Solving equations and optimization problems with uncertainty

Abstract

We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the theoretical work of Franek and Krcal (J ACM 62(4):26:1–26:19, 2015) with explicit algorithms and experiments with an implementation ( https://bitbucket.org/robsatteam/rob-sat ). Further, we show how to use the algorithm for approximating worst-case optima in optimization problems in which the feasible domain is defined by the zero set of a function $$f\colon X\rightarrow {\mathbbm{R}}^n$$ which is only known approximately. The algorithm first identifies a subdomain A where the function f is provably non-zero, a simplicial approximation $$f'\colon A\rightarrow S^{n-1}$$ of f/|f|, and then verifies non-extendability of $$f'$$ to X to certify a zero. Deciding extendability is based on computing the cohomological obstructions and their persistence. We describe an explicit algorithm for the primary and secondary obstruction, two stages of a sequence of algorithms with increasing complexity. Using elements and techniques of persistent homology, we quantify the persistence of these obstructions and hence of the robustness of zero. We provide experimental evidence that for random Gaussian fields, the primary obstruction—a much less computationally demanding test than the secondary obstruction—is typically sufficient for approximating robustness of zero.