DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization

Abstract

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 diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (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 trade-offs 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 to the paper introduces an accompanying MATLAB package for DSOS and SDSOS optimization.

On polyhedral and second-order cone decompositions of semidefinite optimization problems

“…Proposition 2 demonstrates that even if the optimal choice of X is bounded in trace norm by a constant, the quality of Problem (2)'s approximation degrades with n. This result explains why various authors [21,13,1] have found that Problem (2) poorly approximates Problem (1) in a high-dimensional setting.…”

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

“…By combining this theorem with some linear algebraic observations, the authors show in [10] that optimization of a linear objective function over the intersection of the cone of dsos (resp. sdsos) polynomials of a given degree with an affine subspace can be carried out via linear programming (resp.…”

“…In [10], the authors introduce "DSOS and SDSOS optimization" as linear programming and second-order cone programming-based alternatives to sum of squares and semidefinite optimization that allow for a trade off between computation time and solution quality. The following definitions are central to their framework.…”

“…On the theoretical front, DSOS and SDSOS optimization enjoy some of the same guarantees as those enjoyed by SOS optimization. For example, classical theorems in algebraic geometry can be utilized to conclude that any even positive definite homogeneous polynomial is the ratio of two dsos polynomials [10,Sect. 3.2].…”

Review on set-theoretic methods for safety verification and control of power system

“…Diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (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 [80]. In addition, applying different positivity certificates, such as Handelman's representation [69] and Krivine-Stengles certificate [81], will lead to linear program.…”