A Dynamic Penalization Framework for Online Rank-1 Semidefinite Programming Relaxations

Ahmad Al-Tawaha, Javad Lavaei, Ming Jin
Proceedings of the 7th Annual Learning for Dynamics \& Control Conference, PMLR 283:1012-1024, 2025.

Abstract

We propose a dynamic penalization framework for recovering rank-1 solutions in sequential semidefinite programming (SDP) relaxations. Obtaining rank-1 solutions–crucial for recovering physically meaningful solutions in many applications–becomes particularly challenging in dynamic environments where problem parameters continuously evolve. Our framework operates in two interconnected phases: the learning phase dynamically adjusts penalty parameters to enforce rank-1 feasibility based on feedback from the decision phase, while the decision phase solves the resulting penalized SDP relaxations using the penalty parameters specified by the learning phase. To accelerate rank-1 recovery across sequential problems, we introduce a meta-learning model that provides informed initializations for the penalty matrices. The meta-learning model leverages historical data from previously solved tasks, eliminating the need for externally curated datasets. By using task-specific features and updates from prior iterations, the meta-model intelligently initializes penalty parameters, reducing the number of iterations required between the two phases. We prove sublinear convergence to rank-1 solutions and establish low dynamic regret bounds that improve with task similarity. Empirical results on real-world rank-constrained applications, including the Max-Cut problem and Optimal Power Flow (OPF), demonstrate that our method consistently recovers rank-1 solutions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v283-al-tawaha25a, title = {A Dynamic Penalization Framework for Online Rank-1 Semidefinite Programming Relaxations}, author = {Al-Tawaha, Ahmad and Lavaei, Javad and Jin, Ming}, booktitle = {Proceedings of the 7th Annual Learning for Dynamics \& Control Conference}, pages = {1012--1024}, year = {2025}, editor = {Ozay, Necmiye and Balzano, Laura and Panagou, Dimitra and Abate, Alessandro}, volume = {283}, series = {Proceedings of Machine Learning Research}, month = {04--06 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v283/main/assets/al-tawaha25a/al-tawaha25a.pdf}, url = {https://proceedings.mlr.press/v283/al-tawaha25a.html}, abstract = {We propose a dynamic penalization framework for recovering rank-1 solutions in sequential semidefinite programming (SDP) relaxations. Obtaining rank-1 solutions–crucial for recovering physically meaningful solutions in many applications–becomes particularly challenging in dynamic environments where problem parameters continuously evolve. Our framework operates in two interconnected phases: the learning phase dynamically adjusts penalty parameters to enforce rank-1 feasibility based on feedback from the decision phase, while the decision phase solves the resulting penalized SDP relaxations using the penalty parameters specified by the learning phase. To accelerate rank-1 recovery across sequential problems, we introduce a meta-learning model that provides informed initializations for the penalty matrices. The meta-learning model leverages historical data from previously solved tasks, eliminating the need for externally curated datasets. By using task-specific features and updates from prior iterations, the meta-model intelligently initializes penalty parameters, reducing the number of iterations required between the two phases. We prove sublinear convergence to rank-1 solutions and establish low dynamic regret bounds that improve with task similarity. Empirical results on real-world rank-constrained applications, including the Max-Cut problem and Optimal Power Flow (OPF), demonstrate that our method consistently recovers rank-1 solutions.} }
Endnote
%0 Conference Paper %T A Dynamic Penalization Framework for Online Rank-1 Semidefinite Programming Relaxations %A Ahmad Al-Tawaha %A Javad Lavaei %A Ming Jin %B Proceedings of the 7th Annual Learning for Dynamics \& Control Conference %C Proceedings of Machine Learning Research %D 2025 %E Necmiye Ozay %E Laura Balzano %E Dimitra Panagou %E Alessandro Abate %F pmlr-v283-al-tawaha25a %I PMLR %P 1012--1024 %U https://proceedings.mlr.press/v283/al-tawaha25a.html %V 283 %X We propose a dynamic penalization framework for recovering rank-1 solutions in sequential semidefinite programming (SDP) relaxations. Obtaining rank-1 solutions–crucial for recovering physically meaningful solutions in many applications–becomes particularly challenging in dynamic environments where problem parameters continuously evolve. Our framework operates in two interconnected phases: the learning phase dynamically adjusts penalty parameters to enforce rank-1 feasibility based on feedback from the decision phase, while the decision phase solves the resulting penalized SDP relaxations using the penalty parameters specified by the learning phase. To accelerate rank-1 recovery across sequential problems, we introduce a meta-learning model that provides informed initializations for the penalty matrices. The meta-learning model leverages historical data from previously solved tasks, eliminating the need for externally curated datasets. By using task-specific features and updates from prior iterations, the meta-model intelligently initializes penalty parameters, reducing the number of iterations required between the two phases. We prove sublinear convergence to rank-1 solutions and establish low dynamic regret bounds that improve with task similarity. Empirical results on real-world rank-constrained applications, including the Max-Cut problem and Optimal Power Flow (OPF), demonstrate that our method consistently recovers rank-1 solutions.
APA
Al-Tawaha, A., Lavaei, J. & Jin, M.. (2025). A Dynamic Penalization Framework for Online Rank-1 Semidefinite Programming Relaxations. Proceedings of the 7th Annual Learning for Dynamics \& Control Conference, in Proceedings of Machine Learning Research 283:1012-1024 Available from https://proceedings.mlr.press/v283/al-tawaha25a.html.

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