[论文解读] Response to "Counterexample to global convergence of DSOS and SDSOS hierarchies"

Amir Ali Ahmadi, Anirudha Majumdar|arXiv (Cornell University)|Oct 9, 2017

Fault Detection and Control Systems参考文献 7被引用 4

一句话总结

本文回应了一项批评，该批评声称DSOS与SDSOS多项式优化层级不具备全局收敛性。本文澄清，作者从未就该批评中定义的具体层级声称过此类收敛性，而是主张存在收敛的（S)DSOS基础层级——这一结论已通过现有的基于线性规划（LP）与第二类锥规划（SOCP）的层级得到验证。作者进一步反驳了关于可扩展性的质疑，表明SOCP约束的增长为多项式增长而非指数增长，并在大规模问题中展示了显著的计算优势。

ABSTRACT

In a recent note [8], the author provides a counterexample to the global convergence of what his work refers to as "the DSOS and SDSOS hierarchies" for polynomial optimization problems (POPs) and purports that this refutes claims in our extended abstract [4] and slides in [3]. The goal of this paper is to clarify that neither [4], nor [3], and certainly not our full paper [5], ever defined DSOS or SDSOS hierarchies as it is done in [8]. It goes without saying that no claims about convergence properties of the hierarchies in [8] were ever made as a consequence. What was stated in [4,3] was completely different: we stated that there exist hierarchies based on DSOS and SDSOS optimization that converge. This is indeed true as we discuss in this response. We also emphasize that we were well aware that some (S)DSOS hierarchies do not converge even if their natural SOS counterparts do. This is readily implied by an example in our prior work [5], which makes the counterexample in [8] superfluous. Finally, we provide concrete counterarguments to claims made in [8] that aim to challenge the scalability improvements obtained by DSOS and SDSOS optimization as compared to sum of squares (SOS) optimization. [3] A. A. Ahmadi and A. Majumdar. DSOS and SDSOS: More tractable alternatives to SOS. Slides at the meeting on Geometry and Algebra of Linear Matrix Inequalities, CIRM, Marseille, 2013. [4] A. A. Ahmadi and A. Majumdar. DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization. In proceedings of the 48th annual IEEE Conference on Information Sciences and Systems, 2014. [5] A. A. Ahmadi and A. Majumdar. DSOS and SDSOS optimization: more tractable alternatives to sum of squares and semidefinite optimization. arXiv:1706.02586, 2017. [8] C. Josz. Counterexample to global convergence of DSOS and SDSOS hierarchies. arXiv:1707.02964, 2017.

研究动机与目标

• 纠正近期批评中对作者声称DSOS与SDSOS层级具备全局收敛性的误解。
• 重申作者原始工作中从未定义或声称该批评中所列具体层级具备收敛性。
• 证明基于（S)DSOS的收敛层级确实存在，其依据为已知的基于LP与SOCP的层级。
• 反驳（S)DSOS优化因SOCP约束呈指数增长而比SOS更不可行的论断。
• 突出（S)DSOS编程在大规模多项式优化问题中的实际可扩展性与计算优势。

提出的方法

• 基于区分批评中所定义的具体层级与作者实际讨论的（S)DSOS基础层级的广义类别进行反驳。
• 利用现有收敛层级（如Lasserre、Peña-Vera-Zuluaga、Ahmadi-Hall）作为证据，证明（S)DSOS基础的收敛层级确实存在。
• 证明在SDSOS规划中，将半定规划约束替换为SOCP约束后，约束数量的增长为多项式增长，而非指数增长。
• 表明（S)DSOS编程的矩阵规模与SOS编程相同，但将昂贵的SDP约束替换为更便宜的SOCP约束。
• 通过原始论文中的定理10与定理12（[5]）提供理论依据，证明其与多项式时间可解性等价。
• 利用来自不同领域（机器人学、控制、组合优化）的数值示例，验证其可扩展性与实际性能。

实验结果

研究问题

• RQ1 该批评中定义的DSOS与SDSOS层级是否真正反驳了作者先前工作中的任何主张？
• RQ2 是否存在基于DSOS与SDSOS优化的有效全局收敛层级构造？
• RQ3 SDSOS规划中的第二类锥约束数量是否随层级阶数呈指数增长？
• RQ4 （S)DSOS优化在大规模问题中是否能为SOS编程提供实际的可扩展性优势？
• RQ5 （S)DSOS编程的保守性是否大到足以削弱其相对于SOS的实用价值？

主要发现

• 作者从未就该批评中所定义的具体DSOS与SDSOS层级声称过全局收敛性，因此该反例并未反驳其任何主张。
• 存在多个基于DSOS与SDSOS优化的收敛层级，如现有基于LP与SOCP的层级可被转化为（S)DSOS形式所示。
• SDSOS层级中的第二类锥约束数量随层级阶数呈多项式增长，而非如批评所称的指数增长。
• （S)DSOS框架保持与SOS编程相同的矩阵规模，但将昂贵的半定规划约束替换为更廉价的SOCP约束，从而实现显著的计算加速。
• 在实际应用中，例如30维人形机器人控制问题，（S)DSOS编程实现了SOS编程因计算不可行而无法解决的方案。
• （S)DSOS的保守性较小，且可通过迭代改进技术缓解，如原始论文第5节所示。