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59
Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity
 SIAM Journal on Optimization
, 2006
"... Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of ..."
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Cited by 121 (30 self)
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Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.
Convergent SDPRelaxations in Polynomial Optimization with Sparsity
 SIAM Journal on Optimization
"... Abstract. We consider a polynomial programming problem P on a compact semialgebraic set K ⊂ R n, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDPrelaxati ..."
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Cited by 57 (17 self)
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Abstract. We consider a polynomial programming problem P on a compact semialgebraic set K ⊂ R n, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDPrelaxation of order r has the following two features: (a) The number of variables is O(κ 2r) where κ = max[κ1, κ2] witth κ1 (resp. κ2) being the maximum number of variables appearing the monomials of f (resp. appearing in a single constraint gj(X) ≥ 0). (b) The largest size of the LMI’s (Linear Matrix Inequalities) is O(κ r). This is to compare with the respective number of variables O(n 2r) and LMI size O(n r) in the original SDPrelaxations defined in [11]. Therefore, great computational savings are expected in case of sparsity in the data {gj, f}, i.e. when κ is small, a frequent case in practical applications of interest. The novelty with respect to [9] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a byproduct, we also obtain a new representation result for polynomials positive on a basic closed semialgebraic set, a sparse version of Putinar’s Positivstellensatz [16]. 1.
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 54 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
On the complexity of Putinar’s Positivstellensatz
, 2005
"... Abstract. Let S = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = ∑ m i=0 ..."
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Cited by 42 (10 self)
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Abstract. Let S = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = ∑ m i=0 σigi where g0: = 1 and each σi is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms σigi in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre’s procedure for optimization of a polynomial subject to polynomial constraints. 1.
Matrix SumofSquares Relaxations for Robust SemiDefinite Programs
 Math. Program
, 2006
"... We consider robust semidefinite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sumofsquares decompositions, we suggest a systematic procedure to con ..."
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Cited by 33 (0 self)
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We consider robust semidefinite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sumofsquares decompositions, we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart. With a novel matrixversion of Putinar’s sumofsquares representation for positive polynomials on compact semialgebraic sets, we prove asymptotic exactness of the relaxation family under a suitable constraint qualification. If the uncertainty region is a compact polytope, we provide a new duality proof for the validity of Putinar’s constraint qualification with an a priori degree bound on the polynomial certificates. Finally, we point out the consequences of our results for constructing relaxations based on the socalled fullblock Sprocedure, which allows to apply recently developed tests in order to computationally verify the exactness of possibly smallsized relaxations.
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 31 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
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 28 (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
approximations of nonnegative polynomials via simple high degree perturbations
 Math. Z
"... Abstract. We show that every real polynomial f nonnegative on [−1, 1] n can be approximated in the l1norm of coefficients, by a sequence of polynomials {fεr} that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as ..."
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Cited by 27 (15 self)
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Abstract. We show that every real polynomial f nonnegative on [−1, 1] n can be approximated in the l1norm of coefficients, by a sequence of polynomials {fεr} that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semialgebraic set KS ⊂ R n with nonempty interior, then every polynomial nonnegative on KS can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on ɛ as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients. 1.
Global optimization of polynomials using gradient tentacles and sums of squares
 SIAM Journal on Optimization
"... We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n, it is therefore equivalent to look for the greatest low ..."
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Cited by 26 (0 self)
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We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n, it is therefore equivalent to look for the greatest lower bound of f on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when f is bounded from below but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic subsets of R n which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 23 (8 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.