Sums of squares in Macaulay2

SIAM Journal on Optimization

We study sum of squares (SOS) relaxations to optimize polynomial functions over a set V ∩ R n , where V is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents V with a generic set of complex samples. This approach depends only on the geometry of V, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties, we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.

Journal of Symbolic Computation, 2012

Let f 1 , . . . , f p be in Q [X], where X = (X 1 , . . . , X n ) t , that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f ∈ Q[X] bounded below, consider the optimization problem of computing

2011

I thank my colleagues... ... Karim J. Becher for giving me an interesting research question and for his support and advice; ... Julia Hartmann for an inspiring course on the topic covered in [HH10] and [HHK09]; ... Kevin Hutchinson and his colleagues University College Dublin for their hospitality during a predoctoral research stay; ... David Leep for his encouragement and for inspiring discussions beyond the scope of this thesis; ... Claus Scheiderer for carefully reviewing my thesis and spotting non-neglectable mistakes before the final print; ... Jan Van Geel for his ideas and for being my go-to-guy for all geometry-related problems; ... Adrian Wadsworth for helping me find a more conceptual proof of a main result; .

2010

The sum of squares (SOS) decomposition technique allows numerical methods such as semidefinite programming to be used for proving the positivity of multivariable polynomial functions. It is well known that it is not an easy task to find Lyapunov functions for stability analysis of nonlinear systems. An algorithmic tool is used in this work for solving this problem. This approach is presented as SOS programming and solutions were obtained with a Matlab toolbox. Simple examples of SOS concepts, stability analysis for nonlinear polynomial and rational systems with uncertainties in parameters are presented to show the use of this tool. Besides these approaches, an alternative stability analysis for switched systems using a polynomial approach is also presented.

ACM SIGSAM Bulletin, 2003

2007

Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08, 2008

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).

Journal of Pure and Applied Algebra, 2009

Let f , g i , i = 1, . . . , l, h j , j = 1, . . . , m, be polynomials on R n and S := {x ∈ R n | g i (x) = 0, i = 1, . . . , l, h j (x) ≥ 0, j = 1, . . . , m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.

Singular is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, alge-braic geometry, and singularity theory. Singular's main computational objects are polynomials, ideals and modules over a large variety of rings. Singular features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. The new, upcoming version 2-2 includes also algorithms for a wide class of non-commutative algebras (Plural) and the possiblity for dynamic extension of the program at run-time (dynamic modules). Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root– finding, visualisation, and many more related functionalities.

Theoretical Computer Science, 1997

In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field. Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0. The new algorithm is more efficient than existing, deterministic algorithms based on Musser's squarefree algorithm. We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm. The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points. To demonstrate this, we present some timings using implementations in Maple (Char et al., 1991), where the new algorithm is used for Release 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system known to the author to use an implementation of Yun's modular algorithm mentioned above.