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!