Advertisement

Mathematical Methods of Operations Research

, Volume 87, Issue 3, pp 451–483 | Cite as

A primal–dual augmented Lagrangian penalty-interior-point filter line search algorithm

  • Renke Kuhlmann
  • Christof Büskens
Original Article
  • 170 Downloads

Abstract

Interior-point methods have been shown to be very efficient for large-scale nonlinear programming. The combination with penalty methods increases their robustness due to the regularization of the constraints caused by the penalty term. In this paper a primal–dual penalty-interior-point algorithm is proposed, that is based on an augmented Lagrangian approach with an \(\ell 2\)-exact penalty function. Global convergence is maintained by a combination of a merit function and a filter approach. Unlike the majority of filter methods, no separate feasibility restoration phase is required. The algorithm has been implemented within the solver WORHP to study different penalty and line search options and to compare its numerical performance to two other state-of-the-art nonlinear programming algorithms, the interior-point method IPOPT and the sequential quadratic programming method of WORHP.

Keywords

Nonlinear programming Constrained optimization Augmented Lagrangian Penalty-interior-point algorithm Primal–dual method 

Mathematics Subject Classification

49M05 49M15 49M29 49M37 90C06 90C26 90C30 90C51 

References

  1. Armand P, Omheni R (2017a) A globally and quadratically convergent primal–dual augmented Lagrangian algorithm for equality constrained optimization. Optim Methods Softw 32(1):1–21MathSciNetCrossRefzbMATHGoogle Scholar
  2. Armand P, Omheni R (2017b) A mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization. J Optim Theory Appl 173:1–25Google Scholar
  3. Armand P, Benoist J, Omheni R, Pateloup V (2014) Study of a primal–dual algorithm for equality constrained minimization. Comput Optim Appl 59(3):405–433MathSciNetCrossRefzbMATHGoogle Scholar
  4. Benson H, Vanderbei R, Shanno D (2002) Interior-point methods for nonconvex nonlinear programming: filter methods and merit functions. Comput Optim Appl 23(2):257–272MathSciNetCrossRefzbMATHGoogle Scholar
  5. Benson HY, Shanno DF, Vanderbei RJ (2003) A comparative study of large-scale nonlinear optimization algorithms. In: Di Pillo G, Murli A (eds) High performance algorithms and software for nonlinear optimization, applied optimization, vol 82. Springer, New York, pp 95–127CrossRefGoogle Scholar
  6. Benson HY, Sen A, Shanno DF (2007) Convergence analysis of an interior-point method for nonconvex nonlinear programming, Technical reportGoogle Scholar
  7. Boman EG (1999) Infeasibility and negative curvature in optimization, Ph.D. thesis. Stanford UniversityGoogle Scholar
  8. Büskens C, Wassel D (2013) The ESA NLP solver WORHP. In: Fasano G, Pintér JD (eds) Modeling and optimization in space engineering, Springer optimization and its applications, vol 73. Springer, New York, pp 85–110CrossRefGoogle Scholar
  9. Byrd RH, Curtis FE, Nocedal J (2010) Infeasibility detection and SQP methods for nonlinear optimization. SIAM J Optim 20(5):2281–2299MathSciNetCrossRefzbMATHGoogle Scholar
  10. Chen L, Goldfarb D (2006a) Interior-point \(\ell 2\)-penalty methods for nonlinear programming with strong global convergence properties. Math Program 108(1):1–36MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chen L, Goldfarb D (2006b) On the fast local convergence of interior-point \(\ell 2\)-penalty methods for nonlinear programming, Technical report. IEOR Department, Columbia University, New YorkGoogle Scholar
  12. Chen L, Goldfarb D (2009) An interior-point piecewise linear penalty method for nonlinear programming. Math Program 128(1):73–122MathSciNetzbMATHGoogle Scholar
  13. Conn A, Gould N, Toint P (2000) Trust-region methods. SIAM, PhiladelphiaGoogle Scholar
  14. Conn A, Gould G, Toint P (2013) Lancelot: a Fortran package for large-scale nonlinear optimization (Release A), Springer Series in Computational Mathematics. Springer, BerlinGoogle Scholar
  15. Curtis FE (2012) A penalty-interior-point algorithm for nonlinear constrained optimization. Math Program Comput 4(2):181–209MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201–213MathSciNetCrossRefzbMATHGoogle Scholar
  17. Fletcher R (1983) Penalty functions. Springer, Berlin, pp 87–114zbMATHGoogle Scholar
  18. Fletcher R (1985) An \(\ell 1\) penalty method for nonlinear constraints. In: Boggs PT, Byrd RH, Schnabel RB (eds) Numerical optimization 1984. SIAM, Philadelphia, pp 26–40Google Scholar
  19. Fletcher R, Leyffer S (2002) Nonlinear programming without a penalty function. Math Program 91(2):239–269MathSciNetCrossRefzbMATHGoogle Scholar
  20. Forsgren A, Gill PE (1998) Primal–dual interior methods for nonconvex nonlinear programming. SIAM J Optim 8(4):1132–1152MathSciNetCrossRefzbMATHGoogle Scholar
  21. Forsgren A, Gill PE, Wright MH (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):525–597MathSciNetCrossRefzbMATHGoogle Scholar
  22. Gertz EM, Gill PE (2004) A primal–dual trust region algorithm for nonlinear optimization. Math Program 100(1):49–94MathSciNetzbMATHGoogle Scholar
