42 problems

For $1\le p<q\le\infty$, let $B_p^n=\{x\in\mathbb{R}^n:\|x\|_p\le 1\}$. For the class of convex functions $f:B_p^n\to\mathbb{R}$ that are $L$-Lipschitz with respect to…

Let $E_n$ be the explicit family of $n$-dimensional ellipsoids considered in the cited problem, accessed through a membership oracle. Determine the quantum membership-query complex…

Cyclic Douglas–Rachford best approximation conjecture. If a best approximation pair relative to $(C_1,C_2)$ exists, then the two-set cyclic Douglas–Rachford scheme converges weakly…

Let $K>0$. Consider a method whose iterates have the form … where $\boldsymbol{x}_0\in\mathbb{R}^d$ is the initial point, $f$ is a convex function accessed through a gradient oracl…

A homogeneous polynomial $h\in\mathbb{R}[x_1,\ldots,x_n]$ is hyperbolic with respect to $e\in\mathbb{R}^n$ if $h(e)>0$ and, for every $a\in\mathbb{R}^n$, the polynomial $h(te-a)$ h…

Let $n,d\geq 1$, and let $x_1,\ldots,x_n$ be independent random vectors with … For an ellipsoid $\mathcal{E}$, say that it is an ellipsoid fit to $(x_i)_{i=1}^n$ if every point…

Polynomial lower-bound conjecture. There exists $\eta\in[\mu,L]$ such that

Let $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ be finite-dimensional complex Hilbert spaces, let $\rho \in \mathsf{D}(\mathcal{X}\otimes \mathcal{Z})$ and…

The paper considers generalized gradient flows and convex optimization on entanglement polytopes in Hadamard manifolds. In particular, the preceding discussion uses unitary matrice…

Tensor robust PCA recovery conjecture. The factor

In the convex composite problem with closed proper convex functions $f,g$, suppose the Douglas–Rachford operator has a fixed point $w^\star$ and $f$ is $L$-smooth. Let $\rho$ denot…

Consider the convex composite problem … where $f$ and $g$ are closed proper convex functions. Assume that the Douglas–Rachford operator has a fixed point $w^\star$, that $f$ is $L$…

Let $A$ and $B$ be operators satisfying the paper's standing Assumption, let $B$ be $\beta$-cocoercive, and let $A=\partial g$ for a closed proper convex function $g$. Let…

Let $\mathcal H$ be the Hilbert space underlying the compact convex set considered, and let $P$ denote the optimal runtime complexity of its projection operator, while $L(0)$ denot…

Let $f$ and $g$ define a convex constrained optimization problem of the form … For a penalty parameter $\Lambda$, consider the penalty relaxation … where $[g(x)]_+$ denotes the pos…

Log trace-inverse self-concordance conjecture. The function $k(X)$ is $(\nu,M)$-generalized self-concordant for some $\nu\in[2,3]$ and $M>0$ on some subset of the positive-definite…

Spectrum-aware debiasing conjecture. Under suitable conditions, there is a unique solution $\widehat{\mathsf{adj}}$ of this equation and

Let $\ell$ be a lower bound on the information complexity, with respect to a fixed oracle, for a family of continuous, constrained convex optimization instances. Mixed-integer tran…

Non-integer convex-hull conjecture. Every vector satisfying that characterization should belong to

Non-integer characterization conjecture. A vector $\boldsymbol{v}\in\Lambda(v)$ is optimal if and only if it has this decomposition and satisfies, for the relevant indices $l$,

Non-integer optimality conjecture. The set

Integer-case convex-hull conjecture. Every such vector should satisfy

Let $\Omega$ be a bounded convex domain, and let the load $f \in \mathcal{M}(\overline{\Omega};\mathbb{R})$ be finitely supported. An optimal string system consists of a pair…

Closed-form worst-case conjecture. The worst-case performance satisfies

Let $\mathcal{C}_{\mu_g}^{0}$ and $\mathcal{D}_{\mu_g}^{\mu_M}$ be the operator-composed function classes defined in the source, and let $\mathcal{C}_{\mu_g}^{0,(1)}$ and…