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

## TL;DR

This paper introduces DSOS and SDSOS optimization as scalable, tractable alternatives to sum of squares methods, enabling larger problem solving with acceptable accuracy tradeoffs in various fields.

## Contribution

It proposes DSOS and SDSOS as LP and SOCP-based alternatives to SOS, extending scalability while maintaining key theoretical properties.

## Key findings

- Numerical experiments show scalability beyond traditional SOS methods.
- Tradeoffs between accuracy and computational efficiency are demonstrated.
- Theoretical results extend SOS properties to DSOS and SDSOS.

## 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 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.