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Mathematical Programming

, Volume 170, Issue 1, pp 267–292 | Cite as

Competitive online algorithms for resource allocation over the positive semidefinite cone

  • Reza Eghbali
  • James Saunderson
  • Maryam Fazel
Full Length Paper Series B

Abstract

We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing returns (PSD-DR) property (that its gradient is order-reversing with respect to the PSD cone). Using the representation for monotone maps on the PSD cone given by Löwner’s theorem, we obtain a convex parametrization of the family of functions satisfying PSD-DR. We then formulate a convex optimization problem to directly optimize our competitive ratio bound over this set. This design problem can be solved offline before the data start arriving. The online algorithm that uses the designed smoothing is tailored to the given cost function, and enjoys a competitive ratio at least as good as our optimized bound. We provide examples of computing the smooth surrogate for D-optimal and A-optimal experiment design, and demonstrate the performance of the custom-designed algorithm.

Keywords

Online algorithms Competitive ratio Positive semidefinite cone Löwner’s theorem 

Mathematics Subject Classification

68W27 Online Algorithms 90C25 Convex Programming 62K05 Optimal Design 

Notes

Acknowledgements

The authors thank Omid Sadeghi-Meibodi for helpful comments. The work of MF and RE was supported in part by Grants ONR N000141612789, NSF CCF 1409836, NSF Tripods 1740551, and ONR MURI N000141612710. Part of this work was done while RE and MF were visiting the Simons Institute for the Theory of Computing, partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant CCF-1740425.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Tetration AnalyticsCisco Systems, Inc.Palo AltoUSA
  2. 2.Department of Electrical and Computer Systems EngineeringMonash UniversityAustralia
  3. 3.Department of Electrical EngineeringUniversity of WashingtonSeattleUSA

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