12 problems

Let $f$ be a semialgebraic function with a stratification of its graph, let $\cX$ denote the relevant stratum, and let $\partial\cX$ and $\sX_{\dim(\cX)-1}$ denote its boundary and…

Consider the $\ell_1$-norm rank-one symmetric matrix factorization problem and initialize the subgradient method randomly, with an initialization distribution absolutely continuous…

Burke–Chen conjecture. The claim that

Generic convergence conjecture. For a full-measure set of vectors $v\in\mathbf{R}^n$, the subgradient method applied to $f_v$ either diverges or converges to a local minimizer of…

Let $\mathcal{P}_m$ be the space of complex polynomials of degree at most $m$, let $\mathcal M^n$ be the space of $n\times n$ complex matrices, and for $A\in\mathcal M^n$ let…

The paper studies nonconvex, nonsmooth optimization through oracle complexity. In its smoothing results, the algorithm is assumed to be translation invariant with respect to consta…

BFGS for the unit ball. The trial step $s$ converges to zero.

Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be defined by … A BFGS sequence means any sequence generated by the BFGS method for minimizing $f$. BFGS convergence conjecture. Every BFGS s…

The paper considers general nonconvex and nonsmooth programming problems, denoted by NNPs, in the primal formulation introduced earlier in the paper. Optimality conjecture. The the…

Restart conjecture. Restarting could improve the performance of FISTA even in the presence of nonsmooth regularizers.

Given a positively homogeneous function $h:\mathbb{R}^n\to\mathbb{R}$, an upper exhauster is a family $E^*$ of closed convex sets such that … For a finite family of polyhedral sets…

The paper studies solutions of a differential inclusion associated with a convex, possibly nonsmooth objective function, using the subdifferential inequality and a generalized chai…