12 problems

Dedieu–Shub conjecture. The total curvature of the central path is linearly bounded in the dimension of the ambient space.

Linear-growth conjecture. The worst-case total curvature of a central path is $O(n)$.

Let $\mu$ and $\tau$ be the barrier and upper-level complementarity relaxation parameters, respectively, and consider an algorithm that solves an early relaxed barrier problem only…

The paper considers interior-point methods for bound-constrained nonlinear optimization problems with bounded noise, relaxing only the Armijo line-search condition while retaining…

Tangential-direction conjecture.

Quantum speedup conjecture. A quantum speedup for semidefinite optimization should rely on techniques other than the direct quantization of a classical interior point method with l…

Conjecture on structured reformulations. The most common occurrence of the displayed block structure in linear optimization problems is due to the linear reformulation of convex pi…

Self-concordance conjecture. The function $F_{\beta}(X)$ is self-concordant for each $\beta>0$, with this self-concordance depending on the structure of the quantum relative entrop…

A semidefinite programming (SDP) program is optimized using primal-dual interior-point methods, whose centering condition restricts the iterates to “commutative” (central) paths: f…

A sparse structured semidefinite program (SDP) is an optimization problem whose data have a sparse structure, and matrix-completion preprocessing modifies its structure before an i…

Continuous analogue of the Hirsch conjecture. The total curvature of the central path is linearly bounded in the number of constraints.

For a polytope $P$ of dimension $d$ defined by $n$ inequalities and a linear objective function $c$, let $\lambda_c(P)$ be the total curvature of its central path. Let…