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

$\displaystyle t^{*}=argmax\frac{v^TAv}{||v||_2}$

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

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

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

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

Where $\displaystyle {t}$ denotes the $\displaystyle n-1$ dimensional column parameter vector, each entry taking the integer. $\displaystyle t^{*}$ represents the optimal solution. $\displaystyle c_k,x_k,w_k$ denotes the $\displaystyle N$ dimensional column vectors, the entries of which are known. $\displaystyle q$ 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 $\displaystyle m$, $\displaystyle n$ and $\displaystyle N$. What's happiest, you can yield an analytical solution to the optimization problem.