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31
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 58 (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.
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 ∗.
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 51 (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.
SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 29 (4 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank.
EXPLOITING SYMMETRIES IN SDPRELAXATIONS FOR POLYNOMIAL OPTIMIZATION
"... In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investi ..."
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Cited by 9 (2 self)
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In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.
Truncated multivariable moment problems with finite variety
"... Let β ≡ {βi}i∈Zd+,i≤2n denote a real ddimensional multisequence of degree 2n, with moment matrix M(n), and let V ≡ V (M(n)) denote the associated algebraic variety. For the case v ≡ card V < +∞, we prove that β has a representing measure if and only if r ≡ rank M(n) ≤ v and there exists a p ..."
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Cited by 8 (3 self)
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Let β ≡ {βi}i∈Zd+,i≤2n denote a real ddimensional multisequence of degree 2n, with moment matrix M(n), and let V ≡ V (M(n)) denote the associated algebraic variety. For the case v ≡ card V < +∞, we prove that β has a representing measure if and only if r ≡ rank M(n) ≤ v and there exists a positive moment matrix extension M ≡ M(n + v − r + 1) satisfying rank M ≤ card V (M). For the class of recursively determinate moment matrices M(n), we present a computational algorithm for establishing the existence (or nonexistence) of an extensionM as above and, in the positive case, for computing a minimal representing measure for β. We also show that for the case r < v < +∞, it is possible for β to admit a representing measure µ with card supp µ < v; equivalently, in this case supp µ may be a proper subset of V (M(n)).
THE TRUNCATED TRACIAL MOMENT PROBLEM
"... Abstract. We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in noncommuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence ca ..."
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Cited by 6 (3 self)
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Abstract. We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in noncommuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefinite and of finite rank. A truncated tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply this theory via duality to investigate tracepositive polynomials in noncommuting variables. 1.