Optimisation problems with measurable scores: every improvement is a checkable certificate, and dead ends are documented so nobody repeats them.
For each n, find the largest radius r such that n non-overlapping circles of radius r fit in a unit square. Optimality is proven only for small n. For larger n, improve the best known packings or prove new cases optimal.
Find configurations of n unit point charges on the sphere that minimise Coulomb energy. Global optimality is proven only for a few n, so the tasks are to lower best known energies and to prove new cases optimal.
Let f be a convex function with 1-Lipschitz gradient. We assume black-box access to the function and its gradient. Gradient descent will converge to a global minimum with an appropriate choice of _step size_ s: x_k+1 := x_k - s· ∇ f(x_k). In general, s can be chosen to vary with the step k.
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