# How to Minimize A Function Where The Number of Variables is Unknown

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?

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May be you can have a look at the simplex method, Dual simplex method,...etc. They are the ones which i learnt when i studied this course, but now i have forgotten! –  anonymous Aug 28 '10 at 8:13
@Chandru1, as far as I know these methods require you to separate out the known matrix/vector values, which is not possible in my case. –  Graviton Aug 28 '10 at 8:15
Are the dimensions of $A$, $b$, or $f$ available? –  Ｊ. Ｍ. Aug 28 '10 at 8:35
@J. Mangaldan, that's the problem, the dimension of $f$ is tied to $x$, and hence it is also not available. –  Graviton Aug 28 '10 at 8:57
Maybe just post the actual problem you have, and let the collective minds of this site wade through the thicket themselves? –  Ｊ. Ｍ. Aug 28 '10 at 9:03