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 $\{\mathbf{x}|A\mathbf{x} \leq \mathbf{1}\}$.

翻译：我们将随机线性规划问题视为随机优化问题的一个子类，并研究其典型行为。特别关注的是适当的线性目标，这些目标将随机线性规划与随机多面体/多胞体的平均宽度联系起来。利用随机对偶理论的强大框架，我们在高维背景下获得了这些问题目标的精确刻画。具体而言，对于任意$\alpha=\lim_{n\rightarrow\infty}\frac{m}{n}\in(0,\infty)$、任意单位向量$\mathbf{c}\in{\mathbb R}^n$、任意固定向量$\mathbf{a}\in{\mathbb R}^n$以及具有独立同分布标准正态分量的矩阵$A\in {\mathbb R}^{m\times n}$，我们有 \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*} 其中 \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*} 例如，当$\mathbf{a}=\mathbf{1}$时，可得到 \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*} 此外，$2 \xi_{opt}(\alpha)$恰好是多面体$\{\mathbf{x}|A\mathbf{x} \leq \mathbf{1}\}$平均宽度的集中点。