A polynomially solvable case of unconstrained (-1,1)-quadratic fractional optimization

TL;DR

This paper introduces a polynomial-time algorithm for a specific class of unconstrained (-1,1)-quadratic fractional optimization problems, leveraging eigenvalue structure and matrix diagonal properties.

Contribution

It presents the first polynomial-time solution for this class of fractional quadratic problems under certain eigenvalue and diagonal entry conditions.

Findings

The problem can be solved in polynomial time under specified eigenvalue and diagonal entry constraints.

The accelerated Newton-Dinkelbach method achieves the solution with complexity depending on matrix ranks.

The approach extends to cases with logarithmic numbers of positive or negative diagonal entries.

Abstract

In this paper, we consider an unconstrained (-1,1)-quadratic fractional optimization in the following form: $\min_{x\in\{-1,1\}^n}~(x^TAx+\alpha)/(x^TBx+\beta)$, 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.