Experiments with Super-Universal Regularized Newton Method.
(See the paper: https://arxiv.org/abs/2208.05888 by Nikita Doikov, Konstantin Mishchenko, and Yurii Nesterov.)
- Polytope Feasibility (
experiment_polytope_feasibility.ipynb):
$$\min\limits_{x \in \mathbb{R}^n} f(x)
\quad = \quad \sum\limits_{i = 1}^m \max(0, \langle a_i, x \rangle - b_i\bigr)^p,
\qquad p \geq 2.$$
- Soft Max (
experiment_soft_maximum.ipynb):
$$
\min\limits_{x \in \mathbb{R}^n} f(x)
\quad = \quad \mu \ln \biggl( \sum\limits_{i = 1}^m \exp\Bigl( \frac{\langle a_i, x \rangle - b_i}{\mu} \Bigr)\biggr)
\quad \approx \quad \max\limits_{1 \leq i \leq m} \bigl[ \langle a_i, x \rangle - b_i \bigr],
\qquad \mu > 0.
$$
- Worst Instances (
experiment_worst_instances.ipynb)
$$
\min\limits_{x \in \mathbb{R}^n} f(x)
\quad = \quad \frac{1}{q}
\sum\limits_{i = 1}^{n - 1} |x^{(i)} - x^{(i + 1)}|^q + \frac{1}{q} |x^{ (n) }|^q,
\qquad q \geq 2.
$$