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On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

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Abstract

In this paper, we present new convergence properties of the augmented Lagrangian method for nonlinear semidefinite programs (NSDP). Convergence to the approximately global solutions and optimal values of NSDP is first established for a basic augmented Lagrangian scheme under mild conditions, without requiring the boundedness condition of the multipliers. We then propose four modified augmented Lagrangian methods for NSDP based on different algorithmic strategies. We show that the same convergence of the proposed methods can be ensured under weaker conditions.

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References

  • Andreani R., Birgin E.G., Martínez J.M., Schuverdt M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ben-Tal A., Jarre F., Kocvara M., Nemirovski A., Zowe J.: Optimal design of trusses under a nonconvex global buckling constraints. Optim. Eng. 1, 189–213 (2000)

    Article  Google Scholar 

  • Birgin E.G., Castillo R.A., Martínez J.M.: Numerical comparison of augmented Lagrangian algorithms for nonconvex problems. Comput. Optim. Appl. 31, 31–55 (2005)

    Article  Google Scholar 

  • Birgin E.G., Floudas C.A., Martınez J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Program. 125, 139–162 (2010)

    Article  Google Scholar 

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

    Google Scholar 

  • Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory. SIAM Stud. Appl. Math., SIAM, Philadelphia (1994)

  • Chan Z.X., Sun D.: Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J. Optim. 19, 370–396 (2008)

    Article  Google Scholar 

  • Conn A.R., Gould N.I.M., Toint P.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545–572 (1991)

    Article  Google Scholar 

  • Correa R., Ramírez C.H.: A global algorithm for solving nonlinear semidefinite programming. SIAM J. Optim. 15, 303–318 (2004)

    Article  Google Scholar 

  • 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, 348–360 (2001)

    Article  Google Scholar 

  • Fares B., Noll D., Apkarian P.: Robust control via sequential semidefinite programming. SIAM J. Control Optim. 40, 1791–1820 (2002)

    Article  Google Scholar 

  • Forsgren A.: Optimality conditions for nonconvex semidefinite programming. Math. Program. 88, 105–128 (2000)

    Article  Google Scholar 

  • Ghaoui L.E., Niculescu S.I.: Advances in Linear Matrix Inequality Methods in Control, Advances in Design Control. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  • Goh, K.C., Turan, L., Safonov, M.G., Papavassilopoulos, G.P., Ly, J.H.: Biaffine matrix inequality properties and computational methods. In: Proceedings of American Control Conference, pp. 850–851. Baltimore, Maryland (June 1994)

  • Huang X.X., Teo K.L., Yang X.Q.: Approximate augmented Lagrangian functions and nonlinear semidefinite programs. Acta Math. Sin. Engl. Ser. 22, 1283–1296 (2006)

    Article  Google Scholar 

  • Huang X.X., Teo K.L., Yang X.Q.: Lower-order penalization to nonlinear semidefinite programming. J. Optim. Theory Appl. 132, 1–20 (2007)

    Article  Google Scholar 

  • Huang, X.X., Yang, X.Q., Teo, K.L.: Augmented Lagrangian and nonlinear semidefinite programs. In: Variational Analysis and Applications, Nonconvex Optimization and Its Applications, vol. 79, pp. 513–529 (2005)

  • Jarre F.: An interior point method for semidefinite programs. Optim. Eng. 1, 347–372 (2000)

    Article  Google Scholar 

  • Kanzow C., Nagel C.: Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SIAM J. Optim. 13, 1–23 (2002)

    Article  Google Scholar 

  • Lasserre J.B.: On representations of the feasible set in convex optimization. Optim. Lett. 4(1), 1–5 (2010)

    Article  Google Scholar 

  • Luo H.Z., Mastroeni G., Wu H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)

    Article  Google Scholar 

  • Luo H.Z., Sun X.L., Li D.: On the convergence of augmented Lagrangian methods for constrained global optimization. SIAM J. Optim. 18, 1209–1230 (2007)

    Article  Google Scholar 

  • Luo H.Z., Sun X.L., Wu H.X.: Convergence properties of augmented Lagrangian methods for constrained global optimization. Optim. Methods Softw. 23, 763–778 (2008)

    Article  Google Scholar 

  • Luo H.Z., Sun X.L., Xu Y.F.: Convergence properties of modified and partially-augmented Lagrangian methods for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 145, 489–506 (2010)

    Article  Google Scholar 

  • Luo H.Z., Sun X.L., Xu Y.F., Wu H.X.: On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints. J. Global Optim. 46, 217–232 (2010)

    Article  Google Scholar 

  • Luo, H.Z., Wu, H.X., Chen, G.T.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming. J. Optim. Theory Appl. (accepted for publication) (2011)

