Positivity certificates and polynomial optimization on non-compact semialgebraic sets
Mathematical Programming(2021)
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.
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关键词
Nonnegativity certificate, Putinar’s Positivstellensatz, Basic semialgebraic set, Sums of squares, Polynomial optimization, Semidefinite programming, Moment-SOS hierarchy, Uniform denominators, Polynomial systems, 90C22, 65H04
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