Solving equations and optimization problems with uncertainty

TL;DR

This paper introduces algorithms to detect zeros of continuous functions with uncertainty, enabling robust zero detection and worst-case optimization in approximate constraint scenarios, supported by experiments and topological methods.

Contribution

The paper develops explicit algorithms for approximating the robustness of zeros and applies topological obstruction methods to optimization problems with uncertain constraints.

Findings

Primary obstruction often suffices for Gaussian fields

Algorithms effectively approximate zero robustness

Experimental results validate approach

Abstract

We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the earlier theoretical work with explicit algorithms and experiments with an implementation. More formally, the robustness of zero of a continuous map $f: X\to \mathbb{R}^n$ is the maximal $r>0$ such that each $g:X\to\mathbb{R}^n$ with $\|f-g\|_\infty\le r$ has a zero. We develop and implement an efficient algorithm approximating the robustness of zero. Further, we show how to use the algorithm for approximating worst-case optima in optimization problems in which the feasible domain is defined by equations that are only known approximately. An important ingredient is an algorithm for deciding the topological extension problem based on computing cohomological obstructions to extendability and their persistence. We describe an explicit algorithm for the primary and secondary obstruction, two stages of a sequence of algorithms with increasing complexity. 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.