Ivo Fagundes David de Oliveira

arXiv2026

Lonely Solids

A three-dimensional solid has the Rupert property if a congruent copy of the solid can pass through a hole cut through it without splitting it. We extend this idea to pairs of convex solids: two solids are called \textit{friends} if each can pass through a suitable hole in the other. A solid is called \textit{lonely} if it has no friends, including itself. We show that a convex solid is lonely if and only if it has constant width. We also show that every convex solid that does not have constant width has a particularly simple friend: an arbitrarily long and arbitrarily thin rectangular cuboid. Finally, we prove that all non-constant-width convex solids lie in a single connected component of the friendship graph. More precisely, any two such solids are connected by a chain of at most ``three handshakes''.

math.HO↗

arXiv2021

Efficient solvers for Armijo's backtracking problem

Backtracking is an inexact line search procedure that selects the first value in a sequence $x_0, x_0β, x_0β^2...$ that satisfies $g(x)\leq 0$ on $\mathbb{R}_+$ with $g(x)\leq 0$ iff $x\leq x^*$. This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in $(βx^*,x^*]$ and enjoys an upper-bound $\lceil \log_β ε/x_0 \rceil$ on the number of function evaluations to terminate, with $ε$ a lower bound on $x^*$. The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of performing inexact line searches, leading to a new class of inexact line search procedures. The traditional bisection algorithm for root-searching is transposed into a very simple method that completes the same inexact line search in at most $\lceil \log_2 \log_β ε/x_0 \rceil$ function evaluations. A recent bracketing algorithm for root-searching which presents both minmax function evaluation cost (as the bisection algorithm) and superlinear convergence is also transposed, asymptotically requiring $\sim \log \log \log ε/x_0 $ function evaluations for sufficiently smooth functions. Other bracketing algorithms for root-searching can be adapted in the same way. Numerical experiments suggest time savings of 50\% to 80\% in each call to the inexact search procedure.

math.OC↗