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Mathematical Programming

, Volume 168, Issue 1–2, pp 509–531 | Cite as

Spectral operators of matrices

  • Chao Ding
  • Defeng SunEmail author
  • Jie Sun
  • Kim-Chuan Toh
Full Length Paper Series B

Abstract

The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low rank matrices within and beyond the optimization community. This trend can be credited to some extent to the exciting developments in emerging fields such as compressed sensing. The Löwner operator, which generates a matrix valued function via applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for the Löwner operator has become inadequate in these recent applications. The main objective of this paper is to provide necessary theoretical foundations from the perspectives of designing efficient numerical methods for solving MOPs. We achieve this goal by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability and Fréchet-differentiability are systematically studied.

Keywords

Spectral operators Directional differentiability Fréchet differentiability Matrix valued functions Proximal mappings 

Mathematics Subject Classification

90C25 90C06 65K05 49J50 49J52 

Notes

Acknowledgements

We would like to thank the referees as well as the editors for their constructive comments that have helped to improve the quality of the paper. The research of C. Ding was supported by the National Natural Science Foundation of China under projects No. 11301515, No. 11671387 and No. 11531014.

References

  1. 1.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Candès, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56, 2053–2080 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58, 11 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, Z.X., Sun, D.F.: Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming. SIAM J. Optim. 19, 370–396 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21, 572–596 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, C.H., Liu, Y.J., Sun, D.F., Toh, K.C.: A semismooth Newton-CG dual proximal point algorithm for matrix spectral norm approximation problems. Math. Program. 155, 435–470 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, X., Qi, H.D., Tseng, P.: Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complement problems. SIAM J. Optim. 13, 960–985 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chu, M., Funderlic, R., Plemmons, R.: Structured low rank approximation. Linear Algebra Appl. 366, 157–172 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demyanov, V.F., Rubinov, A.M.: On quasidifferentiable mappings. Optimization 14, 3–21 (1983)zbMATHGoogle Scholar
  11. 11.
    Ding, C.: An introduction to a class of matrix optimization problems. PhD thesis, National University of Singapore. http://www.math.nus.edu.sg/~matsundf/DingChao_Thesis_final.pdf (2012)
  12. 12.
    Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming. Math. Program. 144, 141–179 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ding, C., Sun, D.F., Ye, J.J.: First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints. Math. Program. 147, 539–579 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dobrynin, V.: On the rank of a matrix associated with a graph. Discrete Math. 276, 169–175 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flett, T.M.: Differential Analysis. Cambridge University Press, Cambridge (1980)CrossRefzbMATHGoogle Scholar
  16. 16.
    Greenbaum, A., Trefethen, L.N.: GMRES/CR and Arnoldi/Lanczos as matrix approximation problems. SIAM J. Sci. Comput. 15, 359–368 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kotlov, A., Lovász, L., Vempala, S.: The Colin de Verdière number and sphere representations of a graph. Combinatorica 17, 483–521 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lewis, A.S.: The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2, 173–183 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lewis, A.S.: Derivatives of spectral functions. Math. Oper. Res. 21, 576–588 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lewis, A.S., Overton, M.L.: Eigenvalue optimization. Acta Numer. 5, 149–190 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lewis, A.S., Sendov, H.S.: Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23, 368–386 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part I: theory. Set-Valued Anal. 13, 213–241 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: application. Set-Valued Anal. 13, 243–264 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, Y.J., Sun, D.F., Toh, K.C.: An implementable proximal point algorithmic framework for nuclear norm minimization. Math. Program. 133, 399–436 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Löwner, K.: Über monotone matrixfunktionen. Math. Z. 38, 177–216 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Miao, W.M., Sun, D.F., Pan, S.H.: A rank-corrected procedure for matrix completion with fixed basis coefficients. Math. Program. 159, 289–338 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mohebi, H., Salemi, A.: Analysis of symmetric matrix valued functions. Numer. Funct. Anal. Optim. 28, 691–715 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. 40, 226–252 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Moreau, J.-J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 1067–1070 (1965)zbMATHGoogle Scholar
  32. 32.
    Nashed, M.Z.: Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 103–309. Academic Press, New York (1971)CrossRefGoogle Scholar
  33. 33.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (1970)zbMATHGoogle Scholar
  34. 34.
    Qi, H.D., Yang, X.Q.: Semismoothness of spectral functions. SIAM J. Matrix Anal. Appl. 25, 766–783 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  38. 38.
    Sun, D.F.: The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sun, D.F., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Sun, D.F., Sun, J.: Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33, 421–445 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Toh, K.C.: GMRES vs. ideal GMRES. SIAM J. Matrix Anal. Appl. 18, 30–36 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Toh, K.C., Trefethen, L.N.: The Chebyshev polynomials of a matrix. SIAM J. Matrix Anal. Appl. 20, 400–419 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Wright, J., Ma, Y., Ganesh, A., Rao, S.: Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. In: Bengio, Y., Schuurmans, D., Lafferty, J., Williams, C. (eds.), Advances in Neural Information Processing Systems 22 (2009)Google Scholar
  45. 45.
    Wu, B., Ding, C., Sun, D.F., Toh, K.C.: On the Moreau–Yosida regularization of the vector k-norm related functions. SIAM J. Optim. 24, 766–794 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Yang, L.Q., Sun, D.F., Toh, K.C.: SDPNAL\(+\): a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7, 331–366 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Yang, Z.: A study on nonsymmetric matrix-valued functions. Master’s Thesis, National University of Singapore. http://www.math.nus.edu.sg/~matsundf/Main_YZ.pdf (2009)
  48. 48.
    Zhao, X.Y., Sun, D.F., Toh, K.C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeSingapore
  3. 3.Department of Mathematics and StatisticsCurtin UniversityBentleyAustralia
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore

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