Section:
New Results
Gröbner bases and critical points
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,etc.). 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 be polynomials in of degree ,
be their complex variety and be the projection map
. The critical points of the
restriction of to are defined by the vanishing of
and some maximal minors of the Jacobian matrix
Indus associated to . Such a system is algebraically
structured: the ideal it generates is the sum of a determinantal
ideal and the ideal generated by . In [26] ,
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 , the
complexity is polynomial in the generic number of critical points,
i.e.
More particularly, in the
quadratic case , the complexity of such a Gröbner basis
computation is polynomial in the number of variables and
exponential in . We also give experimental evidence supporting
these theoretical results.