Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations

Jisun Park, Ernest K. Ryu
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:27294-27345, 2023.

Abstract

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a $\mathcal{O}(1/k^2)$ and $\mathcal{O}(1/k)$ rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves $\mathcal{O}(1/k^2)$ and $\tilde{\mathcal{O}}(1/k^2)$ rates. We then provide a matching complexity lower bound to establish that $\Theta(1/k^2)$ is indeed the optimal accelerated rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-park23k, title = {Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations}, author = {Park, Jisun and Ryu, Ernest K.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {27294--27345}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/park23k/park23k.pdf}, url = {https://proceedings.mlr.press/v202/park23k.html}, abstract = {As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a $\mathcal{O}(1/k^2)$ and $\mathcal{O}(1/k)$ rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves $\mathcal{O}(1/k^2)$ and $\tilde{\mathcal{O}}(1/k^2)$ rates. We then provide a matching complexity lower bound to establish that $\Theta(1/k^2)$ is indeed the optimal accelerated rate.} }
Endnote
%0 Conference Paper %T Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations %A Jisun Park %A Ernest K. Ryu %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-park23k %I PMLR %P 27294--27345 %U https://proceedings.mlr.press/v202/park23k.html %V 202 %X As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a $\mathcal{O}(1/k^2)$ and $\mathcal{O}(1/k)$ rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves $\mathcal{O}(1/k^2)$ and $\tilde{\mathcal{O}}(1/k^2)$ rates. We then provide a matching complexity lower bound to establish that $\Theta(1/k^2)$ is indeed the optimal accelerated rate.
APA
Park, J. & Ryu, E.K.. (2023). Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:27294-27345 Available from https://proceedings.mlr.press/v202/park23k.html.

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