Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

Abstract

We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sum-of-squares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13).The numeric SOSes produced by the current fixed precision semi-definite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11's exact linear algebra. Therefore, before projection we refine the SOSes by rank-preserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP ("SDP-free SOS"), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.

Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

“…Sum-ofsquares rational lower bound certificates were introduced in [12] to overcome the high algebraic degree in the exact real algebraic minima.…”

“…The bivariate case n = 2 can be solved by Seidenberg's [26] algorithm (see also [9] and [11]), which is generalized to arbitrarily many variables via Lagrangian multipliers in [1,25] or used in nonstandard decision methods [29]. Alternatively, one can use Artin's theorem of sumof-squares and semidefinite programming (see, e.g., [12,14]). …”

The “Seven Dwarfs” of Symbolic Computation

“…The first are the numerical optimization algorithms for semidefinite programming. The second is a symbolic technique for converting an imprecise SOS with floating point coefficients to an exact identity over the rational numbers [Peyrl and Parrilo 2008;Kaltofen et al 2008Kaltofen et al , 2012. Among the recent successes are the proof of the Monotone Column Permanent Conjecture for n = 4 [Kaltofen et al 2009], which was completed shortly before the general conjecture could be established, the Bessis-Moussa-Villani (BMV) conjecture for m ≤ 13 [Klep and Schweighofer 2008], new SOS proofs for many known inequalities, and a deformation analysis approach to Seidenberg's problem of Section 4 [Hutton et al 2010].…”

A model problem for global optimization

“…The lower bounds in Table I are taken from a paper by Kaltofen et al [9] improving the results in [8]. He transforms the problem into a semidefinite programming problem and shows that a certain function is a sum of squares.…”