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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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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Cen, X., Xia, Y. Globally maximizing the sum of squares of quadratic forms over the unit sphere. Optim Lett 14, 1907–1919 (2020). https://doi.org/10.1007/s11590-019-01498-7
- Polynomial optimization
- Sum of squares
- Quadratic programming