Abstract
A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain by its convex hull) termed “LSOP-R”, is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of the feasible set of LSOP-R with different matrix spaces and prove their tightness. The proposed rank bounds recover two well-known results in the literature from a fresh angle and allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle, coupled with a rank-reduction algorithm, which ensures that the output solution always satisfies the theoretical rank bound. Finally, we numerically verify the strength of LSOP-R and the efficacy of the proposed algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Askari, A., d’Aspremont, A., Ghaoui, L.E.: Approximation bounds for sparse programs. SIAM J. Math. Data Sci. 4(2), 514–530 (2022)
Barvinok, A.I.: Problems of distance geometry and convex properties of quadratic maps. Discret. Comput. Geom. 13(2), 189–202 (1995)
Bedoya, J.C., Abdelhadi, A., Liu, C.C., Dubey, A.: A QCQP and SDP formulation of the optimal power flow including renewable energy resources. In: 2019 International Symposium on Systems Engineering (ISSE), pp. 1–8. IEEE (2019)
Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications. SIAM (2001)
Bertsimas, D., Cory-Wright, R., Pauphilet, J.: A new perspective on low-rank optimization. arXiv preprint arXiv:2105.05947 (2021)
Burer, S., Monteiro, R.D.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95(2), 329–357 (2003)
Burer, S., Monteiro, R.D.: Local minima and convergence in low-rank semidefinite programming. Math. Program. 103(3), 427–444 (2005)
Burer, S., Ye, Y.: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Program. 181(1), 1–17 (2020)
Deza, M.M., Laurent, M., Weismantel, R.: Geometry of Cuts and Metrics, vol. 2. Springer, Berlin, Heidelberg (1997). https://doi.org/10.1007/978-3-642-04295-9
Drusvyatskiy, D., Kempton, C.: Variational analysis of spectral functions simplified. arXiv preprint arXiv:1506.05170 (2015)
Eltved, A., Burer, S.: Strengthened SDP relaxation for an extended trust region subproblem with an application to optimal power flow. Math. Program. 1–26 (2022)
Kim, J., Tawarmalani, M., Richard, J.P.P.: Convexification of permutation-invariant sets and an application to sparse principal component analysis. Math. Oper. Res. 47(4), 2547–2584 (2022)
Kulis, B., Sustik, M.A., Dhillon, I.S.: Low-rank kernel learning with bregman matrix divergences. J. Mach. Learn. Res. 10(2) (2009)
Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization, vol. 46. Cambridge University Press, Cambridge (2011)
Li, Y., Xie, W.: On the exactness of Dantzig-Wolfe relaxation for rank constrained optimization problems. arXiv preprint arXiv:2210.16191 (2022)
Li, Y., Xie, W.: On the partial convexification for low-rank spectral optimization: rank bounds and algorithms. arXiv preprint arXiv:2305.07638 (2023)
Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2), 339–358 (1998)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1972)
Samadi, S., Tantipongpipat, U., Morgenstern, J.H., Singh, M., Vempala, S.: The price of fair PCA: one extra dimension. Adv. Neural Inf. Process. Syst. 31 (2018)
Tantipongpipat, U., Samadi, S., Singh, M., Morgenstern, J.H., Vempala, S.: Multi-criteria dimensionality reduction with applications to fairness. Adv. Neural Inf. Process. Syst. 32 (2019)
Yu, H., Lau, V.K.: Rank-constrained schur-convex optimization with multiple trace/log-det constraints. IEEE Trans. Signal Process. 59(1), 304–314 (2010)
Zhang, D., Hu, Y., Ye, J., Li, X., He, X.: Matrix completion by truncated nuclear norm regularization. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2192–2199. IEEE (2012)
Acknowledgements
This research has been supported in part by the National Science Foundation grants 2246414 and 2246417, the Office of Naval Research grant N00014-24-1-2066, and the Georgia Tech ARC-ACO fellowship. The authors would like to thank Prof. Fatma Kılınç-Karzan for her valuable suggestions on the earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Li, Y., Xie, W. (2024). On the Partial Convexification of the Low-Rank Spectral Optimization: Rank Bounds and Algorithms. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham. https://doi.org/10.1007/978-3-031-59835-7_20
Download citation
DOI: https://doi.org/10.1007/978-3-031-59835-7_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-59834-0
Online ISBN: 978-3-031-59835-7
eBook Packages: Computer ScienceComputer Science (R0)