Results 1  10
of
11
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 62 (9 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zerodimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.
Sums of squares and moment problems in equivariant situations
, 2008
"... We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group G over R acting on an affine Rvariety V, we consider the induced dual action on the coordinate ring R[V] and on the linear dual space of R[V]. In this setting, given an invariant closed ..."
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Cited by 7 (3 self)
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We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group G over R acting on an affine Rvariety V, we consider the induced dual action on the coordinate ring R[V] and on the linear dual space of R[V]. In this setting, given an invariant closed semialgebraic subset K of V (R), we study the problem of representation of invariant nonnegative polynomials on K by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on R[V] by invariant measures supported on K. To this end, we analyse the relation between quadratic modules of R[V] and associated quadratic modules of the (finitely generated) subring R[V] G of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional Kmoment problem. Most of our results are specific to the case where the group G(R) is compact.
Projection methods for conic feasibility problems, applications to polynomial sumofsquares decompositions
, 2009
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POSITIVITY AND OPTIMIZATION FOR SEMIALGEBRAIC FUNCTIONS
, 2009
"... 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 semialgebraic functions. In which case the standard global optimization problem with constraints given by element ..."
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Cited by 3 (0 self)
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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 semialgebraic 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.
Classifications of Linear Operators Preserving Elliptic, Positive and NonNegative Polynomials
"... Abstract. We characterize all linear operators on finite or infinitedimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and nonnegative polynomials, respectively. This is done by means of FischerFock dualities, Hankel forms, and convolutions with nonnegat ..."
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Cited by 1 (0 self)
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Abstract. We characterize all linear operators on finite or infinitedimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and nonnegative polynomials, respectively. This is done by means of FischerFock dualities, Hankel forms, and convolutions with nonnegative measures. We also establish higherdimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert’s 17th problem. 1.
unknown title
"... 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, thi ..."
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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 highdimensional 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 wellunderstood 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 interiorpoint algorithms that run in polynomial time. More generally, polyhedra can be
UNDECIDABILITY IN A FREE *ALGEBRA By
, 2007
"... MIHAI PUTINAR Abstract. The integral structure of the maximal weights of the finite dimensional representations of sl2(k) imply that a Tarski type decision principle for free ∗algebras with two or more generators does not hold. 1. ..."
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MIHAI PUTINAR Abstract. The integral structure of the maximal weights of the finite dimensional representations of sl2(k) imply that a Tarski type decision principle for free ∗algebras with two or more generators does not hold. 1.
Convergent relaxations of
, 903
"... polynomial optimization problems with noncommuting variables ..."
The Operator FejérRiesz Theorem
, 903
"... To the memory of Paul Richard Halmos. Abstract. The FejérRiesz theorem has inspired numerous generalizations in one and several variables, and for matrix and operatorvalued functions. This paper is a survey of some old and recent topics that center around Rosenblum’s operator generalization of th ..."
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To the memory of Paul Richard Halmos. Abstract. The FejérRiesz theorem has inspired numerous generalizations in one and several variables, and for matrix and operatorvalued functions. This paper is a survey of some old and recent topics that center around Rosenblum’s operator generalization of the classical FejérRiesz theorem.
NonCommutative Harmonic and Subharmonic Polynomials
, 2009
"... The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x1,...,xg). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h 2 ∆xp where ∆xp is the usual Laplacian. A symmetr ..."
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The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x1,...,xg). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h 2 ∆xp where ∆xp is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h): = Lap[p, h] takes positive semidefinite matrix values whenever matrices X1,...,Xg, H are substituted for the variables x1,..., xg, h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. 1 1