Results 1  10
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474,510
Global Optimization with Polynomials and the Problem of Moments
 SIAM Journal on Optimization
, 2001
"... We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear mat ..."
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Cited by 569 (47 self)
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matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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Cited by 18 (1 self)
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of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
PRESERVING POSITIVE POLYNOMIALS AND BEYOND
, 801
"... Abstract. Following the classical approach of PólyaSchur theory [12] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of nonnegative and elliptic polynomials. Contents ..."
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Abstract. Following the classical approach of PólyaSchur theory [12] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of nonnegative and elliptic polynomials. Contents
A criterion for positive polynomials
, 2004
"... Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semidefinite. In this note we produce a necessary condition for positivity and a sufficient condit ..."
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Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semidefinite. In this note we produce a necessary condition for positivity and a sufficient
SIGNATURE PAIRS OF POSITIVE POLYNOMIALS
"... Abstract. A wellknown theorem of Quillen says that if r(z, z̄) is a bihomogeneous polynomial on Cn positive on the sphere, then there exists d such that r(z, z̄)‖z‖2d is a squared norm. We obtain effective bounds relating this d to the signature of r. We obtain the sharp bound for d = 1, and for d ..."
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Abstract. A wellknown theorem of Quillen says that if r(z, z̄) is a bihomogeneous polynomial on Cn positive on the sphere, then there exists d such that r(z, z̄)‖z‖2d is a squared norm. We obtain effective bounds relating this d to the signature of r. We obtain the sharp bound for d = 1
Optimization over Positive Polynomial Matrices
 Proceedings of the International Symposium on Mathematical Theory of Networks and Systems
, 2000
"... Positive polynomial matrices play a fundamental role in systems and control theory. We give here a simplified proof of the fact that the convex set of positive polynomial matrices can be parameterized using block Hankel and block Toeplitz matrices. We also show how to derive efficient computational ..."
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Cited by 1 (0 self)
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Positive polynomial matrices play a fundamental role in systems and control theory. We give here a simplified proof of the fact that the convex set of positive polynomial matrices can be parameterized using block Hankel and block Toeplitz matrices. We also show how to derive efficient computational
Positive Polynomials and Robust Stabilization with FixedOrder Controllers
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2002
"... Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixedorder controllers robustly stabilizing a linear system subject to polytopic uncertainty is then propos ..."
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Cited by 47 (15 self)
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Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixedorder controllers robustly stabilizing a linear system subject to polytopic uncertainty
The Askeyscheme of hypergeometric orthogonal polynomials and its qanalogue
, 1998
"... We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erent ..."
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Cited by 580 (6 self)
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We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di
On Hilbert’s construction of positive polynomials
"... Abstract. In 1888, Hilbert described how to find real polynomials which take only nonnegative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present ..."
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Cited by 11 (2 self)
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and present many such polynomials. 1. History and Overview A real polynomial f(x1,...,xn) is psd or positive if f(a) ≥ 0 for all a ∈ R n; it is sos or a sum of squares if there exist real polynomials hj so that f = ∑ h 2 j. For forms, we follow the notation of [4] and use Pn,m to denote the cone of real psd
Results 1  10
of
474,510