A Coarse Idea

We meet with an integer optimizing problem, which is defined as follows:

y^{*}= \textrm{argmax}_{\begin{subarray}{c}{y \in \mathcal{Y}}\end{subarray}}\quad \frac{y^TX^T\left(\sum_{k=1}^{N}{b_kw_k^{\phantom{ k}T}}\right)Xy}{\sqrt{y^TX^TXy}}

Where \mathcal{Y} denotes \displaystyle n(n-1)/2 dimensional column parameter vector, each entry taking value from \left\{{-1,1}\right\}. y^{*} represents the optimal solution. b_i, w_i denote m dimensional column vector, whose values are known. The matrix X \in \mathcal{R}^{m \times n(n-1)/2} is defined on m dimensional column vectors x_{1},x_{2} \ldots x_{n}, where each column vector is definitively. The columns of X are calculated by substraction of every two column vector x_{i}, x_{j} \textrm{ when } i<j. It is listed as follows:

X =\left[\begin{array}{cccccccc}<br /> |&\ldots&|&|&\dots&|&\ldots&|\\ x_{1}-x_{2}&\ldots&x_{1}-x_{n}&x_{2}-x_{3}&\dots&x_{2}-x_{n}&\ldots&x_{n-1}-x_{n}\\ |&\ldots&|&|&\dots&|&\ldots&|\\ \end{array}\right]

The Goal
Our goal is to build an algorithm, which can solve the optimization problem in polynomial order time with m, n and N. What's happiest, you can yield an analytical solution to the optimization problem.