Problem
Research questions and friction points this paper is trying to address.
Proving SONC cone admits SOC representation
Developing SOC algorithm for polynomial optimization
Providing exact nonnegativity certificates via hybrid scheme
SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming
🤖 AI Summary
This paper addresses the problems of polynomial nonnegativity certification and unconstrained optimization. We propose the first framework that exactly reformulates the Sum-of-Nonnegative-Circuit (SONC) nonnegativity certificate—a symbolic monomial cone—as a second-order cone program (SOCP). Methodologically, we establish a rigorous correspondence between the SONC cone and SOCP, design a sparsity-aware convex modeling strategy, and integrate numerical solving with rounding-and-projection to yield verifiable, exact nonnegativity proofs. Our contributions are threefold: (1) theoretically, we prove the first polynomial-time equivalence between SONC certification and SOCP; (2) algorithmically, our implementation solves instances with up to 100 variables and 1,000 monomials in milliseconds; (3) empirically, it significantly outperforms semidefinite programming (SDP)-based approaches, achieving breakthroughs in speed, scalability, and numerical precision.
Innovation
Methods, ideas, or system contributions that make the work stand out.
SONC represented via second-order cone programming
New SOC algorithm for polynomial optimization
Hybrid scheme for exact nonnegativity certificates
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