A polynomially solvable case of unconstrained (-1,1)-quadratic fractional optimization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909389189808128 |
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| author | Yang, Meijia Xia, Yong |
| author_facet | Yang, Meijia Xia, Yong |
| contents | In this paper, we consider an unconstrained (-1,1)-quadratic fractional optimization in the following form: $\min_{x\in\{-1,1\}^n}~(x^TAx+α)/(x^TBx+β)$, where $A$ and $B$, given by their nonzero eigenvalues and associated eigenvectors, have ranks not exceeding fixed integers $r_a$ and $r_b$, respectively. We show that this problem can be solved in $O(n^{r_a+r_b+1}\log^2 n)$ by the accelerated Newton-Dinkelbach method when the matrices $A$ has nonpositive diagonal entries only, $B$ has nonnegative diagonal entries only. Furthermore, this problem can be solved in $O(n^{r_a+r_b+2}\log^2 n)$ when $A$ has $O(\log(n))$ positive diagonal entries, $B$ has $O(\log(n))$ negative diagonal entries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_09190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A polynomially solvable case of unconstrained (-1,1)-quadratic fractional optimization Yang, Meijia Xia, Yong Optimization and Control 90C32, 90C20, 90C10 In this paper, we consider an unconstrained (-1,1)-quadratic fractional optimization in the following form: $\min_{x\in\{-1,1\}^n}~(x^TAx+α)/(x^TBx+β)$, where $A$ and $B$, given by their nonzero eigenvalues and associated eigenvectors, have ranks not exceeding fixed integers $r_a$ and $r_b$, respectively. We show that this problem can be solved in $O(n^{r_a+r_b+1}\log^2 n)$ by the accelerated Newton-Dinkelbach method when the matrices $A$ has nonpositive diagonal entries only, $B$ has nonnegative diagonal entries only. Furthermore, this problem can be solved in $O(n^{r_a+r_b+2}\log^2 n)$ when $A$ has $O(\log(n))$ positive diagonal entries, $B$ has $O(\log(n))$ negative diagonal entries. |
| title | A polynomially solvable case of unconstrained (-1,1)-quadratic fractional optimization |
| topic | Optimization and Control 90C32, 90C20, 90C10 |
| url | https://arxiv.org/abs/2411.09190 |