How to solve the coupled integer programming problem? I have the following integer linear programming problem:
$$\begin{equation*}
\begin{aligned}
& \underset{x}{\text{maximize}}
&& \sum_{k=1}^K\sum_{t=1}^Tx_{kt} \\
& \text{subject to}
&& \sum_{k=1}^Kx_{kt} \leqslant N, \; \forall t\\
&&& x_{kt}\cdot \alpha_k(t) \leqslant E_k(t), \; \forall k, t\\
&&& E_k(t+1) = E_k(t) - x_{kt}\cdot\alpha_k(t)+H_k(t), \; \forall k, t\\
&&& x_{kt}\in\{0, 1\}.
\end{aligned}
\end{equation*}$$
I cannot figure out how to solve it. I mean I cannot write it in a solver as :
\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f_0(x) \\
& \text{subject to}
& & Ax\leqslant b.
\end{aligned}
\end{equation*}
because of the dependencies between the inputs. $H_k(t)$ is a random variable that follows Bernoulli distribution which is given as input. $K, T, \alpha_k(t)$ are also given as input. I cannot give $E_k(t)$ as an input because of the last constraint.
How can I write it in a solver as a linear integer programming problem when $E_k(t)$ depends on the variables $x_{kt}$ and on the random variable $H_k(t)$?
 A: Notice that you can write the sequence $\{x_{kt}\}_{K\geq k\geq1,T\geq t\geq 1} $ at a vector as so: $$x := \begin{bmatrix}
    x_{11} \\ x_{21} \\ \vdots \\ x_{K1} \\ x_{12} \\ \vdots \\ x_{K2} \\ \vdots \\ \vdots \\  x_{KT}  \end{bmatrix}$$ 
Next, we can rewrite the last constraint recursively as $$ E_k(t)=E_k(0)-tx_{kt}\alpha_k(t)+tH_k(t).$$
With this, we can rewrite the second constraint as
$$ (t+1)\alpha_k(t)x_{kt}\leq E_k(0)+tH_k(t)\qquad \forall k,t$$
Now to rewrite all constraints into one matrix $A$ such that $Ax\leq b$ we set 
$$ A=\begin{bmatrix}
    1 & \dots & 1 & 0 & \dots & 0 & 0 & \dots & 0 \\
    0 & \dots & 0 & 1 & \dots & 1 & 0 & \dots & 0   \\
    \vdots & \dots & \vdots &  & \ddots & & \ddots   \\
    0 & \dots & 0 & \dots  & 0 & \dots & 1 &\dots & 1\\ 
    2\alpha_1(1) & 0 & \dots  & \dots & \dots & \dots& \dots  & \dots & 0 \\
0 & 2\alpha_2(1) & 0  & \dots & \dots & \dots& \dots  & \dots& 0\\
0 & 0 & \ddots  & 0 & \dots & \dots& \dots  & \dots & 0 \\
0 & \dots & 0  &  2\alpha_K(1) &0& \dots& \dots  & \dots & 0 \\
0 & \dots & 0  &  0 & 3\alpha_1(2)& 0& \dots  & \dots & 0 \\
0 & \dots & 0  &  0 & 0 & \ddots& 0  & \dots & 0 \\
0 & \dots & 0  &  0 & 0 &  0 & \ddots & \dots & 0 \\
0 & \dots & 0  &  0 & 0 &  0 & \dots  & & (T+1)\alpha_K(T) \\
\end{bmatrix} $$
with the vector $$b := \begin{bmatrix}
    N\\ \vdots\\ N\\ E_1(0)+ H_1(1)\\ \vdots\\ E_K(0)+ H_K(1)\\ E_1(0)+ H_1(2) \\ \vdots \\ E_K(0)+ H_K(2) \\  \vdots\\\vdots\\ E_K(0)+ H_K(T) \end{bmatrix}$$
With this the problem can be written as $$ \max_{x \text{ binary vector}} \|x\|_1 \\ \text{subject to } Ax\leq b$$  
