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A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Detecting global optimality and extracting solutions in GloptiPoly
 Chapter in D. Henrion, A. Garulli (Editors). Positive polynomials in control. Lecture Notes in Control and Information Sciences
, 2005
"... GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of po ..."
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Cited by 80 (12 self)
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GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations. 1
Optimization of polynomials on compact semialgebraic sets
 SIAM J. Optim
"... Abstract. 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 conve ..."
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Cited by 59 (4 self)
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Abstract. 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 ∗. 1. Introduction to Lasserre’s method Throughout the paper, we suppose 1 ≤ n ∈ N and abbreviate (X1,..., Xn) by ¯X. We let R [ ¯ X] denote the ring of real polynomials in n indeterminates. Suppose we are given a so called basic closed semialgebraic set, i.e., a set S: = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0}
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 56 (16 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 52 (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.
Minimizing polynomials via sum of squares over the gradient ideal
 Math. Program
"... A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown ..."
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Cited by 52 (16 self)
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A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is strictly positive on the gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.
Truncated Kmoment problems in several variables
 J. OPERATOR THEORY
, 2005
"... Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, w ..."
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Cited by 40 (11 self)
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Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is ratomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an ratomic (minimal) representing measure supported in a semialgebraic set KQ subordinate to a family Q ≡ {qi} m i=1 ⊆ R[t1,..., tN] if and only if M(n) is positive semidefinite and admits a rankpreserving extension M(n + 1) for which the associated localizing matrices Mqi (n + [ 1+deg qi]) are positive semidefinite (1 ≤ i ≤ m); in this case, µ (as 2 above) satisfies supp µ ⊆ KQ, and µ has precisely rank M(n) − rank Mqi (n + [ 1+deg qi]) atoms in 2 Z(qi) ≡ { t ∈ R N: qi(t) = 0} , 1 ≤ i ≤ m.
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...
Sum of squares methods for sensor network localization
, 2006
"... We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of t ..."
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Cited by 30 (3 self)
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We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, and discuss its various properties. When distances are given exactly, this SOS relaxation often returns true sensor locations. At each step of interior point methods solving this SOS relaxation, the complexity is O(n 3), where n is the number of sensors. When the distances have small perturbations, we show that the sensor locations given by this SOS relaxation are accurate within a constant factor of the perturbation error under some technical assumptions. The performance of this SOS relaxation is tested on some randomly generated problems.
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
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