Is it possible that all polynomial optimization problems are equivalent with convex formulations?

Polynomial optimization includes a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. Recently, polynomial optimization problems have been formulated as a relaxed conic program over a convex set. In this report, we study equivalent convex reformulations for general polynomial optimization problems. By the definition of recession cone, we prove that all feasible polynomial optimization problems are equivalent to convex optimization problems with linear objective functions. The reformulated convex optimization problem is a conic problem with tensor variables defined in the sum of two convex sets. Particularly, we show that polynomial optimization problems with linear constraints are equivalent with completely positive programs which are linear programs with completely positive tensor variables. Furthermore, the equivalent copositive optimization problems are given for the case with linear constraints, which are dual problems of the obtained completely positive optimization problems. It is shown that the strong duality holds under some conditions.

报告人简介：陈海滨，曲阜师范大学教授， 博导， 国家自然科学基金 通讯评审专家， 山东省杰青， 中国高等教育学会教育数学专委会理事， 山东省高校青年创新团队发展计划带头人。 主持（结题） 国家自然科学基金 2 项， 省部级项目 5 项。 与祁力群教授在 Springer 出版社出版张量专著 1 部； 发表 SCI 论文 50 余篇。