Linear programming with infinitely many constraints I wish to study the following linear program
$$\begin{array}{ll} \text{minimize} & \mathrm c^{\top} \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\\ & \mathrm x \geq 0\end{array}$$
where 


*

*$\mathrm A$ is an infinite matrix with a finite number of nonzero elements in each row. In other words, each constraint only contains a finite number of variables.

*$\mathrm c$ only contains a finite number of nonzero elements. 


Are there any references on this problem? I would like to know if the standard results of finite linear programming involving basic feasible solutions and extreme points also hold for this situation as well. 
If no references are available, any intuition about how the finite programming results would or would not apply would be also appreciated. Thank you!
 A: It sounds weird.  You essentially only care about the finitely many $x_i$ for which $c_i \neq 0$.  But the constraints involving those variables you like might involves lots of variables you don't care about.  Ultimately, this seems equivalent to a finite linear program in the sense that as far as you care, the matrix $Ax = b$ could be replaced by some finite matrix equation $A' x' = b'.$ whose solution set is the projection of $\{x \ : \ Ax =b\}$ onto the variables you actually care about.
But I'm not sure to what extent that even makes sense since for instance we could have the equations $x_0 = x_1$ and for all $n \geq 1$
$$
x_{n+1} - x_{n} + \frac{(-1)^{n}}{n} = 0.
$$
This matrix would have the finite row condition you want (and a finite column condition), but it's not at all clear how to solve it.  One would be tempted to sum all the equations together yielding the telescoping
$$
x_0 = x_1 + \sum_{n\geq 1} \left [x_{n+1} - x_{n} + \frac{(-1)^{n}}{n} \right ] =  \sum_{n\geq 1} \frac{(-1)^{n}}{n},
$$
but on the other hand, the value of this sum depends on the order that you add the rows together, whereas the formal sum $\sum_{n} ( x_{n+1} - x_{n} )$ seems not to depend on reordering.
So in short, the problem sounds like you should be able to get rid of all the infinite parts you don't care about, but on the other hand, these infinitely many equations don't seem to make a lot of sense to begin with.
Perhaps if you had some sort of absolute convergence condition or something on $A$?
A: *

*H. Edwin Romeijn, Robert L. Smith, Shadow Prices in Infinite-Dimensional Linear Programming, Mathematics of Operations Research, Vol. 23, No. 1, February 1998.



We consider the class of linear programs that can be
  formulated with infinitely many variables and constraints but where
  each constraint has only finitely many variables. This class includes
  virtually all infinite horizon planning problems modeled as infinite
  stage linear programs. Examples include infinite horizon production
  planning under time-varying demands and equipment replacement under
  technological change. We provide, under a regularity condition,
  conditions that are both necessary and sufficient for strong duality
  to hold. Moreover we show that, under these conditions, the Lagrangean
  function corresponding to any pair of primal and dual optimal
  solutions forms a linear support to the optimal value function, thus
  extending the shadow price interpretation of an optimal dual solution
  to the infinite dimensional case. We illustrate the theory through an
  application to production planning under time-varying demands and
  costs where strong duality is established.

