Barrier Algorithms for Constrained Non-Convex Optimization

Pavel Dvurechensky, Mathias Staudigl
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:12190-12214, 2024.

Abstract

In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-dvurechensky24a, title = {Barrier Algorithms for Constrained Non-Convex Optimization}, author = {Dvurechensky, Pavel and Staudigl, Mathias}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {12190--12214}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/dvurechensky24a/dvurechensky24a.pdf}, url = {https://proceedings.mlr.press/v235/dvurechensky24a.html}, abstract = {In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively.} }
Endnote
%0 Conference Paper %T Barrier Algorithms for Constrained Non-Convex Optimization %A Pavel Dvurechensky %A Mathias Staudigl %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-dvurechensky24a %I PMLR %P 12190--12214 %U https://proceedings.mlr.press/v235/dvurechensky24a.html %V 235 %X In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively.
APA
Dvurechensky, P. & Staudigl, M.. (2024). Barrier Algorithms for Constrained Non-Convex Optimization. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:12190-12214 Available from https://proceedings.mlr.press/v235/dvurechensky24a.html.

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