Abstract
The quadratic maximization problem with rank constraint means
where
n and
m are two given positive integers with
n ≤
m,
Q is a given
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real symmetric matrix,
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is the unit sphere in
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and
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stands for the Euclidean inner product. In this note, motivated by Briët, Filho and Vallentin's and Ye's excellent works [J. Briët, F. Filho, and F. Vallentin,
The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.; Y. Ye,
Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226], we first derive the relative approximation ratio
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for the above problem with arbitrary symmetric matrix
Q. While
Q was supposed to be positive semidefinite in [J. Briët, F. Filhom, and F. Vallentin,
The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.]. Secondly, we extend such a relative approximation ratio to a generalization of the above optimization problem. Finally, a slightly improved relative approximation ratio
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for the optimization problem considered in [Y. Ye,
Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226] is presented.