The convex Positivstellensatz in a free algebra

Helton, J. William; Klep, Igor; McCullough, Scott

doi:10.1016/j.aim.2012.04.028

Cited by 53 publications

(57 citation statements)

References 18 publications

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“…Further, this paper gives an efficient algorithm for computing the L-real radical of I. This Positivstellensatz extends and unifies two different lines of results: (1) the free real Nullstellensatz of [CHMN13,Nel] which gives an algebraic certificate corresponding to one polynomial being zero on the free variety where others are zero; this is (P) with L = 1; (2) the convex Positivstellensatz of [HKM12,KS11] which is (P) without I; i.e., I = {0}. The representation (P) has a number of additional consequences which will be presented.…”

Section: Abstract By a Results Of Helton And Mcculloughmentioning

confidence: 99%

Noncommutative polynomials nonnegative on a variety intersect a convex set

Helton¹,

Klep²,

Nelson³

2014

Journal of Functional Analysis

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Abstract. By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D • L of a linear matrix inequality (LMI) L(X) ≻ 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz.For example, given a generic convex free semialgebraic set D • L we determine all "(strong sense) defining polynomials" p for D • L . Such polynomials must have the formwhere q i , r j are matrices of polynomials, and C j are real matrices satisfying C j L = LC j . This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and all ι ∈ I vanish if and only ifwhere each p i , q j and r k are matrices of polynomials of appropriate dimension, and each ι k is an element of the "L-real radical" of I. In this representation, we can restrict p i , q i , ι k and r k to be elements of a low-dimensional subspace of matrices of polynomials, and in particular, their degrees depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. This Positivstellensatz extends and unifies two different lines of results: (1) the free real Nullstellensatz of [CHMN13, Nel] which gives an algebraic certificate corresponding to one polynomial being zero on the free variety where others are zero; this is (P) with L = 1; (2) the convex Positivstellensatz of [HKM12, KS11] which is (P) without I; i.e., I = {0}. The representation (P) has a number of additional consequences which will be presented.

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Section: Abstract By a Results Of Helton And Mcculloughmentioning

confidence: 99%

Noncommutative polynomials nonnegative on a variety intersect a convex set

Helton¹,

Klep²,

Nelson³

2014

Journal of Functional Analysis

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“…In subsequent work [HKMNS] we shall exploit how the ideas presented here apply to operator algebras and complete positivity [ER00, BL04, Pau02, Pis03], by using the results from [HKM,HKM12].…”

Section: Proof Of Theoremmentioning

confidence: 99%

An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

Klep

Schweighofer

2013

Mathematics of OR

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Abstract. Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.

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“…One, free positivity is an analogue of classical real algebraic geometry, a theory of polynomial inequalities embodied in Positivstellensätze. As is the case with the sum of squares result above (Theorem 8.1), generally free Positivstellensätze have cleaner statements than do their commutative counterparts; see, e.g., [53,27,39,33] for a sample. Free convexity, the second branch of free real algebraic geometry, arose in an effort to unify a torrent of ad hoc techniques which came on the linear systems engineering scene in the mid 1990s.…”

Section: Theorem 81 (Helton [27]) a Nonnegative (Suitably Defined) mentioning

confidence: 90%

“…As such it also serves as a point of entry into the larger field of free real algebraic geometry and makes contact with noncommutative real algebraic geometry [27,30,32,33,38,47,48,53,59,62,63], free analysis and free probability (lying at the origins of free analysis; cf. [64]), and free analytic function theory and free harmonic analysis [28,29,34,54,60,69,70,46].…”

Section: Introductionmentioning

confidence: 99%

Chapter 8: Free Convex Algebraic Geometry

Helton¹,

Klep²,

McCullough³

2012

Semidefinite Optimization and Convex Algebraic Geometry

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A new development is extension of the algebraic certificates of real algebraic geometry to noncommutative polynomials, thereby giving a theory of noncommutative polynomial inequalities. Here we shall focus on convexity aspects of noncommutative real algebraic geometry, and we shall see this leads to a very rigid structure. Our subject pertains to optimization problems where the unknowns are matrices.

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The convex Positivstellensatz in a free algebra

Abstract: openAccessArticle: FalsePage Range: 516-516doi: 10.1016/j.aim.2012.04.028Harvest Date: 2016-01-12 15:10:31issueName:cover date: 2012-09-10pubType

Cited by 53 publications

References 18 publications

Noncommutative polynomials nonnegative on a variety intersect a convex set

Noncommutative polynomials nonnegative on a variety intersect a convex set

An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

Chapter 8: Free Convex Algebraic Geometry

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