Globally maximizing the sum of squares of quadratic forms over the unit sphere


We first lift the problem of maximizing the sum of squares of quadratic forms over the unit sphere to an equivalent nonlinear optimization problem, which provides a new standard quadratic programming relaxation. Then we employ a simplicial branch and bound algorithm to globally solve the lifted problem and show that the time-complexity is linear with respect to the number of all nonzero entries of the input matrices under certain conditions. Numerical results demonstrate the efficiency of the new algorithm.

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    There are more selections of a simplex to cover the hyper-rectangle \([ \underline{t}, \overline{t}]\), see [20]. Here we take the simplest one since its diameter is easy to compute and overestimate, as required in Lemma 1.


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This research was supported by National Natural Science Foundation of China under Grants 11822103, 11571029, 11771056 and Beijing Natural Science Foundation Z180005.

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Correspondence to Yong Xia.

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Cen, X., Xia, Y. Globally maximizing the sum of squares of quadratic forms over the unit sphere. Optim Lett 14, 1907–1919 (2020).

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  • Polynomial optimization
  • Sum of squares
  • Quadratic programming
  • Branch-and-bound