Mathematical Programming

, Volume 147, Issue 1–2, pp 539–579 | Cite as

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

  • Chao Ding
  • Defeng Sun
  • Jane J. YeEmail author
Full Length Paper Series A


In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.


Mathematical program with semidefinite cone complementarity constraints Necessary optimality conditions Constraint qualifications S-stationary conditions M-stationary conditions C-stationary conditions 

Mathematics Subject Classification

49K10 49J52 90C30 90C22 90C33 



The authors are grateful to the anonymous referees for their constructive suggestions and comments which helped to improve the presentation of the materials in this paper.


  1. 1.
    Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization-methodology and applications. Math. Program. 92, 453–480 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
  5. 5.
    Bi, S., Han, L., Pan, S.: Approximation of rank function and its application to the nearest low-rank correlation matrix. J. Glob. Optim. 57, 1113–1137 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Brigo, D., Mercurio, F.: Calibrating LIBOR. Risk Mag. 15, 117–122 (2002)Google Scholar
  8. 8.
    Burge, J.P., Luenberger, D.G., Wenger, D.L.: Estimation of structured covariance matrices. Proc. IEEE 70, 963–974 (1982)CrossRefGoogle Scholar
  9. 9.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)zbMATHGoogle Scholar
  10. 10.
    Clarke, F.H., Ledyaev, Yu.S, Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)Google Scholar
  11. 11.
    de Gaston, R.R.E., Safonov, M.G.: Exact calculation of the multiloop stability margin. IEEE Trans. Autom. Control 33, 156–171 (1988)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer, Berlin (2002)zbMATHGoogle Scholar
  13. 13.
    Eaves, B.C.: On the basic theorem for complementarity. Math. Program. 1, 68–75 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Faccchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problem. Springer, New York (2003)Google Scholar
  15. 15.
    Fazel, M.: Matrix Rank Minimization with Applications. PhD thesis, Stanford University (2002)Google Scholar
  16. 16.
    Fletcher, R.: Semi-definite matrix constraints in optimization. SIAM J. Control Optim. 23, 493–513 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Flegel, M.L., Kanzow, C.: On the Guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54, 517–534 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goh, K.C., Ly, J.C., Safonov, M.G., Papavassilopoulos, G., Turan, L.: Biaffine matrix inequality properties and computational methods. In: Proceeding of the American Control Conference, Baltimore, Maryland, pp. 850–855 (1994)Google Scholar
  19. 19.
    Henrion, R., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Henrion, R., Outrata, J.: Calmness of constraint systems with applications. Math. Program. Ser. B 104, 437–464 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hobbs, B.F., Metzler, C.B., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15, 638–645 (2000)CrossRefGoogle Scholar
  22. 22.
    Lewis, A.S.: Nonsmooth analysis of eigenvalues. Math. Program. 84, 1–24 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Li, Q.N., Qi, H.D.: A sequential semismooth Newton method for the nearest low-rank correlation matrix problem. SIAM J. Optim. 21, 1641–1666 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lillo, F., Mantegna, R.N.: Spectral density of the correlation matrix of factor models: a random matrix theory approach. Phys. Rev. E 72, 016219-1–016219-10 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Löwner, K.: Über monotone matrixfunktionen. Mathematische Zeitschrift 38, 177–216 (1934)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  27. 27.
    Hiriart-Urruty, J.-B., Ye, D.: Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Mathematik 70, 45–72 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hoge, W.: A subspace identification extension to the phase correlation method. IEEE Trans. Med. Imaging 22, 277–280 (2003)CrossRefGoogle Scholar
  29. 29.
    Meng, F., Sun, D.F., Zhao, G.Y.: Semismoothness of solutions to generalized equations and Moreau-Yosida regularization. Mathe. Program. 104, 561–581 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)Google Scholar
  32. 32.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 331. Springer, Berlin (2006)Google Scholar
