Skip to main content
SpringerLink
Log in

An inexact modified subgradient algorithm for nonconvex optimization

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang-bang control problem, under several different inexactness schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bazaraa, M.S., Sherali, H.D.: On the choice of step sizes in subgradient optimization. Eur. J. Oper. Res. 7, 380–388 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Birgin, E.G., Floudas, C.A., Martínez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Preprint (2007)

  3. Brännlund, U.: A generalized subgradient method with relaxation step. Math. Program. 71, 207–219 (1995)

    Article  Google Scholar 

  4. Burachik, R.S., Gasimov, R.N., Ismayilova, N.A., Kaya, C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Glob. Optim. 34(1), 55–78 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burachik, R.S., Kaya, C.Y.: An update rule and a convergence result for a penalty function method. J. Ind. Manage. Optim. 3(2), 381–398 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Burachik, R.S., Rubinov, A.M.: On the absence of duality gap for Lagrange-type functions. J. Ind. Manage. Optim. 1(1), 33–38 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Floudas, A.C., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüş, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  8. Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Glob. Optim. 24, 187–203 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gasimov, R.N., Ismayilova, N.A.: The modified subgradient method for equality constrained nonconvex optimization problems. In: Qi, L., et al. (eds.) Optimization and Control with Applications. Applied Optimization, vol. 96, pp. 257–270. Springer, New York (2005)

    Chapter  Google Scholar 

  10. Kaya, C.Y., Noakes, J.L.: Computational method for time-optimal switching control. J. Optim. Theory Appl. 117(1), 69–92 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    MATH  Google Scholar 

  12. Kiwiel, K.: Convergence of approximate and incremental subgradient methods for convex optimization. SIAM J. Optim. 14(3), 807–840 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9(1), 112–147 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. McKinnon, K.I.M.: Convergence of the Nelder-Mead simplex method to a nonstationary point. SIAM J. Optim. 9(1), 148–158 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mijangos, E.: Approximate subgradient methods for nonlinearly constrained network flow problems. J. Optim. Theory Appl. 128(1), 167–190 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Murtagh, B.A., Saunders, M.A.: MINOS 5.4 user’s guide. Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University (1983)

  17. Nedić, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12(1), 109–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Price, J.C., Coope, I.D., Byatt, D.: A convergent variant of the Nelder-Mead algorithm. J. Optim. Theory Appl. 113(1), 5–19 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Polyak, B.T.: Minimization of unsmooth functionals. Z. Vychisl. Mat. Mat. Fiz. 9, 509–521 (1969)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  21. Sherali, H.D., Choi, G., Tuncbilek, C.H.: A variable target value method for nondifferentiable optimization. Oper. Res. Lett. 26, 1–8 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Shor, N.Z.: Minimization Methods for Nondifferentiable Functions. Springer, Berlin (1985)

    Google Scholar 

  23. Tong, X., Qi, L., Yang, Y.: The Lagrangian globalization method for nonsmooth constrained equations. Comput. Optim. Appl. 33, 89–109 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear Lagrange duality theorems and penalty function methods in continuous optimization. J. Glob. Optim 27, 473–484 (2003)

    Article  MATH  Google Scholar 

  25. Wood, A.J., Wollenberg, B.F.: Power Generation and Control. Wiley, New York (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Informatics and Applied Optimization, University of Ballarat, Victoria, Australia

    Regina S. Burachik & Musa Mammadov

  2. School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA, 5095, Australia

    Regina S. Burachik & C. Yalçın Kaya

Authors
  1. Regina S. Burachik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Yalçın Kaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Musa Mammadov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Yalçın Kaya.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burachik, R.S., Kaya, C.Y. & Mammadov, M. An inexact modified subgradient algorithm for nonconvex optimization. Comput Optim Appl 45, 1–24 (2010). https://doi.org/10.1007/s10589-008-9168-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-008-9168-7

Keywords

Search

Navigation