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...

0 downloads 107 Views 163KB Size
Journées SMAI MODE 2018

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

Q

(resp.

R)

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]).

Further,

d

K

dened

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

n

(and polynomial in

d).

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.

13]).

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

f

checked

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

a posteriori.

f

f

n.

over the

is negative.

certicate that can be

Journées SMAI MODE 2018

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 )

L

and

D

(and

are not known exactly.

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

exact

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

intsos,

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

multivsos.

We provide

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

Journées SMAI MODE 2018

Références [1] B. Bank, M. Giusti, J. Heintz, and G.-M. Mbakop. Polar varieties and ecient real equation solving: the hypersurface case. Journal of Complexity, 13(1):527, 1997. [2] B. Bank, M. Giusti, J. Heintz, and G.-M. Mbakop. Polar varieties and ecient real elimination. Mathematische Zeitschrift, 238(1):115144, 2001. [3] B. Bank, M. Giusti, J. Heintz, and L.-M. Pardo. Generalized polar varieties: Geometry and algorithms. Journal of complexity, 2005. [4] B. Bank, M. Giusti, J. Heintz, and L.M. Pardo. On the intrinsic complexity of point nding in real singular hypersurfaces. Information Processing Letters, 109(19):11411144, 2009. [5] S. Basu, R. Pollack, and M.-F. Roy. A new algorithm to nd a point in every cell dened by a family of polynomials. In Quantier elimination and cylindrical algebraic decomposition. Springer-Verlag, 1998. [6] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics). Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006. [7] G. E Collins. Quantier elimination for real closed elds by cylindrical algebraic decompostion. In Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 2023, 1975, pages 134183. Springer, 1975. [8] D. Grigoriev and N. Vorobjov. Solving systems of polynomials inequalities in subexponential time. Journal of Symbolic Computation, 5:3764, 1988. [9] J. Heintz, M.-F. Roy, and P. Solernò. On the theoretical and practical complexity of the existential theory of the reals. The Computer Journal, 36(5):427431, 1993. [10] D. Henrion, S. Naldi, and M. Safey El Din. Exact Algorithms for Linear Matrix Inequalities. SIAM Journal on Optimization, 26(4):25122539, 2016. [11] E. Kaltofen, B. Li, Z. Yang, and L. Zhi. Exact certication of global optimality of approximate factorizations via rationalizing sums-of-squares with oating point scalars. In Proceedings of the twenty-rst international symposium on Symbolic and algebraic computation, pages 155164. ACM, 2008. [12] J.-B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. SIAM Journal on Optimization, 11(3):796817, 2001. [13] M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157270. Springer, 2009. [14] P. A. Parrilo. Structured Semidenite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000. [15] H. Peyrl and P.A. Parrilo. Computing sum of squares decompositions with rational coecients. Theoretical Computer Science, 409(2):269281, 2008. [16] J. Renegar. On the computational complexity and geometry of the rst order theory of the reals. Journal of Symbolic Computation, 13(3):255352, 1992. [17] J. F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, 1998. [18] M. Yamashita, K. Fujisawa, K. Nakata, M. Nakata, M. Fukuda, K. Kobayashi, and K. Goto. A high-performance software package for semidenite programs : SDPA7. Technical report, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 2010.