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Finding the maximum eigenvalue of essentially nonnegative symmetric tensors via sum of squares programming
, 2012
"... Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on ..."
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Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on a PerronFrobenius type theorem for nonnegative tensors. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in turn, can be equivalently rewritten as a semidefinite programming problem. Then, using this sum of squares programming problem, we also provide upper as well as lower estimate for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor.
A QCQP Approach to Triangulation
"... Abstract. Triangulation of a threedimensional point from n ≥ 2 twodimensional images can be formulated as a quadratically constrained quadratic program. We propose an algorithm to extract candidate solutions to this problem from its semidefinite programming relaxations. We then describe a sufficien ..."
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Abstract. Triangulation of a threedimensional point from n ≥ 2 twodimensional images can be formulated as a quadratically constrained quadratic program. We propose an algorithm to extract candidate solutions to this problem from its semidefinite programming relaxations. We then describe a sufficient condition and a polynomial time test for certifying when such a solution is optimal. This test has no false positives. Experiments indicate that false negatives are rare, and the algorithm has excellent performance in practice. We explain this phenomenon in terms of the geometry of the triangulation problem. 1
Global Optimality Principles for Polynomial Optimization Problems over Box or Bivalent Constraints by Separable Polynomial
"... In this paper we present necessary conditions for global optimality for polynomial problems over box or bivalent constraints using separable polynomial relaxations. We achieve this by completely characterizing global optimality of separable polynomial problems with box as well as bivalent constrai ..."
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In this paper we present necessary conditions for global optimality for polynomial problems over box or bivalent constraints using separable polynomial relaxations. We achieve this by completely characterizing global optimality of separable polynomial problems with box as well as bivalent constraints. Then, by employing separable polynomial underestimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. The underestimators are constructed using the sum of squares convex polynomials. The significance of our optimality condition is that they can be numerically checked by solving semidefinite programming problems. We illustrate the versatility of our optimality conditions by simple numerical examples. ∗The authors are grateful to the referees and the editor for their constructive comments and helpful suggestions which have contributed to the final preparation of the paper. Research was partially supported