On the Complexity of p-Order Cone Programs

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.

• The paper presents a refined complexity analysis of p-order cone feasibility problems, demonstrating better bounds than standard SDP reformulations.
• It derives explicit solution bounds and metrics for infeasibility that quantify the log-modulus of feasible solutions.
• The study highlights computational simplifications in special cases, paving the way for more efficient optimization algorithms in applied contexts.

An In-depth Analysis of the Complexity of p-Order Cone Programs

This paper, authored by Víctor Blanco, Victor Magron, and Miguel Martínez-Antón, presents an exploration of the computational complexity associated with solving feasibility problems over p-order cones. The study extends the foundational complexity work by Porkolab and Khachiyan on semidefinite programs (SDP) to a broader class of conic problems characterized by p-order cones.

Key Contributions and Results

The authors leverage the intrinsic structure of p-order cones to derive refined complexity bounds, demonstrating improvements over those obtainable via standard SDP reformulations. Specifically, the paper establishes complexity results by assessing the necessary number of arithmetic operations, and it explores bounds on the feasible solutions if they exist. Using the framework developed, the authors provide the following contributions:

1. Refined Complexity Analysis: The complexity of the p-order feasibility problem is examined, showing that native handling of p-order cones leads to better complexity bounds than full reformulation into SDP constraints, particularly in scenarios beyond the commonly used second-order cones (SOC).
2. Explicit Solution Bounds: For feasible instances, explicit bounds on solutions are derived, allowing for a quantifiable assessment of solution space characteristics. This includes a detailed analysis of the log-modulus of feasible solutions.
3. Discrepancy Bounds for Infeasibility: The paper explores infeasible instances, providing metrics for discrepancy or the degree of infeasibility. This is crucial for determining how "far" a problem is from having a feasible solution.
4. Special Cases and Computational Simplifications: By investigating specific cases, such as even p or SOCs, further computational simplifications are demonstrated, revealing scenarios where computational complexity can be notably reduced.

Tables summarizing these complexity results are included, providing structured insights into various instances and special cases of the p-order cones feasibility problem.

Implications and Applications

This research provides both theoretical and practical insights, impacting fields like optimization, finance, machine learning, and robust optimization. By extending complexity results beyond SOC, this work indicates the potential for efficiency gains in solving optimization problems with non-Euclidean p-norms, which are increasingly relevant in applications like Support Vector Machines (SVMs), multiple criteria optimization, and fair resource allocation.

In practical applications, the implications are far-reaching: by improving complexity bounds, the authors suggest more efficient algorithms for optimization problems that incorporate p-order cones. The insights into feasible solution bounds and discrepancies can also inform robustness assessments in optimization scenarios where stability is crucial.

Future Directions

The paper opens avenues for exploration into Mixed Integer p-Order Cone Optimization (MIPOCO) problems, which combine continuous relaxations with discrete decision variables—an area yet to be examined beyond MISOCO reformulations. This future work could significantly enhance computational approaches to problems where p-order cones are native, rather than converted into SOCs.

Conclusion

The authors have delivered a rigorous treatment of the complexity related to p-order cone feasibility problems, enhancing both the theoretical framework and computational possibilities in conic optimization. This work offers a detailed roadmap for tackling complex conic problems effectively, making significant strides in understanding and leveraging the unique structure of p-order cones. The achievements underscore the potential computational efficiencies that can be realized through a nuanced understanding and manipulation of p-order cones within optimization tasks.