42 problems

Rank-one optimal solution conjecture. If

The truncate-and-relax threshold conjecture. If $I_2(\alpha,\beta) > 1$, then

Let $\delta(m,n,r)$ denote the algebraic degree of semidefinite programming, and let the formula in Theorem be the formula established there under a stated restriction on the range…

Let $G_n$ be the Schrijver graph, and let $G_n^{\boxtimes 2}$ denote its strong square. The Schrijver number of a graph $G$, denoted by $\vartheta'(G)$, is the corresponding streng…

Path-graph robustness conjecture. For all $n\geq 1$,

Let $G$ be an $n$-vertex graph, and let $\xi_f(G)$ denote the projective rank of $G$. Projective-rank energy conjecture. … This is proposed as a weakening of Fajtlowicz's graph ene…

Maximal-gamma conjecture. For every positive integer $\ell$, the maximum value of $\gamma_{\ell-1}(G,a)$ among $\ell$-minimal graphs is attained by a sparse stretched clique…

Enumeration conjecture. There are exactly $49$ non-isomorphic $3$-minimal graphs.

Cohn–de Laat–Leijenhorst conjecture. Three-point semidefinite programming bounds prove that $C$ is a maximal spherical code.

Fix $N$, a $1$-smooth convex function $f$, an initial point $x_0$, a minimizer $x_\star$, and gradient-descent iterates $x_0,\ldots,x_N$ with constant stepsize $\alpha$. Write…

Let $\mathcal{C}$ be the set of $24$ vertices of a snub cube, regarded as unit vectors in $\mathbb{R}^3$. Snub-cube optimality conjecture. Three-point bounds prove that…

For each integer $k e 0$, let a Kerdock spherical code be the spherical code in $\mathbb{R}^{2^{2k}}$ constructed from the Kerdock binary code, with $2^{4k}+2^{2k+1}$ points and…

Saunderson–Chandrasekaran–Parrilo–Willsky conjecture. For arbitrarily small $\varepsilon>0$, if $n<(\frac14-\varepsilon)\cdot d^2$, then with high probability over $z_1,\ldots,z_n$…

Let $\text{\sc Beta}$ denote the convex relaxation of the ball-constrained nonconvex quadratic program defined in the paper, with matrix variable $W$ and vectors $\ell_1,\ell_2$ as…

Asymptotic Gross–Kunze inner-product conjecture. For each $\lambda$ there exists a strictly positive scalar $c$ such that

Asymptotic linear bound conjecture. The optimal objective satisfies

Let $n$ be a positive integer, let $P_x$ be the region associated with a sign vector $x\in\{-1,+1\}^n$, and let $(C,c)$ be a quadratic-programming input. The preceding proposition…

Let $n$ be the dimension parameter, and let $(C,c)$ be the randomly sampled input used by algorithm. The algorithm repeatedly applies Step. The iteration-bound conjecture. terminat…

A lattice packing in $\mathbb{R}^4$ has density bounded above by the lattice three-point bound, obtained from a semidefinite-programming formulation of the three-point bound. Sharp…

Let $X(t)$ be the trajectory of the general first ansatz for the semidefinite program, and write $X(0)\succ F$ when $X(0)-F$ is positive definite. Let $F$ be a feasible solution of…

Let $G$ be a graph, let $M_G$ be its graph matrix, let $F_G(x)=(x^{\circ2})^TM_Gx^{\circ2}$, and let … Define $\vartheta^{(r)}(G)=\min\{t:t(A_G+I)-J\in\mathcal K_n^{(r)}\}$. The fi…

Let $G_1$ and $G_2$ be two graphs with vertex sets … Let $(X_1,S_1)$ and $(X_2,S_2)$ be the primal-dual solution pairs to the max-cut SDPs on $G_1$ and $G_2$, respectively. Let $G$…

Finite-convergence conjecture. For any graph $G$,

Pseudodistribution product inequality.

Let $m$ be the size of the symmetric matrices and let $d$ denote the dimension parameter for the central curve. Write $\mathrm{genus}(m,d)$ for its genus. The complementary paramet…