9 problems

Pataki's conjecture. Facial exposedness and niceness are equivalent: every facially exposed closed convex cone is nice.

Let $B^n$ be the unit ball in $\mathbb{R}^n$, and let $A\subseteq B^n$ be a measurable set containing no pair of points at distance $1$. Write $\mu$ for Lebesgue measure and let…

Let $G(\mathbb{R}^2,\{1\})$ be the graph whose vertices are points of $\mathbb{R}^2$, with two points adjacent when their distance is $1$, and let $\alpha_{\bar{\delta}}$ denote it…

Outer-approximation conjecture. The formulation might not be particularly challenging to solve via interior point methods, but it might be difficult to construct a good-quality out…

Let $\Gamma$ and $\Gamma^\circ$ be dual cones, let ${\cal G}$ be a face of $\Gamma^\circ$, and let ${\cal G}^\triangle$ be the face of $\Gamma$ conjugate to ${\cal G}$. Write…

Let $s\in\mathbb{Z}_{++}$ and let $\widehat{\psi}_{N,2s}(y)$ denote the Taylor-based approximation introduced in the paper for the exponential function. A function is second-order…

A conic linear optimization problem has a feasible region determined by its conic constraints; its vertices are the extreme points of that feasible region. Finiteness conjecture. T…

Existence conjecture. There exist conic mixed-integer programs whose subadditive dual is feasible but whose conic dual of the continuous relaxation is infeasible.

For each $n$, let $f(n)=(f_0(n),\ldots,f_n(n))$ be the intrinsic-volume sequence of the $n$-dimensional Lorentz cone, and let $*$ denote convolution of sequences. Lorentz-cone conv…