SOSTOOLS Version 4.00 Sum of Squares Optimization Toolbox for MATLAB (1310.4716v2)

Abstract: The release of SOSTOOLS v4.00 comes as we approach the 20th anniversary of the original release of SOSTOOLS v1.00 back in April, 2002. SOSTOOLS was originally envisioned as a flexible tool for parsing and solving polynomial optimization problems, using the SOS tightening of polynomial positivity constraints, and capable of adapting to the ever-evolving fauna of applications of SOS. There are now a variety of SOS programming parsers beyond SOSTOOLS, including YALMIP, Gloptipoly, SumOfSquares, and others. We hope SOSTOOLS remains the most intuitive, robust and adaptable toolbox for SOS programming. Recent progress in Semidefinite programming has opened up new possibilities for solving large Sum of Squares programming problems, and we hope the next decade will be one where SOS methods will find wide application in different areas. In SOSTOOLS v4.00, we implement a parsing approach that reduces the computational and memory requirements of the parser below that of the SDP solver itself. We have re-developed the internal structure of our polynomial decision variables. Specifically, polynomial and SOS variable declarations made using sossosvar, sospolyvar, sosmatrixvar, etc now return a new polynomial structure, dpvar. This new polynomial structure, is documented in the enclosed dpvar guide, and isolates the scalar SDP decision variables in the SOS program from the independent variables used to construct the SOS program. As a result, the complexity of the parser scales almost linearly in the number of decision variables. As a result of these changes, almost all users will notice a significant increase in speed, with large-scaleproblems experiencing the most dramatic speedups. Parsing time is now always less than 10% of time spent in the SDP solver. Finally, SOSTOOLS now provides support for the MOSEK solver interface as well as the SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+, and SDPA solvers.

• The paper introduces SOSTOOLS v4.00 with an innovative dpvar structure that scales parsing complexity almost linearly with problem size.
• It reduces computational overhead by integrating multiple SDP solvers such as SeDuMi, SDPT3, and MOSEK, boosting efficiency in control system stability analysis.
• The toolbox bridges theory and practice by providing flexible SOS representations for robust applications in control theory, global minimization, and NP-hard problem relaxations.

A Technical Review of SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB

The paper under review introduces SOSTOOLS, a comprehensive MATLAB toolbox designed for parsing and solving sum of squares (SOS) optimization problems. The tool leverages the SOS representation to provide flexible solutions to polynomial optimization challenges, making it a critical asset in control theory and related fields.

Key Features and Capabilities

SOSTOOLS provides the ability to transform polynomial non-negativity conditions into tractable semidefinite programming (SDP) problems, offering a robust framework for a variety of applications. The toolbox supports integration with several SDP solvers, including SeDuMi, SDPT3, and MOSEK, providing flexibility and improved performance. Recent enhancements have been directed towards reducing computational overhead, specifically in the parsing phase, which historically posed limitations in handling large-scale SOS problems.

A distinctive feature of SOSTOOLS v4.00 is the introduction of a new polynomial decision structure, dpvar, which provides efficient handling of decision variables. This allows the parser’s complexity to scale almost linearly with the number of decision variables, a significant improvement over previous versions, particularly for large-scale problems. Furthermore, integration with MOSEK and the inclusion of a sosquadvar function enhances support for customized decision variables.

Practical Implications and Use Cases

SOSTOOLS has demonstrated its broad applicability in areas such as control theory, where it aids in deriving Lyapunov functions, and facilitates stability analysis of dynamical systems. The ability to handle SOS polynomials efficiently extends to optimizations such as the global minimization of polynomials, structured singular value calculation, and robustness analysis. Moreover, SOSTOOLS provides a reliable method for convex relaxations of various NP-hard problems, offering both theoretical groundwork and numerical experimentation.

The parser’s reduced memory and computational demand enable its use in larger SOS programming problems without the risk of infeasibility stemming from pre-solution parsing tasks. This advancement widens the scope of feasible problems, potentially encouraging the uptake of SOS methodologies in new application fields.

Future Developments and Speculation

There is potential for SOSTOOLS to further exploit structural properties and sparsity in polynomial optimization problems, potentially incorporating state-of-the-art numerical techniques to optimize solver performance. Research into alternative data structures and further integration with emerging SDP solvers could provide enhancements in solution accuracy and computation time.

Exploration into more advanced applications and domain-specific customizations could see SOSTOOLS being integrated as a fundamental tool in the researcher’s toolkit, especially as challenges in control applications continue to grow in complexity.

In conclusion, SOSTOOLS serves as a powerful bridge between polynomial mathematical theory and practical engineering applications. As the field of SOS optimization evolves, the toolbox’s adaptability and robust feature set position it as a pivotal tool for researchers tackling complex optimization problems.