Abstract: |
The global constrained optimization of polynomials problem is the following:given polynomials f1; : : : ; fs and f in Q[X1; : : : ;Xn] we look for computing the in mum f? of f over the reals under the constraints f1 = = fs = 0.A way to get a lower bound is to nd an algebraic set V 0 such that f? is also the in mum of the restriction of f to V 0 \ Rn; if a polynomial is positive over V 0 \Rn then it can be written as a sum of squares of polynomials modulo the polynomial ideal associated to V 0. Thus, it suces to nd the greatest a such that f .. a can be written as a sum of squares modulo V 0, which can be done eciently using semide nite programming. Moreover, it gives a numerical certi cate for the lower bound. In this talk, we present the algebraic part of the work, that is a generalization of a result by Safey El Din and Schost using polar varieties in order to construct a variety satisfying the previous points. This is a joint work with Feng Guo, Mohab Safey El Din and Lihong Zhi. |