  • Mosheyev L., Zibulevsky M.: Penalty/barrier multiplier algorithm for semidefinite programming. Optim. Methods Softw. 13, 235–261 (2000)

    Article  Google Scholar 

  • Noll D.: Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optim. Methods Softw. 22, 777–802 (2007)

    Article  Google Scholar 

  • Noll D., Torki M., Apkarian P.: Partially augmented Lagrangian method for matrix inequality constraints. SIAM J. Optim. 15, 161–184 (2004)

    Article  Google Scholar 

  • Ortiz C., Meziat R.: Semidefinite relaxations of dynamical programs under discrete constraints. Optim. Lett. 4(4), 567–583 (2010)

    Article  Google Scholar 

  • Pardalos, P.M., Wolkowicz, H.: Novel approaches to hard discrete optimization, Fields Institute Communications Series vol. 37. American Mathematical Society (2003)

  • Pardalos, P.M., Wolkowicz, H.: Topics in semidefinite and interior-point methods, Fields Institute Communications Series vol. 18. American Mathematical Society (1998)

  • Di Pillo G., Lucidi S.: An augmented Lagrangian function with improved exactness properties. SIAM J. Optim. 12, 376–406 (2001)

    Article  Google Scholar 

  • Qi H.D.: Local duality of nonlinear semidefinite programming. Math. Oper. Res. 34, 124–141 (2009)

    Article  Google Scholar 

  • Ringertz U.T.: Eigenvalues in optimal structural design. In: Biegler, L.T., Coleman, T.F., Conn, A.R., Santosa, F.N. (eds) Large Scale Optimization and Applications, Part I: Optimization in Inverse Problems and Design, Vol. 92 of the IMA Volumes in Mathematics and its Applications, pp. 135–149. Springer, New York (1997)

    Google Scholar 

  • Rockafellar R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Shapiro A.: First and second order analysis of nonlinear semidefinite programs. Math. Program. Ser. B 77, 301–320 (1997)

    Google Scholar 

  • Shapiro A., Sun J.: Some properties of the augmented Lagrangian in cone-constrained optimization. Math. Oper. Res. 29, 479–491 (2004)

    Article  Google Scholar 

  • Stingl, M.: On the solution of nonlinear semidefinite programs by augmented Lagrangian methods. PhD thesis, Institute of Applied Mathematics, Universitytat Erlangen-Nurnberg (2005)

  • 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  Google Scholar 

  • Sun D., Sun J., Zhang L.W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349–391 (2008)

    Article  Google Scholar 

  • Sun J., Zhang L.W., Wu Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 129, 437–456 (2006)

    Article  Google Scholar 

  • Todd M.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)

    Article  Google Scholar 

  • Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  Google Scholar 

  • Waki, H: How to generate weakly infeasible semidefinite programs via Lasserre’s relaxations for polynomial optimization. Optim. Lett. (2011). doi:10.1007/s11590-011-0384-1 (online)

  • Wang C.Y., Li D.: Unified theory of augmented Lagrangian methods for constrained global optimization. J. Global Optim. 44, 433–458 (2009)

    Article  Google Scholar 

  • Wolkowicz, H., Saigal, R., Vandenberghe, L (eds): Handbook of Semidefinite Programming. Kluwer Academic Publishers, Boston (2000)

    Google Scholar 

  • Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds): Handbook of Semidefinite Programming, Theory, Algorithms and Applications, International Series in Operations Research and Management Science, vol. 27. Kluwer Academic Publishers, Boston, MA (2000)

    Google Scholar 

  • Wu H.X., Luo H.Z., Li S.L.: The global convergence of augmented Lagrangian methods based on NCP function in constrained nonconvex optimization. Appl. Math. Comput. 207, 124–134 (2009)

    Article  Google Scholar 

  • Wu, H.X., Luo, H.Z.: A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization. Optimization (2011). doi:10.1080/02331934.2011.564620 (online)

  • Wu, H.X., Luo, H.Z.: Saddle points of general augmented Lagrangians for constrained nonconvex optimization. J. Global Optim. (2011). doi:10.1007/s10898-011-9731-0 (online)

  • Ye Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, People’s Republic of China

    H. Z. Luo

  2. Department of Mathematics, College of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, People’s Republic of China

    H. X. Wu & G. T. Chen

Authors
  1. H. Z. Luo
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  2. H. X. Wu
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  3. G. T. Chen
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Correspondence to H. Z. Luo.

Additional information

This work was jointly supported by the National Natural Science Foundation of China under grant 11071219, the Postdoctoral Key Research Foundation of China under grant 201003242, the Zhejiang Provincial Natural Science Foundation of China under grant Y6090080.

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Luo, H.Z., Wu, H.X. & Chen, G.T. On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. J Glob Optim 54, 599–618 (2012). https://doi.org/10.1007/s10898-011-9779-x

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