6 problems

Compatibility conjecture. For $\alpha\in[2,\infty)$, the function $-\Psi_\alpha$ is $(\mathbb{H}^n_+,(2\alpha-1)/3)$-compatible with the domain $\mathbb{H}^n_+\times\mathbb{H}^n_+$…

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…

Let $g$ be the function considered in the generalized self-concordance condition, with parameter $\nu$ and constant $M_f$. Tighter-parameter conjecture. There is another value of…

For $a,b\in\mathbbm R$, define … where $a=\log(q_k/q_m)$ and $b=\log(q_l/q_m)$. Let $H$ be the coefficient function appearing in the preceding derivative formulas, so that the ineq…

Let $\mathrm{PD}(n)$ be the complex positive-definite $n\times n$ matrices with their affine-invariant Riemannian metric. For $P_0\in\mathrm{PD}(n)$, define the squared distance fu…

Symmetry-free self-concordance conjecture. The lower bound