Advertisement

Optimality conditions and global convergence for nonlinear semidefinite programming

  • Roberto Andreani
  • Gabriel Haeser
  • Daiana S. Viana
Full Length Paper Series A

Abstract

Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush–Kuhn–Tucker (AKKT), well known in nonlinear optimization. The second one, called Trace-AKKT, is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided. We propose an augmented Lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed, weaker than previously considered ones, which are sufficient for the global convergence of the algorithm to a stationary point.

Keywords

Nonlinear semidefinite programming Optimality conditions Constraint qualifications Practical algorithms 

Mathematics Subject Classification

90C22 90C46 90C30 

Notes

References

  1. 1.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraint. SIAM J. Optim. 18(4), 1286–1309 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5–32 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Andreani, R., Fazzio, N.S., Schuverdt, M.L., Secchin, L.D.: A sequential optimality condition related to the quasinormality constraint qualification and its algorithmic consequences. Optimization online (2017). http://www.optimization-online.org/DB_HTML/2017/09/6194.html
  4. 4.
    Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60(5), 627–641 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Andreani, R., Haeser, G., Ramos, A., Silva, P.J.S.: A second-order sequential optimality condition associated to the convergence of algorithms. IMA J. Numer. Anal. 37(4), 1902–1929 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22(3), 1109–1135 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135(1–2), 255–273 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26(1), 96–110 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: Strict constraint qualifications and sequential optimality conditions for constrained optimization. Math. Oper. Res. 43(3), 693–717 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequencial optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. 20(6), 3533–3554 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Andreani, R., Martínez, J.M., Santos, L.T.: Newton’s method may fail to recognize proximity to optimal points in constrained optimization. Math. Program. 160, 547–555 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Andreani, R., Secchin, L.D., Silva, P.J.S.: Convergence properties of a second order augmented Lagrangian method for mathematical programs with complementarity constraints. SIAM J. Optim. 28(3), 2574–2600 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Practical Methods of Optimization: Theory and Algorithms. Wiley, NJ (2006)zbMATHGoogle Scholar
  14. 14.
    Birgin, E., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM, Philadelphia (2014)zbMATHCrossRefGoogle Scholar
  15. 15.
    Birgin, E.G., Gardenghi, J.L., Martínez, J.M., Santos, S.A., Toint, PhL: Evaluation complexity for nonlinear constrained optimization using unscaled KKT conditions and high-order models. SIAM J. Optim. 26, 951–967 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Birgin, E.G., Haeser, G., Ramos, A.: Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points. Comput. Optim. Appl. 69(1), 51–75 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Birgin, E.G., Krejic, N., Martínez, J.M.: On the minimization of possibly discontinuous functions by means of pointwise approximations. Optim. Lett. 11(8), 1623–1637 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Bolte, J., Daniilidis, A., Lewis, A.S.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2007)zbMATHCrossRefGoogle Scholar
  19. 19.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)zbMATHCrossRefGoogle Scholar
  20. 20.
    Correa, R., Ramírez, H.: A global algorithm for nonlinear semidefinite programming. SIAM J. Optim. 15(1), 303–318 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. 56(4), 1463–1499 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fares, B., Apkarian, P., Noll, D.: An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory. Int. J. Control 74(4), 348–360 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fares, B., Noll, D., Apkarian, P.: Robust control via sequential semidefinite programming. SIAM J. Control Optim. 40(6), 1791–1820 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fiacco, A.V., McCormick, G.P.: Nonlinear Programming Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)zbMATHGoogle Scholar
  25. 25.
    Forsgren, A.: Optimality conditions for nonconvex semidefinite programming. Math. Program. 88(1), 105–128 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Freund, R.W., Jarre, F., Vogelbusch, C.H.: Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling. Math. Program. 109, 581–611 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Giorgi, G., Jiménez, B., Novo, V.: Approximate Karush–Kuhn–Tucker condition in multiobjective optimization. J. Optim. Theory Appl. 171(1), 70–89 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gómez, W., Ramírez, H.: A filter algorithm for nonlinear semidefinite programming. Comput. Appl. Math. 29(2), 297–328 (2010)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Haeser, G.: A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms. Comput. Optim. Appl. 70(2), 615–639 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Haeser, G., Melo, V.V.: Convergence detection for optimization algorithms: approximate-KKT stopping criterion when Lagrange multipliers are not available. Oper. Res. Lett. 43(5), 484–488 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Haeser, G., Schuverdt, M.L.: On approximate KKT condition and its extension to continuous variational inequalities. J. Optim. Theory Appl. 149(3), 528–539 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)zbMATHCrossRefGoogle Scholar
