177 problems

Let $T$ be the number of iterations and let $a,b\to 1,\ldots,n$ index replicas. For $1\leq s\leq t\leq T$, define the order parameters $Q_{u,ab}^{st}$ and $Q_{v,ab}^{st}$ as the pa…

Exactness conjecture. The proposed relaxation is tight and yields the exact solution whenever the graph $(V,E)$ has no cycle.

Let $\Omega$ be an irreducible symmetric cone, and let $H$ be a self-scaled barrier on $\Omega$. A self-scaled barrier is isotropic when it is rotationally invariant under the rele…

Oracle-call optimality conjecture. For the class of methods that require an optimal number of communication rounds, these bounds are also optimal, up to polylogarithmic factors, in…

Let $\bar{L}$ be a continuously differentiable, smooth, and strongly convex function, and consider constant regressors. Let $\{\theta_k\}_{k=0}^{\infty}$ denote the iterates genera…

Let $0<\mu<L$, let $p\in\mathbb{N}$, and let $\mathcal{A}$ be a consistent $p$-SCLI optimization algorithm with diagonal inversion matrix and linear coefficient matrices. Let…

Let $a(x)$ be the continuous-case function defined by the preceding maximization problem, and let $(a_i)_{i\geq 1}$ be the sequence defined by $a_1=1$, $a_2=2$, $a_3=4$, and, for…

Optimality conjecture. For $\Theta=\Theta_{M,\,\lambda}$, an optimal proxy function minimizing the ratio $\overline{V}/\alpha$ is the entropy-type function defined in equation (2).

Let $D(\theta):=\sup_{y\in\mathbb R^m}\psi_1(y)$ be the Case 1.2 dual value, where $\theta>0$ is the regularization parameter. Let $V$ be the primal value and $W$ the limiting lowe…

Consider the aggregation schemes for the Hamiltonian-flow trajectory used to choose the next iterate in convex optimization, including the uniform average and the minimizer along t…

Let ${\boldsymbol{M}}_{l}$ be the matrices associated with the user-placement scenario in which ${\boldsymbol{M}}_{l}$ is not an inverse Z-matrix for every $l\in\{1,\ldots,N\}$. Al…

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…

Unconditional convergence conjecture. The algorithm $$ converges unconditionally for $(\alpha,\beta,\theta)\in\mathbb{R}^2_{++}\times(\mathbb{R}\setminus\{0\})$ over $\mathcal{A}\t…

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

The paper studies safeguarded augmented Lagrangian methods for fully convex composite optimization, including an elastic safeguarding mechanism that enlarges the safeguarding set i…

Mixed-integer oracle lower-bound conjecture. Under these assumptions, at least

Bilateral facial reduction conjecture. Many aspects of Facial Reduction and the Extended Dual System can be recovered—and potentially generalized—from the results developed in this…

Tensor robust PCA recovery conjecture. The factor

Conjectured bounds.

Consider the one-bit precoding scheme … where $\hat{\mathbf{x}}$ solves a convex relaxation with objective … $R(\mathbf{x})$ is a convex regularization function, and…

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…

Power-objective divergence conjecture. For every integer exponent $c\geq 2$, the problems with objective functions $\varphi(z)=z^c$ admit distinct sets of optimal acyclic orientati…

Logarithm-free last-iterate conjecture. It should be possible to eliminate the $ln(T)$ term from the best possible last-iterate convergence bound for convex smooth problems; in par…