Computing the global optimum of a multivariate polynomial over the reals

• Authors: Mohab Safey El Din
• Venue: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
• Year: 2008

Let f be a polynomial in Q[X1, ..., Xn] of degree D. We provide an efficient algorithm in practice to compute the global supremum supx∈ Rn f(x) of f (or its infimum inf{x∈ Rn}f(x)). The complexity of our method is bounded by DO(n)}. In a probabilistic model, a more precise result yields a complexity bounded by O(n7D4n) arithmetic operations in Q. Our implementation is more efficient by several orders of magnitude than previous ones based on quantifier elimination. Sometimes, it can tackle problems that numerical techniques do not reach. Our algorithm is based on the computation of generalized critical values of the mapping x- f(x), i.e. the set of points {c∈ C mid exists (xll)ll∈ N}⊂ Cn ;f(xll)- c, ;||xll||||dxll f||- 0 { when }ll- ∞}. We prove that the global optimum of f lies in its set of generalized critical values and provide an efficient way of deciding which value is the global optimum.