Computational Optimization and Applications

, Volume 37, Issue 2, pp 157–176 | Cite as

Smoothed penalty algorithms for optimization of nonlinear models



We introduce an algorithm for solving nonlinear optimization problems with general equality and box constraints. The proposed algorithm is based on smoothing of the exact l 1-penalty function and solving the resulting problem by any box-constraint optimization method. We introduce a general algorithm and present theoretical results for updating the penalty and smoothing parameter. We apply the algorithm to optimization problems for nonlinear traffic network models and report on numerical results for a variety of network problems and different solvers for the subproblems.


Penalty methods Traffic networks Non-convex optimization methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ben Tal, A., Melman, A., Zowe, J.: Curved search methods for unconstrained optimization. Optimization 21, 669 (1990) MATHMathSciNetGoogle Scholar
  2. 2.
    Bertsekas, D.P.: Necessary and sufficient conditions for a penalty method to be exact. Math. Program. 9, 87 (1975) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic, New York (1982) MATHGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Nondifferentiable Optimization via Approximation. In: Balinski, M., Wolfe, P. (eds.) Mathematical Programming Study, vol. 3. North-Holland, Amsterdam (1993) Google Scholar
  5. 5.
    Bonnans, J.F., Gilbert, J.C., Lemarechal, C., Sagastizabal, C.A.: Numerical Optimization. Springer, Berlin (1997) MATHGoogle Scholar
  6. 6.
    Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Opt. 29, 968 (1991) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190 (1995) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Byrd, R.H., Nocedal, J., Schnabel, R.: Representations of quasi-Newton matrices and their use in limited memory methods. Math. Program. 63, 129 (1994) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chen, C., Mangasarin, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 1 (1995) CrossRefGoogle Scholar
  10. 10.
    Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on road networks. SIAM J. Math. Anal. 36, 1862–1886 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Conn, A.R., Sinclair, J.W.: Quadratic programming via a nondifferentiable penalty function. University of Waterloo, Report CORR 75/15. Google Scholar
  12. 12.
    DiPillo, G., Grippo, L.: A new class of augmented Lagrangians in nonlinear programming. SIAM J. Control Optim. 17, 618 (1979) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Felkel, R.: On a bound constrained optimization technique using second order information. Ph.D. thesis, TU Darmstadt (1999) Google Scholar
  14. 14.
    Fügenschuh, A., Herty, M., Klar, A., Martin, A.: Combinatorial and continuous models for the optimization of traffic flow networks. SIAM J. Optim. 16, 1155–1176 (2006) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Garavello, M., Piccoli, B.: Traffic flow on a road network using the Aw-Rascle model. Preprint (2004). Commun. Partial Differ. Equations (in press) Google Scholar
  16. 16.
    Gonzaga, C.C., Castillo, R.A.: A nonlinear programming algorithm based on non-coercive penalty functions. Math. Program. Ser. A 96 (2003) Google Scholar
  17. 17.
    Han, S.-P., Mangasarin, O.L.: Exact penalty functions in nonlinear programming. Math. Program. 17, 251 (1979) MATHCrossRefGoogle Scholar
  18. 18.
    Heinrich, N.: Eine neue Modifikation des Newtonverfahrens für nichtrestringierte und linear restringierte Optimierungsprobleme mit Mehrfachinaktivierung im linear restringierten Fall. Ph.D. thesis, TU Darmstadt (1995) Google Scholar
  19. 19.
    Herty, M., Klar, A.: Simplified dynamics and optimization of large scale traffic networks. Math. Models Methods Appl. Sci. 14(4), 1 (2004) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Herty, M., Klar, A.: Modelling and optimization of traffic networks. SIAM J. Sci. Comput. 25, 1066 (2004) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Herty, M., Klar, A., Singh, A.K.: Flow Optimization on Traffic Networks. Applied Mathematics and Mechanics, vol. 4. Shanghai Univ. Techn., Shanghai (2004), 624 p. Google Scholar
  22. 22.
    Holden, H., Risebro, N.H.: A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26, 999 (1995) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jahn, O., Möhring, R., Schulz, A.S.: Optimal routing of traffic flows with length restrictions in networks with congestion. In: Inderfurth, K., et al.(eds.) Operations Research Proceedings, p. 437. Springer, Berlin (2000) Google Scholar
  24. 24.
    Jahn, O., Möhring, R., Schulz, A.S., Stier-Moses, N.E.: System-optimal routing of traffic flows with user constraints in networks with congestion. Oper. Res. 53, 600 (2005) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Janesch, S.M.H., Santos, L.T.: Exact penalty methods with constrained subproblems. Investigacion Operativa (1997) Google Scholar
  26. 26.
    Kelley, C.T.: Iterative Methods for Optimization. SIAM Frontiers in Applied Mathematics. SIAM, Philadelphia (1999) MATHGoogle Scholar
  27. 27.
    Köhler, E., Skutella, M., Möhring, R.H.: Traffic networks and flows over time. Preprint No. 752/2002 (2002) Google Scholar
  28. 28.
    Lee, Y.J., Mangasarin, O.L.: SSVM: A smooth support vector machine for classification. Comput. Optim. Appl. 20, 5 (2001) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lighthill, M., Whitham, J.: On kinematic waves, Proc. Roy. Soc. Edinb. A 229, 281 (1983) MathSciNetGoogle Scholar
  30. 30.
    Madsen, K., Nielsen, H.B.: A finite smoothing algorithm for Linear l 1 estimation, SIAM J. Optim. 3 (1993) Google Scholar
  31. 31.
    Mayne, D.Q., Maratos, N.: A first order exact penalty function algorithm for equality constrained optimization problems. Math. Program. 16, 303 (1979) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Mayne, D.Q., Polak, E.: A Superlinearly Convergent Algorithm for Constrained Optimization Problems. Mathematical Programming Study, vol. 16 (1979) Google Scholar
  33. 33.
    Nocedal, J., Wright, St.J.: Numerical Optimization. Springer, Berlin (1999) MATHCrossRefGoogle Scholar
  34. 34.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM Classics in Applied Mathematics. SIAM, Philadelphia (1997) Google Scholar
  35. 35.
    Pinar, M.C., Zenios, S.A.: On smoothing exact penalty functions for convex constrained optimization. SIAM J. Optim. 4 (1994) Google Scholar
  36. 36.
  37. 37.
    Spellucci, P.: Solving QP problems by penalization and smoothing. Preprint, TU Darmstadt (2002) Google Scholar
  38. 38.
    Zhangwill, W.I.: Nonlinear Programming. Prentice-Hall, Englewood Cliffs (1969) Google Scholar
  39. 39.
    Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. Technical report, Northwestern University (1994) Google Scholar
  40. 40.
    Wu, Z.Y., Bai, F.S., Yang, X.Q., Zhang, L.S.: An exact lower order penalty function and its smoothing in nonlinear programming. Optimization 53 (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.FB MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.FB MathematikTU DarmstadtDarmstadtGermany

Personalised recommendations