  33. 33.
    Huang, X.X., Teo, K.L., Yang, X.Q.: Approximate augmented Lagrangian functions and nonlinear semidefinite programs. Acta Math. Sin. 22(5), 1283–1296 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Janin, R.: Directional Derivative of the Marginal Function in Nonlinear Programming, pp. 110–126. Springer, Berlin (1984)zbMATHGoogle Scholar
  35. 35.
    Jarre, F.: Elementary optimality conditions for nonlinear SDPs. In: Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science (2012)Google Scholar
  36. 36.
    Kočvara, M., Stingl, M.: PENNON—a generalized augmented Lagrangian method for semidefinite programming. In: Di Pillo, G., Murli, A. (eds.) High Performance Algorithms and Software for Nonlinear Optimization, pp. 297–315. Kluwer, Dordrecht (2003)zbMATHGoogle Scholar
  37. 37.
    Kočvara, M., Stingl, M.: On the solution of large-scale SDP problems by the modified barrier method using iterative solvers. Math. Program. 109, 413–444 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kočvara, M., Stingl, M.: PENNON—a code for convex nonlinear and semidefinite programming. Optim. Methods Softw. 18(3), 317–333 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kanno, Y., Takewaki, I.: Sequential semidefinite program for maximum robustness design of structures under load uncertainty. J. Optim. Theory Appl. 130, 265–287 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Konno, H., Kawadai, N., Wu, D.: Estimation of failure probability using semi-definite Logit model. Comput. Manag. Sci. 1(1), 59–73 (2003)zbMATHCrossRefGoogle Scholar
  41. 41.
    Lewis, A.S.: Convex analysis on the Hermitian matrices. SIAM J. Optim. 6(1), 164–177 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Lourenço, B.F., Fukuda, E.H., Fukushima, M.: Optimality conditions for nonlinear semidefinite programming via squared slack variables. Math. Program. 166, 1–24 (2016)zbMATHGoogle Scholar
  43. 43.
    Lovász, L.: Semidefinite Programs and Combinatorial Optimization, pp. 137–194. Springer, New York (2003)zbMATHGoogle Scholar
  44. 44.
    Luo, H.Z., Wu, H.X., Chen, G.T.: On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. J. Glob. Optim. 54(3), 599–618 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Martínez, J.M., Pilotta, E.A.: Inexact restoration algorithm for constrained optimization. J. Optim. Theory Appl. 104(1), 135–163 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118(1), 117–133 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Qi, H., Sun, D.: A quadratically convergent newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28(2), 360–385 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Qi, L., Wei, Z.: On the constant positive linear dependence conditions and its application to SQP methods. SIAM J. Optim. 10(4), 963–981 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Ramos, A.: Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences. Optimization online (2016). http://www.optimization-online.org/DB_HTML/2016/04/5423.html
  51. 51.
    Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. SIAM J. Optim. 77(1), 301–320 (1997)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479–491 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Stingl, M.: On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods. PhD thesis, University of Erlangen (2005)Google Scholar
  54. 54.
    Stingl, M., Kočvara, M., Leugering, G.: A sequential convex semidefinite programming algorithm with an application to multiple-load free material optimization. SIAM J. Optim. 20(1), 130–155 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Sun, D., Sun, J., Zhang, L.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114(2), 349–391 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Sun, J., Zhang, L.W., Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 12(3), 437–456 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Theobald, C.M.: An inequality for the trace of the product of two symmetric matrices. Math. Proc. Camb. Philos. Soc. 77(2), 265–267 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2003)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Tuyen, N.V., Yao, J., Wen, C.: A Note on Approximate Karush–Kuhn–Tucker Conditions in Locally Lipschitz Multiobjective Optimization. ArXiv:1711.08551 (2017)
  60. 60.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 549–95 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Vandenberghe, L., Boyd, S., Wu, S.P.: Determinant maximization with linear matrix inequality constraints. SIAM J. Matrix Anal. Appl. 19(2), 499–533 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Wu, H., Luo, H., Ding, X., Chen, G.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programmings. Comput. Optim. Appl. 56(3), 531–558 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Yamashita, H., Yabe, H.: Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Math. Program. 132(1–2), 1–30 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Yamashita, H., Yabe, H.: A survey of numerical methods for nonlinear semidefinite programming. J. Oper. Res. Soc. Jpn. 58(1), 24–60 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Yamashita, H., Yabe, H., Harada, K.: A primal-dual interior point method for nonlinear semidefinite programming. Math. Program. 135(1–2), 89–121 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Zhu, Z.B., Zhu, H.L.: A filter method for nonlinear semidefinite programming with global convergence. Acta Math. Sin. 30(10), 1810–1826 (2014)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of CampinasCampinasBrazil
  2. 2.Department of Applied MathematicsUniversity of São PauloSão PauloBrazil
  3. 3.Center of Exact and Technological SciencesFederal University of AcreRio BrancoBrazil

Personalised recommendations