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!



Linear programming with infinitely many constraints
* H. Edwin Romeijn, Robert L. Smith, [Shadow Prices in Infinite-Dimensional Linear Programming](https://www.researchgate.net/publication/266425025_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.
> 
> 
> 


