# Hessian barrier algorithms for non-convex conic optimization

arXiv: https://arxiv.org/abs/2111.00100 · HTML: https://mlsanity.com/paper/2111.00100/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2021.

## Built on by (most cited first, 5 of 5)

- [On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem](https://mlsanity.com/paper/2507.15264/) (2025)
- [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)
- [Stochastic interior-point methods for smooth conic optimization with applications](https://mlsanity.com/paper/2412.12987/) (2024)
- [A general optimization solver based on OP-to-MaxSAT reduction](https://mlsanity.com/paper/2604.21961/) (2026)

## Abstract (arXiv)

We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier algorithms for linearly constrained optimization problems, SIAM Journal on Optimization, 2019]. Our approach is based on a potential-reduction mechanism and attains a suitably defined class of approximate first- or second-order KKT points with the optimal worst-case iteration complexity O(varepsilon-2) (first-order) and O(varepsilon-3/2) (second-order), respectively. A key feature of our methodology is the use of self-concordant barrier functions to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained optimization problems.