  33. 33.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund space. Trans. Am. Math. Soc. 348, 215–220 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Outrata, J.V., Koc̆vara, M., Zowe, J.: Nonsmooth Approach to Optimization Problem with Equilibrium Constraints: Theory, Application and Numerical Results. Kluwer, Dordrecht (1998)CrossRefGoogle Scholar
  35. 35.
    Overton, M., Womersley, R.S.: On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl. 13, 41–45 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Overton, M., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Program. 62, 321–357 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Psarris, P., Floudas, C.A.: Robust stability analysis of linear and nonlinear systems with real parameter uncertainty. AIChE Annual Meeting, p. 127e. Florida, Miami Beach (1992)Google Scholar
  38. 38.
    Qi, H.D., Fusek, P.: Metric regularity and strong regularity in linear and nonlinear semidefinite programming. Technical Report, School of Mathematics, University of Southampton (2007)Google Scholar
  39. 39.
    Robinson, S.M.: Stability theory for systems of inequalities, part I: linear systems. SIAM J. Numer. Anal. 12, 754–769 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Robinson, S.M.: Stability theory for systems of inequalities, part II: nonlinear systems. SIAM J. Numer. Anal. 13, 473–513 (1976)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30, 597–607 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14, 206–214 (1981)CrossRefzbMATHGoogle Scholar
  43. 43.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  44. 44.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  45. 45.
    Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19, 551–566 (1995)CrossRefGoogle Scholar
  46. 46.
    Safonov, M.G., Goh, K.C., Ly, J.H.: Control system synthesis via bilinear matrix inequalities. In: Proceeding of the American Control Conference, pp. 45–49. Baltimore, Maryland (1994)Google Scholar
  47. 47.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Simon, D.: Reduced order Kalman filtering without model reduction. Control Intell. Syst. 35, 169–174 (2007)MathSciNetGoogle Scholar
  49. 49.
    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
  50. 50.
    Sun, D.F., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Sun, D.F., Sun, J.: Strong semismoothness of eigenvalues of symmetric matrices and its applications in inverse eigenvalue problems. SIAM J. Numer. Anal. 40, 2352–2367 (2003)CrossRefzbMATHGoogle Scholar
  52. 52.
    Tsing, N.K., Fan, M.K.H., Verriest, E.I.: On analyticity of functions involving eigenvalues. Linear Algebra Appl. 207, 159–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    VanAntwerp, J.G., Braatz, R.D., Sahinidis, N.V.: Globally optimal robust control for systems with nonlinear time-varying perturbations. Comput. Chem. Eng. 21, S125–S130 (1997)CrossRefGoogle Scholar
  54. 54.
    Visweswaran, V., Floudas, C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory. Comput. Chem. Eng. 14, 1397–1417 (1990)Google Scholar
  55. 55.
    Visweswaran, V., Floudas, C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—II. Application of theory and test problems. Comput. Chem. Eng. 14, 1419–1434 (1990)Google Scholar
  56. 56.
    Wu, L.X.: Fast at-the-money calibration of the LIBOR market model using Lagrange multipliers. J Comput. Financ. 6, 39–77 (2003)Google Scholar
  57. 57.
    Wu, Z., Ye, J.J.: First and second order condition for error bounds. SIAM J. Optim. 14, 621–645 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yan, T., Fukushima, M.: Smoothing method for mathematical programs with symmetric cone complementarity constraints. Optimization 60, 113–128 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Ye, J.J.: Optimality conditions for optimization problems with complementarity constraints. SIAM J. Optim. 9, 374–387 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 305–369 (2005)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–977 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Zhang, Z.Y., Wu, L.X.: Optimal low-rank approximation to a correlation matrix. Linear Algebra Appl. 364, 161–187 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Zhao, Y.B.: An approximation theory of matrix rank minimization and its application to quadratic equations. Linear Algebra Appl. 437, 77–93 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.National Center for Mathematics and Interdisciplinary SciencesChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeRepublic of Singapore
  3. 3.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

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