On the Complexity of p-Order Cone Programs

2025-01-16

AI Summary

This work investigates the computational complexity of the *p*-order cone feasibility problem. To address its intrinsic intractability, we propose a novel analytical framework integrating the Blum–Shub–Smale (BSS) real-number computation model with convex cone geometry. Our contributions are threefold: (i) We derive the first tight, refined upper complexity bound—substantially improving upon standard semidefinite programming (SDP)-based reconstruction; (ii) We introduce the first inconsistency measure explicitly leveraging *p*-order cone structure and establish an explicit upper bound on the norm of feasible solutions; and (iii) We identify multiple geometric and structural subclasses—such as those with sparse or low-rank constraints—that admit polynomial-time solvability. Methodologically, our approach unifies tools from real algebraic geometry, *p*-norm optimization, and feasibility certification algorithms, thereby extending the classical Porkolab–Khachiyan framework. The results advance both theoretical understanding and practical feasibility criteria for conic programming.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Sustainability of Web economics

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.

Problem

• p-order conic programming
• computational complexity
• structural exploitation

Innovation

• p-order cone
• computational efficiency
• geometric properties