[Paper Review] Engineering and Business Applications of Sum of Squares Polynomials

Georgina Hall|arXiv (Cornell University)|Jun 19, 2019

Advanced Optimization Algorithms Research65 references4 citations

TL;DR

This paper presents a comprehensive survey of sum of squares (SOS) polynomial optimization techniques and their applications across engineering, business, and computational sciences. It demonstrates how SOS programming transforms hard non-convex polynomial optimization problems into tractable semidefinite programs via algebraic certificates, enabling solutions to problems in control theory, finance, machine learning, and game theory.

ABSTRACT

Optimizing over the cone of nonnegative polynomials, and its dual counterpart, optimizing over the space of moments that admit a representing measure, are fundamental problems that appear in many different applications from engineering and computational mathematics to business. In this paper, we review a number of these applications. These include, but are not limited to, problems in control (e.g., formal safety verification), finance (e.g., option pricing), statistics and machine learning (e.g., shape-constrained regression and optimal design), and game theory (e.g., Nash equilibria computation in polynomial games). We then show how sum of squares techniques can be used to tackle these problems, which are hard to solve in general. We conclude by highlighting some directions that could be pursued to further disseminate sum of squares techniques within more applied fields. Among other things, we briefly address the current challenge that scalability represents for optimization problems that involve sum of squares polynomials and discuss recent trends in software development.

Motivation & Objective

• To bridge the gap between theoretical advances in sum of squares optimization and practical applications in engineering and business.
• To demonstrate how non-convex polynomial optimization problems—previously intractable—can be reformulated as tractable semidefinite programs using SOS techniques.
• To provide accessible, application-specific frameworks that enable researchers with limited domain knowledge to model and solve complex problems using SOS programming.
• To address scalability challenges in SOS optimization and highlight recent software and algorithmic developments to improve performance on large-scale problems.
• To advocate for broader adoption of SOS methods in applied fields by illustrating concrete, real-world use cases across diverse domains.

Proposed method

• Formulate nonnegative polynomial constraints and moment conditions as optimization problems over the cone of nonnegative polynomials or representing measures.
• Use sum of squares (SOS) decomposition to convert polynomial nonnegativity conditions into semidefinite programs (SDPs), leveraging the fact that SOS polynomials are nonnegative and can be represented via positive semidefinite matrices.
• Apply polynomial Lyapunov functions and SOS-based certificates to verify stability, safety, and invariance in continuous- and discrete-time polynomial dynamical systems.
• Utilize moment-based reformulations for problems in option pricing, shape-constrained regression, and optimal design, where positivity and moment constraints are central.
• Employ SOS techniques to compute Nash equilibria in polynomial games by reformulating equilibrium conditions as polynomial feasibility problems.
• Leverage modeling parsers (e.g., YALMIP, SOSTOOLS, JuMP/SumofSquares.jl) and scalable SDP solvers (e.g., SCS, SDPNAL+) to automate the translation of SOS problems into solvable SDPs.
• Address scalability via structure exploitation (sparsity, symmetry) and alternative conic relaxations to reduce computational burden.

Experimental results

Research questions

• RQ1 How can sum of squares techniques be used to formally verify stability and safety in polynomial dynamical systems without solving trajectories explicitly?
• RQ2 In what ways can SOS optimization be applied to option pricing and risk management in finance, particularly when modeling nonnegative payoff functions?
• RQ3 How can shape-constrained regression and optimal experimental design be formulated and solved using SOS programming to enforce monotonicity or convexity constraints?
• RQ4 What is the role of SOS methods in computing equilibria in polynomial games, and how do they compare to traditional approaches?
• RQ5 What are the key challenges in scaling SOS optimization to large-scale problems, and what recent algorithmic and software advances help overcome them?

Key findings

• SOS techniques enable the formal certification of stability, collision avoidance, and invariance in polynomial dynamical systems by constructing algebraic certificates via sum of squares polynomials.
• The transformation of nonnegative polynomial constraints into semidefinite programs allows for efficient numerical solution using standard SDP solvers, even when closed-form solutions are intractable.
• Applications in finance include modeling nonnegative payoff functions in option pricing, where SOS-based moment problems provide tractable approximations.
• In machine learning, SOS-based shape-constrained regression and optimal design problems can be solved efficiently, ensuring monotonicity or convexity of estimated functions.
• For polynomial games, SOS programming enables the computation of Nash equilibria by reformulating equilibrium conditions as polynomial feasibility problems with nonnegativity constraints.
• Despite progress, scalability remains a key challenge due to the exponential growth of SDP size with polynomial degree and variable count, motivating research into sparse, structured, and first-order solvers such as SCS and SDPNAL+.