# Active-set Newton methods and partial smoothness

## TL;DR

This paper explores active-set Newton methods and the concept of partial smoothness, providing a unified framework for algorithms that identify active constraints in optimization and variational inequality problems.

## Contribution

It formalizes a linearization scheme for intersecting manifolds, extending active-set methods to broader classes of partly smooth operators.

## Key findings

- Active-set identification occurs in finite time.
- The linearization scheme applies to variational inequalities.
- Framework unifies optimization and variational inequality approaches.

## Abstract

Diverse optimization algorithms correctly identify, in finite time, intrinsic constraints that must be active at optimality. Analogous behavior extends beyond optimization to systems involving partly smooth operators, and in particular to variational inequalities over partly smooth sets. As in classical nonlinear programming, such active-set structure underlies the design of accelerated local algorithms of Newton type. We formalize this idea in broad generality as a simple linearization scheme for two intersecting manifolds.