164 problems

Let $\Lambda_1\in\mathbb{R}^{m\times n}$ and $\Lambda_2\in\mathbb{R}^{n\times n}$ be diagonal matrices, with $\Lambda_2\succ0$ having distinct diagonal entries, and let…

Rank-one optimal solution conjecture. If

Let $\mathbb{R}\langle \xi\rangle$ be the real closed field of algebraic Puiseux series, let $\mathcal{P}=\{P_1,\ldots,P_s\}\subset \mathbb{R}[X_1,\ldots,X_n]$, and set … For a bou…

Let $(x_k)$ be the iterates generated by the mini-batch stochastic approximation framework under the hypotheses stated in Remark 12 of Bolte and Pauwels (2021), with definable pote…

Let $T$ be a finite metric tree of total length $L$, and let $r>0$ satisfy $L\ge 2r$. Then there exists a probability distribution on $0$-good ball covers of $T$, with all ball rad…

Given a modular polynomial equation $f(x_1,\ldots,x_n)\equiv 0\pmod N$ and the associated lattice-based Coppersmith method, determine the optimal asymptotic region of exponents…

Consider the separable constrained optimization problem $\min_{x_1,x_2,x_3}\ f_1(x_1)+f_2(x_2)+f_3(x_3)$ subject to $A_1x_1+A_2x_2+Ix_3=b$, where the first two objective functions…

Determine whether there exist universal constants $C,c>0$ and a polynomial-time online signing algorithm such that, for every dimension $n\ge 1$ and every finite horizon $T$, when…

For every stochastic transition matrix $M=(m_{ij})_{i,j\in\{1,2\}}$, every $\alpha_1,\alpha_2\in[0,1)$, and every $C_1,C_2>0$, the two-site moving-target search problem has an opti…

Let $\rho\in\mathbb{N}$, and assume (A1) obtained from the preceding lemma. Let $(a_k)_{k\geqslant 0}$ be the sequence defined by the two Nesterov equations referred to in the sour…

For a finite point configuration $P\subset\mathbb{R}^2$, let its Steiner ratio be the ratio of the cost of an optimal Steiner tree for $P$ to the cost of a minimum spanning tree fo…

Let $\operatorname{FP}_{p,5,2}(X)$ be the $p$th frame potential of a five-point configuration $X$ in $\mathbb{R}^2$, and let $X_5^\perp$, $X_5^{(h)}$, $Y(\alpha)$, and $Z(\alpha)$…

Let $Q^{c1}_6$ be the positive quadratic form defining the lattice covering studied in the paper, and let lattice coverings in dimension $6$ be compared by their covering density.…

Extension conjecture. Theorem and Corollary remain valid (in substance) even under local oracles.

Middle-regime doubling conjecture. For every $m\ge4$ and every $n$ in this middle regime,

Shortest 2-design curve conjecture. The curve $\gamma$ is the shortest $2$-design curve in $S^2$, up to isometry and reparameterization. The conjecture is motivated by the precedin…

Let $\mathcal{A}\to\mathsf{WfMd}$ be a Lie algebroid of controlled intervention modes over a typed Markdown workflow space, with anchor $\rho:\mathcal{A}\to T\mathsf{WfMd}$. For co…

Global monotonicity conjecture. The Kubo–Mori update can only increase the log-likelihood:

Consider the minimum-energy problem for $k$ points on the unit interval with pairwise kernel $e^{-|x-y|^q}$, and write an odd number of points as … Let the global critical exponent…

Optimal condition-number threshold conjecture. The optimization landscape of the low-rank factorization problem is benign if

Let $N$ be the number of rotor directions and let $K$ denote the number of distinct one-dimensional solution branches. The numerical exhaustion method and solver resolution are the…

Boundary-point conjecture. For any optimal solution with at least nine points, each edge of the unit square $[0,1]^2$ contains exactly two points.

Let $g$ be a $mu_g$-strongly convex and $L_g$-smooth function, and let $M$ belong to the class of linear operators with singular values in $[mu_M,L_M]$. Write…

Let $\mathcal{S}^\kappa_{P,R,s^3}$ and $\mathcal{S}^\kappa_{P,R,S}$ denote the relaxations associated respectively with the PRs$^3$ and PRS formulations. Equality conjecture. … The…

Convergence of SVS-SPRING for LLQ. Under these assumptions and the SVS$(J,k,\lambda)$ sampling distribution, SPRING should satisfy the displayed convergence bound, where the hidden…