On the Complexity of Computing Critical Points with Gr\"obner Bases

Computing the critical points of a polynomial function $q\in\mathbb Q[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb R^n$ of polynomials $f_1,\ldots, f_p\in\mathbb Q[X_1,\ldots, X_n]$ is of first importance in several applications in optimization and in real algebraic geometry. These points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a Jacobian matrix. We investigate the complexity of solving this problem by using Gr\"obner basis algorithms under genericity assumptions on the coefficients of the input polynomials. The main results refine known complexity bounds (which depend on the maximum $D=\max(deg(f_1),\ldots,deg(f_p),deg(q))$) to bounds which depend on the list of degrees $(deg(f_1),\ldots,deg(f_p),deg(q))$: we prove that the Gr\"obner basis computation can be performed in $\delta^{O(\log(A)/\log(G))}$ arithmetic operations in $\mathbb Q$, where $\delta$ is the algebraic degree of the ideal vanishing on the critical points, and $A$ and $G$ are the arithmetic and geometric average of a multiset constructed from the sequence of degrees. As a by-product, we prove that solving such generic optimization problems with Gr\"obner bases requires at most $D^{O(n)}$ arithmetic operations in $\mathbb Q$, which meets the best known complexity bound for this problem. Finally, we illustrate these complexity results with experiments, giving evidence that these bounds are relevant for applications.