We meet with an integer optimizing problem, which is defined as follows:
t^{*}=argmax\frac{v^TAv}{||v||_2}

v=\sum_{k=1}^{n-1}{t_k\left(x_k-x_{k+1}\right)}+\sum_{k=1}^{n}\left(2k-n-1\right)x_k

A=\sum_{k=1}^{m}{c_kw_k^T}

t_k \ge q_k,\quad k=1,2 \ldots n-1

q_k \in \left[0,k-1\right],\quad k=1,2 \ldots n-1

Where {t} denotes the n-1 dimensional column parameter vector, each entry taking the integer. t^{*} represents the optimal solution. c_k,x_k,w_k denotes the N dimensional column vectors, the entries of which are known. q denotes [tex]n-1[tex] dimensional integral column vector.

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