# A Log-Barrier Newton-CG Method for Bound Constrained Optimization with Complexity Guarantees

arXiv: https://arxiv.org/abs/1904.03563 · HTML: https://mlsanity.com/paper/1904.03563/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2019.

## Built on by (most cited first, 7 of 7)

- [Escaping strict saddle points of the Moreau envelope in nonsmooth optimization](https://mlsanity.com/paper/2106.09815/) (2021)
- [A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guarantees](https://mlsanity.com/paper/2301.03139/) (2023)
- [A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees](https://mlsanity.com/paper/2207.05697/) (2022)
- [A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization](https://mlsanity.com/paper/2301.04204/) (2023)
- [Randomized Feasibility Methods for Constrained Optimization with Adaptive Step Sizes](https://mlsanity.com/paper/2601.20076/) (2026)
- [HOUDINI: Escaping from Moderately Constrained Saddles](https://mlsanity.com/paper/2205.13753/) (2022)
- [First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods](https://mlsanity.com/paper/2307.07605/) (2023)

## Abstract (arXiv)

We describe an algorithm based on a logarithmic barrier function, Newton's method, and linear conjugate gradients that obtains an approximate minimizer of a smooth function over the nonnegative orthant. We develop a bound on the complexity of the approach, stated in terms of the required accuracy and the cost of a single gradient evaluation of the objective function and/or a matrix-vector multiplication involving the Hessian of the objective. The approach can be implemented without explicit calculation or storage of the Hessian.