13 problems

Consider the homogeneous quadratically constrained quadratic optimization problem denoted by … has at most four strict local non-global minimizers. The statement is presented after…

Let $f$ be an $L$-smooth function. A Goldstein approximate second-order stationary point (SOSP) is an approximate second-order stationary point in the Goldstein sense, as defined i…

Let $f\in\mathcal{F}_{\mu,L}$ with $\mu\in(-\infty,0)$, and consider $N$ gradient-descent iterations from $x_0$ with stepsizes satisfying $\gamma_iL\in[1,\gbari[1]]$ for…

Let $U_r\in\mathbb{R}^{d\times r}$ be an orthonormal matrix, and let $\mathrm{Gr}(d,r)$ denote the Grassmann manifold of $r$-dimensional subspaces of $\mathbb{R}^d$. Define … as in…

Let $\mathcal{C}_n$ be the space of full-order dynamic policies for $\mathcal{H}_\infty$ control. A policy $\mathsf{K}\in\mathcal{C}_n$ is Clarke stationary when…

Let $\mathcal{C}_n$ be the space of full-order dynamic policies, and let $\mathcal{C}_{\mathrm{nd}}$ be the subset of non-degenerate policies, namely those admitting a certificate…

Let $p$ denote the factorization rank in the Burer–Monteiro factorization … is benign with high probability in each of the following settings: the high-dimensional Kuramoto model,…

Fixed-weight-matrix conjecture. There exists a large constant $\Gamma(\epsilon)>0$ such that every instance $\mathcal{MC}(C,u^*)$ satisfying

A CDT subproblem is a cubic-regularized optimization subproblem, and a nonconvex quadratic optimization problem with a quartic constraint is an optimization problem whose objective…

Let … hold; and (ii) there exists a nonzero vector $\bar v$ satisfying $-$. This conjecture seeks a complete characterization of non-strict local non-global minimizers; the supplie…

Let $V_{\mathsf{ct}}(\cdot)$ be the circuit-to-unitary map, let $U$ be a target unitary, and define … A stationary point ${\mathbold{\theta}}$ is a point where the gradient vanishe…

Partial-decoupling conjecture. The kernels of the partial decouplings of $Z$ are the only critical points of $\Phi_{L^*}$.

Let $X$ be an $N$-dimensional Hilbert space, let $S:=\{x\in X\colon\|x\|=1\}$ be the unit sphere, and let $L:=\{x=\lambda a+\alpha b\colon\lambda\in\mathbb{R}\}$, where $a,b$ are o…