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39
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 ma ..."
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Cited by 577 (48 self)
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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 matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided.
Optimization of polynomials on compact semialgebraic sets
 SIAM J. OPTIM
"... A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxat ..."
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Cited by 57 (4 self)
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A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ∗ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x ∗ , we prove that every sequence of “nearly ” optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x ∗.
Positivity sums of squares and the multidimensional moment problem
 II, Adv. Geom
, 2005
"... Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive soluti ..."
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Cited by 39 (8 self)
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Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the KMoment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n ≥ 2, we show that the KMoment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (‡) ∀f ∈ R[X],f ≥ 0onK ⇒∃q ∈ T such that ∀ real ɛ>0,f + ɛq ∈ T. This property is strictly weaker than the one given in Schmüdgen (1991), but in turn it implies a positive solution to the KMoment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which (‡) holds. Finally, we provide a list of 8 open problems.
Sums of squares on real algebraic surfaces
 Manuscripta Math. 119
, 2006
"... Abstract. Consider real polynomials g1,..., gr in n variables, and assume that the subset K = {g1 ≥ 0,..., gr ≥ 0} of Rn is compact. We show that a polynomial f has a representation f = e∈{0,1} r se · g e1 1 · · · ger r in which the se are sums of squares, if and only if the same is true in every ..."
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Cited by 25 (7 self)
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Abstract. Consider real polynomials g1,..., gr in n variables, and assume that the subset K = {g1 ≥ 0,..., gr ≥ 0} of Rn is compact. We show that a polynomial f has a representation f = e∈{0,1} r se · g e1 1 · · · ger r in which the se are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim(K) = 2 and every polynomial f with fK ≥ 0 has a representation (∗). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given.
Optimization of polynomial functions
 Can. Math. Bull
"... Recently progress has been made in the development of algorithms for optimizing polynomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing metho ..."
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Cited by 22 (4 self)
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Recently progress has been made in the development of algorithms for optimizing polynomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing methods. The idea traces back to work of Shor [16][17] and is further developed by Parrilo [10] and by Parrilo and Sturmfels [11] and by Lasserre [7][8]. In [7][8] Lasserre describes an extension of the method to minimizing a polynomial on an arbitrary basic closed semialgebraic set and uses a result due to Putinar [13] to prove that the method produces the exact minimum in the compact case. In the general case it produces a lower bound for the minimum. The ideas involved come from three branches of mathematics: algebraic geometry (positive polynomials), functional analysis (the moment problem) and optimization. This makes the area an attractive one not only from the computational but also from the theoretical point of view. In Section 1 we define three lower bounds for a polynomial and point out relationships
Representation of nonnegative polynomials, degree bounds and applications to optimization
, 2006
"... Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are giv ..."
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Cited by 22 (2 self)
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Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.
Distinguished representations of nonnegative polynomials
"... Abstract. Let g1,..., gr ∈ R[x1,..., xn] such that the set K = {g1 ≥ 0,..., gr ≥ 0} in R n is compact. We study the problem of representing polynomials f with fK ≥ 0 in the form f = s0 + s1g1 + · · · + srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assu ..."
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Cited by 18 (2 self)
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Abstract. Let g1,..., gr ∈ R[x1,..., xn] such that the set K = {g1 ≥ 0,..., gr ≥ 0} in R n is compact. We study the problem of representing polynomials f with fK ≥ 0 in the form f = s0 + s1g1 + · · · + srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assuming that the quadratic module of all such sums is archimedean, we establish a localglobal condition for f to have such a representation, visàvis the zero set of f in K. This criterion is most useful when f has only finitely many zeros in K. We present a number of concrete situations where this result can be applied. As another application we solve an open problem from [8] on onedimensional quadratic modules.
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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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 the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account 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.
NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY  SOME BASIC CONCEPTS AND FIRST IDEAS
, 2008
"... We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensätze) from real algebraic geometry can be generalized to noncommutative ∗algebras. A version of Stengle’s Positivstellensatz for n×n matrices of rea ..."
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Cited by 16 (3 self)
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We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensätze) from real algebraic geometry can be generalized to noncommutative ∗algebras. A version of Stengle’s Positivstellensatz for n×n matrices of real polynomials is proved.
Approximating positive polynomials using sums of squares, Canad
 Math. Bull
"... 2. Approximation theorems for quadratic modules 3. Representation of positive linear functionals ..."
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Cited by 14 (6 self)
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2. Approximation theorems for quadratic modules 3. Representation of positive linear functionals