Critical points and Gröbner bases: the unmixed case

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in non-convex polynomial optimization which occurs in a wide range of scientific applications (control theory, chemistry, economics,...). Critical points also play a central role in recent algorithms of effective real algebraic geometry. Experimentally, it has been observed that Gröbner basis algorithms are efficient to compute such points. Therefore, recent software based on the so-called Critical Point Method are built on Gröbner bases engines. Let <i>f</i>₁,..., <i>f</i>_<i>p</i> be polynomials in Q[<i>x</i>₁,...,<i>x</i>_<i>n</i>] of degree <i>D</i>, <i>V</i> ⊂ C^<i>n</i> be their complex variety and π₁ be the projection map (<i>x</i>₁,...,<i>x</i>_<i>n</i>) -> <i>x</i>₁. The critical points of the restriction of π₁ to <i>V</i> are defined by the vanishing of <i>f</i>₁,...,<i>f</i>_<i>p</i> and some maximal minors of the Jacobian matrix associated to <i>f</i>₁,...,<i>f</i>_<i>p</i>. Such a system is algebraically structured: the ideal it generates is the sum of a determinantal ideal and the ideal generated by <i>f</i>₁,...,<i>f</i>_<i>p</i>. We provide the first complexity estimates on the computation of Gröbner bases of such systems defining critical points. We prove that under genericity assumptions on <i>f</i>₁,...,<i>f</i>_<i>p</i>, the complexity is polynomial in the generic number of critical points, i.e. <i>D</i>^<i>p</i>(<i>D</i> - 1)^<i>n−p</i>(n-1/p-1). More particularly, in the quadratic case <i>D</i> = 2, the complexity of such a Gröbner basis computation is polynomial in the number of variables <i>n</i> and exponential in <i>p</i>. We also give experimental evidence supporting these theoretical results.