$-order cones, extending the foundational work of Porkolab and Khachiyan. By leveraging the intrinsic structure of $p$-order cones, we derive refined complexity bounds that surpass those obtained via standard semidefinite programming reformulations. Our analysis not only improves theoretical bounds but also provides practical insights into the computational efficiency of solving such problems. In addition to establishing complexity results, we derive explicit bounds for solutions when the feasibility problem admits one. For infeasible instances, we analyze their discrepancy quantifying the degree of infeasibility. Finally, we examine specific cases of interest, highlighting scenarios where the geometry of $p$-order cones or problem structure yields further computational simplifications. These findings contribute to both the theoretical understanding and practical tractability of optimization problems involving $p$-order cones.