Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Shenglong Hu (), Guoyin Li (), Liqun Qi () and Yisheng Song ()
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Shenglong Hu: The Hong Kong Polytechnic University
Guoyin Li: University of New South Wales
Liqun Qi: The Hong Kong Polytechnic University
Yisheng Song: The Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2013, vol. 158, issue 3, No 5, 717-738

Abstract: Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of 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 its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.

Keywords: Symmetric tensors; Maximum eigenvalue; Sum of squares of polynomials; Semi-definite programming problem (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10957-013-0293-9

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