Abstract
We describe a branch-and-bound (b&b) method aimed at searching for an exact solution of the fundamental problem of decomposing a matrix into the sum of a sparse matrix and a low-rank matrix. Previous heuristic techniques employed convex and nonconvex optimization. We leverage and extend those ideas, within a spatial b&b framework, aimed at exact global optimization. Our work may serve to (i) gather evidence to assess the true quality of the previous heuristic techniques, and (ii) provide software to routinely calculate global optima or at least better solutions for moderate-sized instances coming from applications.
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Notes
The nuclear norm is also known as the trace norm, the Ky Fan \(n\) -norm, and the Schatten \(1\) -norm. It has the alternative definition (for a real matrix) as \(\Vert B\Vert _*:=\hbox {Tr}(\sqrt{B^tB})\), where the matrix square-root is well defined, because \(B^tB\) is a symmetric positive-semidefinite matrix.
Even the 3-year–old usage statistics are impressive: http://cvxr.com/cvx/usage-statistics/.
References
Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.S.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21(2), 572–596 (2011)
de Farias, J., Ismael, R.: A polyhedral study of the cardinality constrained knapsack problem. Math. Program. A 96, 439–467 (2003)
Fazel, Maryam.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University, 2002
Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)
Lee, J.: Maximum entropy sampling. In: El-Shaarawi, A.H., Piegorsch, W.W. (eds.) Encyclopedia of Environmetrics’. Wiley, 2001. Revised and updated for the Second Edition, 2012
Mohan, K., Fazel, M.: Iterative reweighted algorithms for matrix rank minimization. J. Mach. Learn. Res. 13, 3441–3473 (2012)
Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory Algorithms, Software, and Applications. Kluwer, Boston (2002)
Raymond, H., Mathias, K., Jon, L., Robert, W.: Nonlinear integer programming. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L. (eds.) 50 Years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys, pp. 561–618. Springer, Heidelberg (2010)
Lee, J., Leyffer, S. (eds.): Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications Vol. 154., 1st edn. Springer, New York (2012)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: Extended formulations. Math. Program. B 124(1–2), 383–411 (2010)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations. Math. Program. A 130(2), 359–413 (2011)
Acknowledgments
The work of the first author was partially supported by NSF Grant CMMI–1160915.
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Lee, J., Zou, B. Optimal rank-sparsity decomposition. J Glob Optim 60, 307–315 (2014). https://doi.org/10.1007/s10898-013-0128-0
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DOI: https://doi.org/10.1007/s10898-013-0128-0