Abstract
We propose in this paper a fixed parameter polynomial algorithm for the cardinality-constrained quadratic optimization problem, which is NP-hard in general. More specifically, we prove that, given a problem of size n (the number of decision variables) and s (the cardinality), if the n−k largest eigenvalues of the coefficient matrix of the problem are identical for some 0 < k ≤ n, we can construct a solution algorithm with computational complexity of \({\mathcal{O}\left(n^{2k}\right)}\) . Note that this computational complexity is independent of the cardinality s and is achieved by decomposing the primary problem into several convex subproblems, where the total number of the subproblems is determined by the cell enumeration algorithm for hyperplane arrangement in \({\mathbb{R}^k}\) space.
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This work was supported by Research Grants Council of Hong Kong, under grants 414207 and 414808.
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Gao, J., Li, D. A polynomial case of the cardinality-constrained quadratic optimization problem. J Glob Optim 56, 1441–1455 (2013). https://doi.org/10.1007/s10898-012-9853-z
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DOI: https://doi.org/10.1007/s10898-012-9853-z