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The Zeigenvalues of a Symmetric Tensor and its Application to Spectral Hypergraph Theory
 NUMER. LINEAR ALGEBRA APPL.
, 2013
"... In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Zeigenvalues of a real symmetric tensor with even order. We first establish that the maximum Zeigenvalue function is a continuous and convex function on the symmetric tensor sp ..."
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Cited by 23 (19 self)
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In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Zeigenvalues of a real symmetric tensor with even order. We first establish that the maximum Zeigenvalue function is a continuous and convex function on the symmetric tensor space, and so, provide formulas of the convex conjugate function and ϵsubdifferential of the maximum Zeigenvalue function. Consequently, for anmth order ndimensional tensorA, we show that the normalized eigenspace associated with maximum Zeigenvalue function is ρthorder Hölder stable at A with ρ = 1 m(3m−3)n−1−1. As a byproduct, we also establish that the maximum Zeigenvalue function is always at least ρthorder semismooth at A. As an application, we introduce the characteristic tensor of a hypergraph and show that the maximum Zeigenvalue function of the associated characteristic tensor provides a natural link for the combinatorial structure and the analytic structure of the underlying hypergraph. Finally, we establish a variational formula for the second largest Zeigenvalue for the characteristic tensor of a hypergraph and use it to provide lower bounds for the bipartition width of a hypergraph. Some numerical examples are also provided to show how one can compute the largest/2ndlargest Zeigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula.
Semidefinite relaxations for best rank1 tensor approximations
 SIAM JOUNRAL ON MATRIX ANALYSIS AND APPLICATIONS
, 2014
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LINEAR OPTIMIZATION WITH CONES OF MOMENTS AND NONNEGATIVE POLYNOMIALS
, 2013
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Optimization on linear matrix inequalities for polynomial systems control
, 2013
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RankConstrained Fundamental Matrix Estimation by Polynomial Global Optimization Versus the EightPoint Algorithm
, 2012
"... The fundamental matrix can be estimated from point matches. The current gold standard is to bootstrap the eightpoint algorithm and twoview projective bundle adjustment. The eightpoint algorithm first computes a simple linear least squares solution by minimizing an algebraic cost and then computes ..."
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The fundamental matrix can be estimated from point matches. The current gold standard is to bootstrap the eightpoint algorithm and twoview projective bundle adjustment. The eightpoint algorithm first computes a simple linear least squares solution by minimizing an algebraic cost and then computes the closest rankdeficient matrix. This article proposes a singlestep method that solves both steps of the eightpoint algorithm. Using recent result from polynomial global optimization, our method finds the rankdeficient matrix that exactly minimizes the algebraic cost. The current gold standard is known to be extremely effective but is nonetheless outperformed by our rankconstrained method boostrapping bundle adjustment. This is here demonstrated on simulated and standard real datasets. With our initialization, bundle adjustment consistently finds a better local minimum (achieves a lower reprojection error) and takes less iterations to converge.
INVERSE POLYNOMIAL OPTIMIZATION
, 2012
"... Abstract. We consider the inverse optimization problem associated with the polynomial program f ∗ = min{f(x) : x ∈ K} and a given current feasible solution y ∈ K. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial ˜ f (which may be of s ..."
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Abstract. We consider the inverse optimization problem associated with the polynomial program f ∗ = min{f(x) : x ∈ K} and a given current feasible solution y ∈ K. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial ˜ f (which may be of same degree as f if desired) with the following properties: (a) y is a global minimizer of ˜ f on K with a Putinar’s certificate with an a priori degree bound d fixed, and (b), ˜ f minimizes ‖f − ˜ f ‖ (which can be the ℓ1, ℓ2 or ℓ∞norm of the coefficients) over all polynomials with such properties. Computing ˜ fd reduces to solving a semidefinite program whose optimal value also provides a bound on how far is f(y) from the unknown optimal value f ∗. The size of the semidefinite program can be adapted to computational capabilities available. Moreover, if one uses the ℓ1norm, then ˜ f takes a simple and explicit canonical form. Some variations are also discussed. 1.
Semidefinite relaxations for semiinfinite polynomial programming
 INFINITE MOMENT PROBLEM 19 1Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK. S7N 5E6, Canada Email address: mehdi.ghasemi@usask.ca, marshall@math.usask.ca 2Fachbereich Mathematik und Statistik, Universität Konstanz 78
, 2014
"... Abstract. This paper studies how to solve semiinfinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP ..."
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Abstract. This paper studies how to solve semiinfinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.
Robust SOSConvex polynomial programs: Exact SDP relaxations
 Optim. Lett
, 2014
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