# Using Negative Curvature in Solving Nonlinear Programs

## TL;DR

This paper extends negative curvature methods from unconstrained to equality constrained nonlinear optimization, proving convergence to second-order stationary points and discussing potential generalizations.

## Contribution

It introduces a negative curvature approach for constrained nonlinear programs and proves its convergence to second-order stationary points.

## Key findings

- Proves convergence to second-order stationary points.
- Extends negative curvature methods to equality constrained problems.
- Discusses potential for broader nonlinear program applications.

## Abstract

Minimization methods that search along a curvilinear path composed of a non-ascent nega- tive curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent appli- cations, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to nonlinear ones. We focus on equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions. A possible way to extend our proposed negative curvature method to general nonlinear programs is also briefly discussed.