How to project onto the intersection of a closed affine subspace and a hyperplane

TL;DR

This paper derives explicit formulas for projecting onto the intersection of a closed affine subspace and a hyperplane in a Hilbert space, including cases where the intersection is empty, supported by examples and numerical experiments.

Contribution

It introduces new formulas for projections onto intersections of affine subspaces and hyperplanes, extending existing methods to cases with empty intersections.

Findings

Derived explicit projection formulas for affine subspace and hyperplane intersections

Provided formulas applicable even when the intersection is empty

Included numerical experiments demonstrating the formulas' effectiveness

Abstract

Let $A$ be a closed affine subspace and let $B$ be a hyperplane in a Hilbert space. Suppose we are given their associated nearest point mappings $P_A$ and $P_B$, respectively. We present a formula for the projection onto their intersection $A\cap B$. As a special case, we derive a formula for the projection onto the intersection of two hyperplanes. These formulas provides useful information even if $A\cap B$ is empty. Examples and numerical experiments are also provided.