New Certificates of Nonnegativity and Their Application in Science and Engineering

• Applicant: Professor Dr. Timo de Wolff
• Subject Area: Mathematics
• Term: from 2017 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 341488811

The aim of this proposed Emmy Noether research group is to establish a new type of certificate for nonnegativity which is independent of sums of squares and to apply this new type of certificate in practice.The investigation of nonnegativity of real multivariate polynomials and of sums of squares (SOS) is a classical subject. Initial results by Hilbert and others were already achieved in the 19th century and culminated in the solution of Hilbert's 17th problem. The subject remains a highly active field of research, particularly due to its close connection to polynomial optimization. A polynomial can be certified to be nonnegative via an SOS certificate if and only if a certain semidefinite optimization problem (SDP) is feasible. SDPs can be solved efficiently in practice and are known to be solvable up to an epsilon-error in polynomial time complexity. SDP methods, however, have natural limits as shown in theory by Blekherman in 2006 as well as shown in practice. Therefore, new certificates of nonnegativity which are independent of SOS and SDPs are strongly desired.Sums of nonnegative circuit polynomials (SONC) are such a new type of nonnegativity certificate. They were recently developed by Iliman and myself in 2013. We have already shown that these certificates are highly applicable in practice via geometric programming (GP) and relative entropy programing (REP), resulting in both reduced runtime and in certain cases better minimization bounds compared to SDPs.I propose to further develop these promising SONC certificates both in theory in practice. I intend to deepen the understanding of the SONC cone, its relation to other mathematical subjects as amoebas and sums of squares, to implement existing and develop stronger GPs and REPs applicable to general polynomial optimization problems as well as to use these methods to attack real world application problems.The proposed work lies at the interface of real algebraic geometry, nonlinear optimization and applied problems from science and engineering. It will improve our understanding of the cones of nonnegative polynomials, the SOS cone and the recently developed SONC cone as well as the interplay between those three, and their connection to other areas of mathematics. During the proposed work we will moreover attack general polynomial optimization problems drawn from engineering and science via our new type of nonnegativity certificate using geometric and relative entropy programming.

• DFG Programme: Independent Junior Research Groups
• International Connection: Denmark, USA
• Cooperation Partners: Professorin Dr. Elisenda Feliu; Dr. Sadik Iliman; Josè Israel Rodriguez, Ph.D.; Professorin Dr. Anne Shiu