181 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…

For every number of points $n\ge 1$, integers $\ell,k\ge 1$ with $k>\ell$, and configurations $x_1,\ldots,x_n\in\mathbb{R}^{\ell}$ and $y_1,\ldots,y_n\in\mathbb{R}^{k}$, define … I…

Let $n\ge 3$ and $m\ge 3$, let $\{v_1,\ldots,v_n\}\subset\mathbb{R}^{n-1}$ be the regular simplex frame, and let $T_{m,n}$ be the associated regular simplex tensor of order $m$. Th…

Consider a convex multi-objective optimization problem $\min (f_1(x),\ldots,f_m(x))$ over $X=\{x\in\mathbb{R}^n:g_j(x)\le 0\ \text{for }j=1,\ldots,r\}$, where the objective functio…

For integers $n\ge 2$ and $1\le p\le\infty$, let $\lambda_3(n,k,p,q)$ be the optimal constant in the inequality $\|u^{(k)}\|_{L^q(0,1)}\le \lambda_3(n,k,p,q)\|u^{(n)}\|_{L^p(0,1)}$…

For even $k$, let $X_1,\ldots,X_n$ be $k$-wise independent Bernoulli random variables with common parameter $p\in(0,1)$, and consider the problem of maximizing…

Let a ship start at the origin and follow an arbitrary deterministic unit-speed path $\gamma:[0,\infty)\to\mathbb{R}^2$. For every straight line $L\subset\mathbb{R}^2$ with…

Let $A$ be maximally monotone and let $B$ be monotone and $L$-Lipschitz on a Hilbert space. The forward-reflected-backward iteration is…

For every integer $n\geq 1$, let $r_n\in(0,1)$ be the unique solution of $r_n^n\bigl(1+n(1+r_n)\bigr)=1$, and set $\bar h_n=1+r_n$. For every $L$-smooth convex function $f$ and gra…

For every finite horizon $N\ge 1$, in each of the symmetric recursive frameworks called the primitive and OBS-S constructions for predetermined-step gradient descent on smooth conv…

For each admissible tilt parameter $\tau$, consider every nonterminating execution of BFGS applied to the one-dimensional tilted absolute-value objective $f_{\tau}$ using the Lewis…

For each horizon $T\ge 2$, consider the Heavy-Ball iteration $x_{t+1}=x_t-\eta_t\nabla f(x_t)+\beta_t(x_t-x_{t-1})$, with predetermined horizon-dependent parameters satisfying…

Let $\mathcal{P}_3$ be the class of convex three-dimensional parallelohedra, namely convex polyhedra that tile $\mathbb{R}^3$ by translations. For every $V>0$ and every…

For every finite simple graph $G$, the semidefinite-hierarchy bound satisfies $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$, where $\alpha(G)$ is the stability number of $G$.

For each fixed rational $p\in(1,\infty)$, given vectors $v_1,\ldots,v_m\in\mathbb{Q}^d$ defining the zonotope $Z=\left\{\sum_{i=1}^m\lambda_i v_i: -1\leq\lambda_i\leq 1\right\}$, d…

Given a finite energy bound $E<\infty$, does there exist a prepare-and-measure semi-device-independent protocol such that the observed data certify strictly positive extractable ra…

Fix $1<p<\infty$ and let $q=p/(p-1)$. Given a first-order local value--gradient oracle for an objective $f:\mathbb{R}^n\to\mathbb{R}$ whose gradient is $L$-Lipschitz from…

For every pair of integers $n,k$ satisfying $3\leq k\leq n-1$, the optimal leading coefficient in the high-SNR maximum-likelihood error-probability expansion for SNR-dependent…

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…

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…

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…

Let $n$ be a positive integer, $A_1,\dots, A_n$ be symmetric positive semidefinite matrices, and let $\lVert\,\cdot\,\rVert$ be the spectral norm. For $m\leq n$, compare the average…

Scaling conjecture. For all $k,m,s\in\mathbb{N}$,