On box-constrained total least squares problem

Abstract

<p style='text-indent:20px;'>We study box-constrained total least squares problem (BTLS), which minimizes the ratio of two quadratic functions with lower and upper bounded constraints. We first prove that (BTLS) is NP-hard. Then we show that for fixed number of dimension, it is polynomially solvable. When the constraint box is centered at zero, a relative <inline-formula><tex-math id="M1">\begin{document}$ 4/7 $\end{document}</tex-math></inline-formula>-approximate solution can be obtained in polynomial time based on SDP relaxation. For zero-centered and unit-box case, we show that the direct nontrivial least square relaxation could provide an absolute <inline-formula><tex-math id="M2">\begin{document}$ (n+1)/2 $\end{document}</tex-math></inline-formula>-approximate solution. In the general case, we propose an enhanced SDP relaxation for (BTLS). Numerical results demonstrate significant improvements of the new relaxation.