This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small ” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented, which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems. Lastly, we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization. Key words: Polynomials, sum of squares (SOS), sparsity, nonlinear least squares, polynomial system, nonlinear differential equations, sensor network localization 1

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.

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Abstract. We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis to algebras generated by semi-algebraic functions. In which case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization. A collection of simple examples and numerical experiments complement the theoretical parts of the article. 1.

We propose a general algorithm to enumerate all solutions of a zero-dimensional polynomial system with respect to a given cost function. The algorithm is developed and is used to study a polynomial system obtained by discretizing the steady cavity flow problem in two dimensions. The key technique on which our algorithm is based is to solve polynomial optimization problems via sparse semidefinite programming relaxations (SDPR) [20], which has been adopted successfully to solve reaction-diffusion boundary value problems in [13]. The cost function to be minimized is derived from discretizing the fluid’s kinetic energy. The enumeration algorithm’s solutions are shown to converge to the minimal kinetic energy solutions for SDPR of increasing order. We demonstrate the algorithm with SDPR of first and second order on polynomial systems for different scenarios of the cavity flow problem and succeed in deriving the k smallest kinetic energy solutions. The question whether these solutions converge to solutions of the steady cavity flow problem is discussed, and we pose a conjecture for the minimal energy solution for increasing Reynolds number.

The notion of convexity underlies important results in many parts of mathematics such as optimization, analysis, combinatorics, probability and number theory. The geometric foundations of the theory of convex sets date back to work of Minkowski, Carathéodory, and Fenchel around 1900. Since then, this area has expanded into a large number of directions and now includes topics such as high-dimensional spaces, convex analysis, polyhedral geometry, computational convexity, approximation methods and others. In the context of optimization, both theory and empirical evidence show that problems with convex constraints allow efficient algorithms. Many applications in the sciences and engineering involve optimization, and it is always extremely advantageous when the underlying feasible regions are convex and have practically useful representations as convex sets. A situation in which convexity has been well-understood is the study of convex polyhedra, which are the solution sets of finitely many linear inequalities [27, 86]. A context in algebraic geometry in which convexity arises is the theory of toric varieties. These are algebraic varieties derived from polyhedra [49, 73]. Both convex polyhedra and toric varieties have satisfactory computational techniques associated to them. Linear optimization over polyhedra is linear programming which admits interior-point algorithms that run in polynomial time. More generally, polyhedra can be

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Suppose that f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present a deterministic algorithm of complexity polynomial in logD that, for most inputs, counts the number of real roots of f. The best previous algorithms have complexity super-linear in D. We also discuss connections to sums of squares and A-discriminants, including explicit obstructions to expressing positive definite sparse polynomials as sums of squares of few sparse polynomials. Our key theoretical tool is the introduction of efficiently computable chamber cones, which bound regions in coefficient space where the number of real roots of f can be computed easily. Much of our theory extends to n-variate(n+3)-nomials.

We consider the problem of reconstructing the root ancestral state for a binary character on a fixed-topology binary phylogenetic tree, and compare the methods of maximum parsimony and maximum likelihood with the goal of checking if the methods are in agreement. For the likelihood method, we consider the problem under the Neyman N2 model with a molecular clock constraint. We show that finding the maximum likelihood assignment requires solving a constrained polynomial optimization problem. We review recently developed techniques of polynomial optimization, exploring some theoretical and practical aspects. Using these optimization techniques, we are able to find, within a small numerical error, the optimal likelihood values and certify the global optimality, thereby allowing us to reconstruct the ancestral state. Using both methods, we solve the reconstruction problem on all four-leaf and five-leaf topologies. We find that the two methods agree on all fourleaf

Based on the convergent sequence of SDP relaxations for a multivariate polynomial optimization problem (POP) by Lasserre, Waki et al. constructed a sequence of sparse SDP relaxations to solve sparse POPs efficiently. Nevertheless, the size of the sparse SDP relaxation is the major obstacle in order to solve POPs of higher degree. This paper proposes an approach to transform general POPs to quadratic optimization problems (QOPs), which allows to reduce the size of the SDP relaxation substantially. We introduce different heuristics resulting in equivalent QOPs and show how sparsity of a POP is maintained under the transformation procedure. As the most important issue, we discuss how to increase the quality of the SDP relaxation for a QOP. Moreover, we increase the accuracy of the solution of the SDP relaxation by applying additional local optimization techniques. Finally, we demonstrate the high potential of this approach through numerical results for large scale POPs of higher degree.