Closed-form expressions for projectors onto polyhedral sets in Hilbert spaces

TL;DR

This paper derives explicit formulas for projecting onto polyhedral sets in Hilbert spaces by formulating the problem as a convex optimization task and solving the optimality conditions, with potential extensions to Banach spaces.

Contribution

It provides closed-form expressions for projections onto polyhedral sets and extends the approach to more general Banach spaces.

Findings

Formulas for projections onto polyhedral sets are derived.

Optimality conditions for the projection problem are explicitly characterized.

Potential generalizations to Banach spaces are discussed.

Abstract

We provide formulas for projectors onto a polyhedral set, i.e. the intersection of a finite number of halfspaces. To this aim we formulate the problem of finding the projection as a convex optimization problem and we solve explicitly sufficient and necessary optimality conditions. This approach has already been successfully applied in deriving formulas for projection onto the intersection of two halfspaces. We also discuss possible generalizations to Banach spaces.