  We consider \emph{random linear programs} (rlps) as a subclass of
\emph{random optimization problems} (rops) and study their typical behavior.
Our particular focus is on appropriate linear objectives which connect the rlps
to the mean widths of random polyhedrons/polytopes. Utilizing the powerful
machinery of \emph{random duality theory} (RDT) \cite{StojnicRegRndDlt10}, we
obtain, in a large dimensional context, the exact characterizations of the
program's objectives. In particular, for any
$\alpha=\lim_{n\rightarrow\infty}\frac{m}{n}\in(0,\infty)$, any unit vector
$\mathbf{c}\in{\mathbb R}^n$, any fixed $\mathbf{a}\in{\mathbb R}^n$, and $A\in
{\mathbb R}^{m\times n}$ with iid standard normal entries, we have
  \begin{eqnarray*}
  \lim_{n\rightarrow\infty}{\mathbb P}_{A} \left ( (1-\epsilon)
\xi_{opt}(\alpha;\mathbf{a})
  \leq \min_{A\mathbf{x}\leq \mathbf{a}}\mathbf{c}^T\mathbf{x} \leq
(1+\epsilon) \xi_{opt}(\alpha;\mathbf{a}) \right ) \longrightarrow 1,
\end{eqnarray*}
  where
  \begin{equation*} \xi_{opt}(\alpha;\mathbf{a}) \triangleq \min_{x>0}
\sqrt{x^2- x^2 \lim_{n\rightarrow\infty} \frac{\sum_{i=1}^{m} \left (
\frac{1}{2} \left (\left ( \frac{\mathbf{a}_i}{x}\right )^2 + 1\right )
\mbox{erfc}\left( \frac{\mathbf{a}_i}{x\sqrt{2}}\right ) -
\frac{\mathbf{a}_i}{x} \frac{e^{-\frac{\mathbf{a}_i^2}{2x^2}}}{\sqrt{2\pi}}
\right )
  }{n} }. \end{equation*}
  For example, for $\mathbf{a}=\mathbf{1}$, one uncovers
  \begin{equation*}
  \xi_{opt}(\alpha)
  =
  \min_{x>0} \sqrt{x^2- x^2 \alpha \left ( \frac{1}{2} \left ( \frac{1}{x^2} +
1\right ) \mbox{erfc} \left ( \frac{1}{x\sqrt{2}}\right ) - \frac{1}{x}
\frac{e^{-\frac{1}{2x^2}}}{\sqrt{2\pi}} \right ) }. \end{equation*}
  Moreover, $2 \xi_{opt}(\alpha)$ is precisely the concentrating point of the
mean width of the polyhedron $\{\mathbf{x}|A\mathbf{x} \leq \mathbf{1}\}$.
