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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.
A Representation Theorem for Certain Partially Ordered Commutative Rings
"... Let A be a commutative ring with 1, let P A be a preordering of higher level (i.e. 0; 1 2 P , 1 62 P , P + P P , P P P and A 2n P for some n 2 N) and let M A be an archimedean Pmodule (i.e. 1 2 M , 1 62 M , M +M M , P M M and 8 a 2 A 9n 2 N n a 2 M ). We endow X(M) := f' 2 Hom(A ..."
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Cited by 39 (1 self)
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Let A be a commutative ring with 1, let P A be a preordering of higher level (i.e. 0; 1 2 P , 1 62 P , P + P P , P P P and A 2n P for some n 2 N) and let M A be an archimedean Pmodule (i.e. 1 2 M , 1 62 M , M +M M , P M M and 8 a 2 A 9n 2 N n a 2 M ). We endow X(M) := f' 2 Hom(A;R) j '(M) R+ g with the weak topology with respect to all mappings ba : X(M) ! R, ba(') := '(a) and consider the representation M : A ! C(X(M);R), a 7! ba. We nd that X(M) is a nonempty compact Hausdorff space. Further we prove that 1 M (C + (X(M);R)) = fa 2 A j 8n 2 N 9 k 2 N k(1 + na) 2 Mg and ker( M ) = fa 2 A j 8n 2 N 9 k 2 N k(1 na) 2 Mg. (By C + (X(M);R) we denote the set of all continuous functions which do not take negative values.)
A sum of squares approximation of nonnegative polynomials
 SIAM J. Optim
, 2006
"... Abstract. We show that every real nonnegative polynomial f can be approximated as closely as desired (in the l1norm of its coefficient vector) by a sequence of polynomials {fɛ} that are sums of squares. The novelty is that each fɛ has a simple and explicit form in terms of f and ɛ. Key words. Real ..."
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Cited by 27 (5 self)
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Abstract. We show that every real nonnegative polynomial f can be approximated as closely as desired (in the l1norm of its coefficient vector) by a sequence of polynomials {fɛ} that are sums of squares. The novelty is that each fɛ has a simple and explicit form in terms of f and ɛ. Key words. Real algebraic geometry; positive polynomials; sum of squares; semidefinite programming. AMS subject classifications. 12E05, 12Y05, 90C22 1. Introduction. The
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
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
Convergent LMI relaxations for nonconvex quadratic programs
"... We consider the general nonconvex quadratic programming problem and provide a series of convex positive semidefinite programs (or LMI relaxations) whose sequence of optimal values is monotone and converges to the optimal value of the original problem. It improves and includes as a special case the w ..."
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Cited by 3 (0 self)
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We consider the general nonconvex quadratic programming problem and provide a series of convex positive semidefinite programs (or LMI relaxations) whose sequence of optimal values is monotone and converges to the optimal value of the original problem. It improves and includes as a special case the wellknown Shor's LMI formulation. Often, the optimal value is obtained at some particular early relaxation as shown on some nontrivial test problems from Floudas and Pardalos [9].
Bounds for Representations of Polynomials Positive on Compact SemiAlgebraic Sets
"... . By Schmudgen's Theorem, polynomials f 2 R[X1 ; : : : ; Xn ] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares oe i from R[X1 ; : : : ; Xn ]. We show the existence of effective bounds on the degrees of the oe i by ..."
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Cited by 2 (0 self)
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. By Schmudgen's Theorem, polynomials f 2 R[X1 ; : : : ; Xn ] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares oe i from R[X1 ; : : : ; Xn ]. We show the existence of effective bounds on the degrees of the oe i by proving first a suitable version of Schmudgen's Theorem over nonarchimedean real closed fields, and then applying the Compactness and Completeness Theorem from Model Theory. 1. Statement of the Results In this paper we deal with `effectivity problems' in connection with the following theorem proved by K. Schmudgen in [Sch]. Let R[X] = R[X 1 ; : : : ; Xn ] be the ring of real polynomials in X 1 ; : : : ; Xn . For h 1 ; : : : ; hm 2 R[X], the set S(h) = S(h 1 ; : : : ; hm ) = fa 2 R n j h 1 (a) 0; : : : ; hm (a) 0g is a basic closed semialgebraic subset of R n . Schmudgen's Theorem states that if S(h) is bounded, then every f 2 R[X] that is strictly positive on S(h) admits a represent...
Likelihood Maximization on Phylogenetic Trees
, 2010
"... We consider the problem of reconstructing the root ancestral state for a binary character on a fixedtopology binary phylogenetic tree, and compare the methods of maximum parsimony and maximum likelihood with the goal of checking if the methods are in agreement. For the likelihood method, we conside ..."
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We consider the problem of reconstructing the root ancestral state for a binary character on a fixedtopology binary phylogenetic tree, and compare the methods of maximum parsimony and maximum likelihood with the goal of checking if the methods are in agreement. For the likelihood method, we consider the problem under the Neyman N2 model with a molecular clock constraint. We show that finding the maximum likelihood assignment requires solving a constrained polynomial optimization problem. We review recently developed techniques of polynomial optimization, exploring some theoretical and practical aspects. Using these optimization techniques, we are able to find, within a small numerical error, the optimal likelihood values and certify the global optimality, thereby allowing us to reconstruct the ancestral state. Using both methods, we solve the reconstruction problem on all fourleaf and fiveleaf topologies. We find that the two methods agree on all fourleaf
Complexity Estimates for Representations of Schmüdgen Type
, 2000
"... Let F={f1,..., fj} and let K be a closed basic set in Rn given by the polynomial inequalities f1 \ 0,..., fj \ 0. Let S{F} be the semiring generated by the fk and the squares in R[x1,..., xn]. For example, if F={f1} then S{F}=s1+s2f1, where s1, s2 are sums of squares of polynomials. Schmüdgen has sh ..."
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Let F={f1,..., fj} and let K be a closed basic set in Rn given by the polynomial inequalities f1 \ 0,..., fj \ 0. Let S{F} be the semiring generated by the fk and the squares in R[x1,..., xn]. For example, if F={f1} then S{F}=s1+s2f1, where s1, s2 are sums of squares of polynomials. Schmüdgen has shown that if K is compact then any polynomial strictly positive on K belongs to S{F}. This paper develops a result of Schmüdgen type for functions in one dimension merely nonnegative on K. For this, it is necessary to add additional hypotheses, such as the proximity of complex zeros, to compensate for the loss of strict positivity necessary for Schmüdgen’s result. © 2001 Elsevier Science (USA) 1.