On Exact Polynomial Optimization

and Cholesky's decomposition. Next, we apply this algorithm to compute exact Polya and Putinar's repre-sentations respectively for positive de nite fo...

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On Exact Polynomial Optimization Victor MAGRON CNRS Verimag, Inria Paris Center, PolSys team

Mohab SAFEY EL DIN Sorbonne Universités, Inria Paris Center, PolSys team

Résumé. We consider the problem of nding exact sums of squares (SOS)

decompositions for certain classes of non-negative multivariate polynomials, while relying on semidenite programming (SDP) solvers. We start by providing a hybrid symbolic-numeric algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. This algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms.

We prove that bit complexity estimates on output size and

runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. This analysis is based on quantier elimination as well as bounds on the cost of the ellipsoid method and Cholesky's decomposition. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive denite forms and polynomials positive over basic compact semialgebraic sets.

We also compare the implemen-

tation of our algorithms with existing methods based on CAD or critical points.

Mots-clefs : optimization algorithms, positive polynomials, Polya's Positivstellensätz, Putinar's Positivstellensätz, sums of squares decomposition, semidenite programming, real algebraic geometry. Let




be the eld of rational (resp. real) numbers.

f ∈ Q[x] g1 (x) ≥ 0, . . . , gm (x) ≥ 0

deciding the non-negativity of by some constraints

We consider the problem of

n either over R or over a semi-algebraic set (with

gj ∈ Q[x]).





will denote the

maximum of the total degrees of these polynomials. As many other algorithmic problems in eective real algebraic geometry, this one is known to be NP hard [13]. The famous Cylindrical Algebraic Decomposition algorithm [7] allows to solve such problems in time doubly exponential in


(and polynomial in


This complexity

result has been improved later on, through the so-called critical point method, starting from [8] and a series of works [16, 9] which culminates with [5] to establish that this decision problem can be solved in time

((s + 1)d)O(n)

(see also [6, Chap.


These latter ones have been

extensively developed and optimized to obtain implementations which reect the complexity gain (see e.g. [2, 1, 4, 3]). However, all in all, these techniques are singly exponential in Besides, these algorithms are root nding algorithms: to decide the positivity of considered domain, one asks these algorithms to nd a point therein at which When



is positive, such algorithms will return an empty list without a

a posteriori.




over the

is negative.

certicate that can be

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To bypass the curse of exponential algorithms while computing certicates of non-negativity for the problems considered here, an approach based on

sums of squares (SOS) decompositions

(and their variants) has been popularized by Lasserre [12] and Parillo [14]. We refer to [13] and references therein for detailed surveys on this approach. In a nutshell, the idea is as follows.

f is non-negative over Rn if it can be written as an SOS s21 + · · · + s2r with si ∈ R[x] for 1 ≤ i ≤ r. P Also f is non-negative over the semi-algebraic set K if it can be m 2 2 written as s1 + · · · + sr + j=1 σj gj where σi is a sum of squares in R[x] for 1 ≤ j ≤ m. It

A polynomial

turns out that, thanks to the Gram matrix method, following e.g. [12, 14], computing such decompositions can be reduced to solving Linear Matrix Inequalities (LMI), which boils down to considering a semidenite programming (SDP) problem.

d = 2k , the decomposition f = s21 + · · · + s2r is T T a by-product of a decomposition of the form f (x) = vk (x) L DLvk (x) where vk is the vector of all monomials of degree ≤ k in Q[x], L is a lower triangular matrix with non-negative entries on the diagonal and D is a diagonal matrix with non-negative entries. The matrices L and D are obtained after computing a symmetric matrix G (the Gram matrix), semidenite T positive (all its eigenvalues are non-negative), such that f = vk Gvk . Such a matrix G is found For instance, on input

f ∈ Q[x]

using solvers for LMIs.

of even degree

Even if such inequalities can be solved symbolically (see [10]), the

degrees of the extensions are prohibitive on large examples. Besides, there exist fast numerical solvers for solving LMIs, e.g. SeDuMi [17], SDPA [18]. But using uniquely numerical solvers yields approximate non-negativity certicates. On our example, the matrices consequently the polynomials

s1 , . . . , s r )





are not known exactly.

This raises topical questions. The rst one is how to let interact symbolic computation with these numerical solvers to get


certicates? What to do when SOS certicates do not

exist? Also, given inputs with rational coecients, can we obtain certicates with rational coecients? For these questions, we inherit from previous contributions from Parillo and Peyrl [15] and next Kaltofen, Li, Yang and Zhi [11]. This work provides a new algorithmic framework to handle (un)-constrained polynomial optimization problems with exact rational SOS decompositions. The rst contribution is a hybrid symbolic-numeric algorithm, called


providing rational SOS decompositions for poly-

nomials belonging to the interior of the SOS cone. The main idea is to perturbate the input polynomial then to obtain an approximate Gram matrix of the perturbation by solving an SDP problem and eventually to recover an exact decomposition with the perturbation terms.

intsos to compute SOS of rational functions for positive denite Polyasos. Finally, we rely on Algorithm intsos to compute weighted SOS decompositions for polynomials positive on basic compact semialgebraic sets, yielding a third algorithm, called Putinarsos. When the input is an n-variate polynomial of degree d with integer coecients of maximum 2 O (n) and outputs SOS bitsize τ , we prove that Algorithm intsos runs in boolean time τ d O (n) polynomials of bitsize bounded by τ d . This also yields bit complexity analysis for Algorithm Polyasos and Algorithm Putinarsos. To the best of our knowledge, this is the rst Then, we rely on Algorithm

forms, based on Polya's representations, yielding a second algorithm, called

complexity estimates for the output of algorithms providing exact multivariate SOS decompositions. The three algorithms are implemented within a Maple library, called


We provide

benchmarks evaluating the performance against existing methods based on CAD or critical points.

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