# Exact augmented Lagrangian functions for nonlinear semidefinite programming

## TL;DR

This paper develops a unified framework for constructing exact augmented Lagrangian functions for nonlinear semidefinite programming, enabling reformulation into unconstrained problems with proven differentiability and exactness.

## Contribution

It generalizes previous work to NSDP, introduces a practical exact augmented Lagrangian function, and proves its properties under nondegeneracy conditions.

## Key findings

- The proposed augmented Lagrangian is continuously differentiable.
- The function is exact under nondegeneracy conditions.
- Preliminary numerical experiments demonstrate its effectiveness.

## Abstract

In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi's work from 1996, that was aimed at solving nonlinear programming problems. Then, through our framework, we propose a practical augmented Lagrangian function for NSDP, proving that it is continuously differentiable and exact under the so-called nondegeneracy condition. We also present some preliminary numerical experiments.