Exact objectives of random linear programs and mean widths of random polyhedrons

TL;DR

This paper addresses the exact asymptotic behavior of random linear programs (rlps) by connecting the optimal objective to the mean width of Gaussian polytopes. Using random duality theory (RDT), it derives a closed-form, α-dependent characterization $\xi_{opt}(\alpha;\mathbf{a})$ for the optimal value in the large-n regime and shows that the randomized primal and dual objectives converge to this value (i.e., $\xi_{rp}=\xi_{rd}=\xi_{opt}$). For the special case $\mathbf{a}=\mathbf{1}$, the result yields a concrete expression whose double of the optimum equals the mean width concentration of the polyhedron $\{\mathbf{x}\;|\;A\mathbf{x}\leq\mathbf{1}\}$, with the limit $\alpha\to\infty$ recovering known scaling $\xi_{opt}(\alpha) \sim \sqrt{2\log\alpha}$. The framework is robust to non-Gaussian data via Lindeberg-type arguments and opens pathways to analyze other random structures beyond rlps.

Abstract

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 $α=\lim_{n\rightarrow\infty}\frac{m}{n}\in(0,\infty)$, 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-ε) ξ_{opt}(α;\mathbf{a}) \leq \min_{A\mathbf{x}\leq \mathbf{a}}\mathbf{c}^T\mathbf{x} \leq (1+ε) ξ_{opt}(α;\mathbf{a}) \right ) \longrightarrow 1, \end{eqnarray*} where \begin{equation*} ξ_{opt}(α;\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{2π}} \right ) }{n} }. \end{equation*} For example, for $\mathbf{a}=\mathbf{1}$, one uncovers \begin{equation*} ξ_{opt}(α) = \min_{x>0} \sqrt{x^2- x^2 α\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{2π}} \right ) }. \end{equation*} Moreover, $2 ξ_{opt}(α)$ is precisely the concentrating point of the mean width of the polyhedron $\{\mathbf{x}|A\mathbf{x} \leq \mathbf{1}\}$.

Figures (1)

• Figure 1: Gaussian polyhedron -- Optimal objective $\xi_{opt}(\alpha)={\mathbb E}_A\xi$

Theorems & Definitions (8)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof