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Variational Analysis of the Ky Fan k-norm

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Abstract

In this paper, we will study some variational properties of the Ky Fan k-norm 𝜃 = ∥⋅∥(k) of matrices, which are closed related to a class of basic nonlinear optimization problems involving the Ky Fan k-norm. In particular, for the basic nonlinear optimization problems, we will introduce the concept of nondegeneracy, strict complementarity and the critical cones associated with the generalized equations. Finally, we present the explicit formulas of the conjugate function of the parabolic second order directional derivative of 𝜃, which will be referred to as the sigma term of the second order optimality conditions. The results obtained in this paper provide the necessary theoretical foundations for future work on sensitivity and stability analysis of the nonlinear optimization problems involving the Ky Fan k-norm.

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References

  1. Bhatia, R.: Matrix Analysis. Springer Science & Business Media (1997)

  2. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  3. Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing markov chain on a graph. SIAM Rev. 46, 667–689 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Candès, E., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Candès, E., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56, 2053–2080 (2010)

    Article  MathSciNet  Google Scholar 

  6. Candès, E., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58, 11 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21, 572–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chu, M., Funderlic, R., Plemmons, R.: Structured low rank approximation. Linear Algebra Appl. 366, 157–172 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  10. Ding, C., Sun, D., Toh, K.-C.: An introduction to a class of matrix cone programming. Math. Program. 144, 141–179 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dobrynin, V.: On the rank of a matrix associated with a graph. Discret. Math. 276, 169–175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Facchinei, F., Pang, J.S. (eds.): Finite-dimensional variational inequalities and complementarity problems, vol. 1. Springer Science & Business Media (2003)

  13. Fan, K.: On a theorem of Weyl concerning eigenvalues of linear transformations. Proc. Natl. Acad. Sci. U. S. A. 35, 652–655 (1949)

    Article  MathSciNet  Google Scholar 

  14. Gao, Y.Y., Sun, D.: A majorized penalty approach for calibrating rank constrained correlation matrix problems. Preprint available at http://www.math.nus.edu.sg/∼matsundf/MajorPen_May5.pdf (2010)

  15. Greenbaum, A., Trefethen, L.N.: GMRES/CR and Arnoldi/Lanczos as matrix approximation problems. SIAM J. Sci. Comput. 15, 359–368 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kotlov, A., Lovász, L., Vempala, S.: The Colin de Verdière number and sphere representations of a graph. Combinatorica 17, 483–521 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods, and Applications. Kluwer Academic Publishers (2002)

  18. Lancaster, P.: On eigenvalues of matrices dependent on a parameter. Numer. Math. 6, 377–387 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: applications. Set-Valued Analysis 13, 243–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Miao, W.M., Sun, D., Pan, S.: A rank-corrected procedure for matrix completion with fixed basis coefficients. Preprint available at arXiv:1210.3709 (2012)

  22. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  23. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moreau, J.-J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société, Mathématique de France 93, 1067–1070 (1965)

    MATH  Google Scholar 

  25. von Neumann, J.: Some matrix-inequalities and metrization of matric-space, Tomsk University Review 1, 286–300 (1937). In: Collected Works, vol. IV, pp. 205–218. Pergamon, Oxford (1962)

    Google Scholar 

  26. Overton, M., Womersley, R.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Program. 62, 321–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30, 597–607 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Robinson, S.M.: Local structure of feasible sets in nonlinear programming. II: nondegeneracy. Mathematical Programming Study 22, 217–230 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. Robinson, S.M.: Local structure of feasible sets in nonlinear programming. III: stability and sensitivity. Mathematical Programming Study 30, 45–66 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)

  32. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  33. Sun, D.: The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Toh, K.-C.: GMRES Vs. ideal GMRES. SIAM J. Matrix Anal. Appl. 18, 30–36 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Toh, K.-C., Trefethen, L.N.: The Chebyshev polynomials of a matrix. SIAM J. Matrix Anal. Appl. 20, 400–419 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Torki, M.: Second-order directional derivatives of all eigenvalues of a symmetric matrix. Nonlinear Anal. 46, 1133–1150 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Watson, G.A.: Linear best approximation using a class of polyhedral norms. Numerical Algorithms 2, 321–336 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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 

  39. Wu, B., Ding, C., Sun, D., Toh, K.-C.: On the Moreau–Yosida regularization of the vector k-norm related functions. SIAM J. Optim. 24, 766–794 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, L., Zhang, N., Xiao, X.: On the second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions. Set-Valued and Variational Analysis 21(3), 557–586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Chao Ding

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  1. Chao Ding
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Correspondence to Chao Ding.

Additional information

This work is supported in part by the National Natural Science Foundation of China (Grant No. 11301515). This work was initiated while C. Ding was with Department of Mathematics, National University of Singapore during 2007 to 2012.

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Ding, C. Variational Analysis of the Ky Fan k-norm. Set-Valued Var. Anal 25, 265–296 (2017). https://doi.org/10.1007/s11228-016-0378-3

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  • DOI: https://doi.org/10.1007/s11228-016-0378-3

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