I am considering a linear program with some equality constraints and some inequality constraints. (I will give the precise form at the end of this question. It comes from reinforcement learning.)
I am interested in the case when the constraints of the linear program are unknown. Specifically, I want to apply the results from this paper to my linear program when the constraints are unknown.
However, I am concerned that the assumptions listed in the paper might not apply to my linear program (or rather, the class of linear program I am considering).
I would like to list the assumptions from the linked paper and then get feedback about whether the type of linear program I am considering can, in principle, satisfy the listed assumptions.
Assumption 1: The linear program should define a polytope that is the bounded intersection of finitely many halfspaces.
Assumption 2: The optimal solution to the LP is unique.
Assumption 3: The (unknown) polytope the linear program defines lies inside the unit $\ell_{\infty}$-ball.
Assumption 4: The coordinates of each vertex of the polytope the linear program defines can be written with $N$ bits of precision.
Assumption 5: Any subset of $d - 1$ rows of the matrix $A$ (of the linear program) have rank $d - 1$.
Assumption 6: Each vertex of the polytope the linear program defines is the intersection of exactly $d$ hyperplanes.
Questions: Are any of these assumptions inherently problematic for the type of linear program I am considering? I guess I'm concerned that the fact that I have equality constraints mean that my optimal solution will lie on a hyperplane, and then this will mean I violate the assumptions (for example, Assumption 1) somehow.
I appreciate any help.
Precise form of the linear program:
\begin{align*} &\text{maximize } \frac{\langle \mu, r \rangle}{1 - \gamma}\\ &\text{subject to } \gamma \sum_{s \in \mathcal S} \sum_{a \in \mathcal A} \mathcal P(s' \mid s, a) \mu(s, a) + (1 - \gamma)\rho(s') = \sum_{a \in \mathcal A} \mu(s', a)\\ &\qquad \text{for all } s' \in \mathcal S,\\ &\text{and } \mu(s, a) \geq 0 \text{ for all } (s, a) \in \mathcal S \times \mathcal A,\\ &\text{and } \langle \mu, c_k \rangle \leq C_k \text{ for } k = 1, \dots, K. \end{align*}