DSOS和SDSOS优化：比平方和优化和半定优化更可行的替代方法
DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization
AI总结：
本文提出DSOS和SDSOS优化作为替代SOS优化的方法，通过线性规划和二次锥规划在计算时间和解的质量之间进行权衡，适用于半定规划中的可扩展性限制问题。
摘要：
近年来，平方和（SOS）优化的出现极大地影响了优化理论。然而，这种技术对大规模半定规划的依赖限制了其应用问题的规模。本文介绍DSOS和SDSOS优化作为SOS优化的替代方法，基于线性规划和二次锥规划，在计算时间和解的质量之间进行权衡。这些优化问题针对某些平方和多项式（或等价的半正定矩阵）的子集，适用于半定规划的一般应用，其中可扩展性是一个限制因素。我们证明了一些基本的SOS优化定理，这些定理依赖于实代数几何的结果，仍然适用于DSOS和SDSOS优化。此外，我们通过来自多项式优化、统计和机器学习、衍生品定价和控制理论等不同应用领域的数值实验，展示了在准确度上有合理权衡的情况下，可以处理目前传统SOS方法无法处理的更大规模的问题。最后，我们提供了一种最近的技术回顾，以弥补我们的DSOS/SDSOS方法与SOS方法之间的差距，但需要额外的运行时间。论文的补充材料介绍了一个用于DSOS和SDSOS优化的MATLAB工具包。
英文摘要：
In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs however, has limited the scale of problems to which it can be applied. In this paper, we introduce DSOS and SDSOS optimization as linear programming and second-order cone programming-based alternatives to sum of squares optimization that allow one to trade off computation time with solution quality. These are optimization problems over certain subsets of sum of squares polynomials (or equivalently subsets of positive semidefinite matrices), which can be of interest in general applications of semidefinite programming where scalability is a limitation. We show that some basic theorems from SOS optimization which rely on results from real algebraic geometry are still valid for DSOS and SDSOS optimization. Furthermore, we show with numerical experiments from diverse application areas---polynomial optimization, statistics and machine learning, derivative pricing, and control theory---that with reasonable tradeoffs in accuracy, we can handle problems at scales that are currently significantly beyond the reach of traditional sum of squares approaches. Finally, we provide a review of recent techniques that bridge the gap between our DSOS/SDSOS approach and the SOS approach at the expense of additional running time. The Supplementary Material of the paper introduces an accompanying MATLAB package for DSOS and SDSOS optimization.
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