Positivity certificates and polynomial optimization on non-compact semialgebraic sets

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) provide an alternative proof with effective degree bound the sums squares weights in such certificates. As consequence, it allows one define hierarchy semidefinite relaxations for general polynomial optimization problem. Convergence this neighborhood optimal value as well strong duality analysis are guaranteed. second introduce new numerical method solving systems inequalities equalities possibly uncountably many solutions. bonus, can apply obtain approximate global optimizers optimization.