Some concrete aspects of Hilbert's 17th Problem

By Bruce Reznick. 2000

190 citations recorded by the source corpus.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/744fa20ff9c4782be81c4df0387785c826e80f7c

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In Applied Mathematics

In Atomic and Molecular Physics, and Optics

In Geometry and Topology

Citations in the corpus

Concepts and techniques of optimization on the sphere (2014, background): “, [19, 25, 26]) as well as the Bi-Quadratic Optimization problem over unit spheres (see, e.”
Optimal Inequalities in Probability Theory: A Convex Optimization Approach (2005, background): “Motzkin (see Reznick [35]) has shown that the polynomial M(x, y, z) = xy + xy + z − 3xyz is nonnegative without being a sum of squares of polynomials.”
Introducing SOSTOOLS: a general purpose sum of squares programming solver (2002, background): “multivariate quadratic, and binary quartic polynomials (see [13] and the references therein).”
Optimality conditions and finite convergence of Lasserre’s hierarchy (2012, unknown): “n problem: ˆ min x 6 1 +x 2 +x63 +3x2 1x 2 2x 2 3 −x4 1(x2 2 +x 3) −x42(x2 3 +x2 1) −x4 3(x 1 +x 2) s.t. x2 1 +x2 2 +x2 3 = 1. The objective is the Robinson form which is nonnegative but not SOS (cf. [24]). The minimum fmin = 0, and the global minimizers are 1 √ 3 (±1,±1,±1), 1 √ 2 (±1,±1,0), 1 √ 2 ”
Symmetry groups, semidefinite programs, and sums of squares (2002, methodology): “Examples from the literature [1, 26 ,22,20,29] have been computed with SOSTOOLS [24].”
A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates (2007, methodology): “framework based on sum of squares decompositions of multivariate polynomials [43] and semidefinite programming [51].”
Nonparametric Instrumental Variable Estimation Under Monotonicity (2015, methodology): “…thank A. Belloni for pointing out this possibility. represent any non-negative polynomial f : R→ R as a sum of squares of polynomials (see the survey by Reznick (2000), for example), i.e. f(x) = p̃(x)′Mp̃(x) where p̃(x) is the vector of monomials up to some order and M a matrix of coefficients.”
Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems (2009, unknown): “32 30 v2 [ m at h. FA ] 1 6 Se Keywords: truncated moment sequence, Riesz functional, (strict) K-positivity, determining set, moment matrix, representing measure AMS subject classifications: 47A57, 44A60, 47N40, 47A20”