I have a standard linear programming problems I want to solve:
$$ \min_x f^T x \text{ such that } \left\{ \begin{aligned} A\cdot x &\le b, \\ A_{eq}\cdot x &= b_{eq}, \\ lb \le x &\le ub. \end{aligned} \right. $$
$F^T$ is a vector that doesn't involve $x$.
The problem with the above minimization is that, I don't know what $n$, the number of variable $x$ is — it is also a part of the minimization. There are a few constraints governing how the vector $x$ should behave. Additionally, there are constraints linking from $x_i$ to $x_{i+1}$. This means that given $x_i$, we know how to form the constraint for $x_{i+1}$. Also, even though we don't know about $n$, but in my problem I can easily construct the corresponding $f_i$ term for each $x_i$.
The constraints are complicated in the sense that it is not easy to express the constraints in the following form:
$$ Ax \leq a $$
and
$$ Bx = b $$
What I can do, at best, is to express $a$ and $b$ involving a first order of $x$ ( i.e., no $x^2$ and above). The matrix $A$ and $B$ are known values with no involvement of $x$.
I understand that linear programming can be used to tackle problems such as this. But the two problems I mention above ( don't know what $n$ is, and cannot separate out the matrix/vector easily) stop me from proceeding.
Is there any other techniques I can use to solve the problem?