Positivity certificates and polynomial optimization on non-compact semialgebraic sets

• 成果类型: Article
• 署名作者: Mai, Ngoc Hoang Anh; Lasserre, Jean-Bernard; Magron, Victor
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01634-1
• 发表日期: 2022
• 页码: 443-485
• 关键词: nonnegative polynomials; uniform denominators; global optimization; moment problems; squares; sums; REPRESENTATIONS; approximations; THEOREM

摘要

In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu (C R Acad Sci Ser I Math 328(6):495-499, 1999). We use Jacobi's technique from (Math Z 237(2):259-273, 2001) to provide an alternative proof with an effective degree bound on the sums of squares weights in such certificates. As a consequence, it allows one to define a hierarchy of semidefinite relaxations for a general polynomial optimization problem. Convergence of this hierarchy to a neighborhood of the optimal value as well as strong duality and analysis are guaranteed. In a second contribution, we introduce a new numerical method for solving systems of polynomial inequalities and equalities with possibly uncountably many solutions. As a bonus, one can apply this method to obtain approximate global optimizers in polynomial optimization.