  23. Gill PE, Robinson DP (2010) A primal–dual augmented Lagrangian. Comput Optim Appl 51(1):1–25MathSciNetCrossRefzbMATHGoogle Scholar
  24. Gill PE, Robinson DP (2013) A globally convergent stabilized SQP method. SIAM J Optim 23(4):1983–2010MathSciNetCrossRefzbMATHGoogle Scholar
  25. Gill PE, Kungurtsev V, Robinson DP (2017a) A stabilized SQP method: global convergence. IMA J Numer Anal 37(1):407–443MathSciNetCrossRefzbMATHGoogle Scholar
  26. Gill PE, Kungurtsev V, Robinson DP (2017b) A stabilized SQP method: superlinear convergence. Math Program 163(1):369–410MathSciNetCrossRefzbMATHGoogle Scholar
  27. Goldfarb D, Polyak R, Scheinberg K, Yuzefovich I (1999) A modified barrier-augmented Lagrangian method for constrained minimization. Comput Optim Appl 14(1):55–74MathSciNetCrossRefzbMATHGoogle Scholar
  28. Gould NIM, Toint PL (2010) Nonlinear programming without a penalty function or a filter. Math Program 122(1):155–196MathSciNetCrossRefzbMATHGoogle Scholar
  29. Gould NIM, Orban D, Toint PL (2005) Numerical analysis and optimization: NAO-III, Muscat, Oman, January 2014, chap. In: An interior-point \(\ell 1\)-penalty method for nonlinear optimization. Springer, Cham, pp 117–150Google Scholar
  30. Gould NIM, Loh Y, Robinson DP (2014) A filter method with unified step computation for nonlinear optimization. SIAM J Optim 24(1):175–209MathSciNetCrossRefzbMATHGoogle Scholar
  31. Gould NIM, Loh Y, Robinson DP (2015a) A nonmonotone filter SQP method: local convergence and numerical results. SIAM J Optim 25(3):1885–1911MathSciNetCrossRefzbMATHGoogle Scholar
  32. Gould NIM, Orban D, Toint PL (2015b) Cutest: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput Optim Appl 60(3):545–557MathSciNetCrossRefzbMATHGoogle Scholar
  33. Greif C, Moulding E, Orban D (2014) Bounds on eigenvalues of matrices arising from interior-point methods. SIAM J Optim 24(1):49–83MathSciNetCrossRefzbMATHGoogle Scholar
  34. Hogg J, Scott JA (2012) New parallel sparse direct solvers for engineering applications, Technical report. Science and Technology Facilities CouncilGoogle Scholar
  35. Izmailov AF, Solodov MV (2011) On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-newton SQP methods to critical multipliers. Math Program 126(2):231–257MathSciNetCrossRefzbMATHGoogle Scholar
  36. Izmailov AF, Solodov MV (2012) Stabilized SQP revisited. Math Program 133(1):93–120MathSciNetCrossRefzbMATHGoogle Scholar
  37. Liu X, Yuan Y (2011) A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization. SIAM J Optim 21(2):545–571MathSciNetCrossRefzbMATHGoogle Scholar
  38. Morales JL, Nocedal J, Waltz RA, Liu G, Goux JP (2003) Assessing the potential of interior methods for nonlinear optimization. In: Biegler LT, Heinkenschloss M, Ghattas O, van Bloemen Waanders B (eds) Large-scale PDE-constrained optimization, lecture notes in computational science and engineering, vol 30. Springer, Berlin, pp 167–183Google Scholar
  39. Nocedal J, Wright S (2006) Numerical optimization. Springer, BerlinzbMATHGoogle Scholar
  40. Nocedal J, Wächter A, Waltz RA (2009) Adaptive barrier update strategies for nonlinear interior methods. SIAM J Optim 19(4):1674–1693MathSciNetCrossRefzbMATHGoogle Scholar
  41. Parikh N, Boyd S (2014) Proximal algorithms. Found Trends Optim 1(3):127–239CrossRefGoogle Scholar
  42. Shen C, Zhang LH, Liu W (2016) A stabilized filter SQP algorithm for nonlinear programming. J Glob Optim 65(4):677–708MathSciNetCrossRefzbMATHGoogle Scholar
  43. Tits AL, Wächter A, Bakhtiari S, Urban TJ, Lawrence CT (2003) A primal–dual interior-point method for nonlinear programming with strong global and local convergence properties. SIAM J Optim 14(1):173–199MathSciNetCrossRefzbMATHGoogle Scholar
  44. Ulbrich M, Ulbrich S, Vicente NL (2003) A globally convergent primal–dual interior-point filter method for nonlinear programming. Math Program 100(2):379–410MathSciNetCrossRefzbMATHGoogle Scholar
  45. Vanderbei RJ (1999) LOQO: an interior point code for quadratic programming. Optim Methods Softw 11(1–4):451–484MathSciNetCrossRefzbMATHGoogle Scholar
  46. Vanderbei RJ, Shanno DF (1999) An interior-point algorithm for nonconvex nonlinear programming. Comput Optim Appl 13(1–3):231–252MathSciNetCrossRefzbMATHGoogle Scholar
  47. Wächter A, Biegler LT (2000) Failure of global convergence for a class of interior point methods for nonlinear programming. Math Program 88(3):565–574MathSciNetCrossRefzbMATHGoogle Scholar
  48. Wächter A, Biegler LT (2006) On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57MathSciNetCrossRefzbMATHGoogle Scholar
  49. Waltz R, Morales J, Nocedal J, Orban D (2006) An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math Program 107(3):391–408MathSciNetCrossRefzbMATHGoogle Scholar
  50. Yamashita H (1998) A globally convergent primal–dual interior point method for constrained optimization. Optim Methods Softw 10(2):443–469MathSciNetCrossRefzbMATHGoogle Scholar
  51. Yamashita H, Yabe H (2003) An interior point method with a primal–dual quadratic barrier penalty function for nonlinear optimization. SIAM J Optim 14(2):479–499MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Optimization and Optimal Control, Center for Industrial Mathematics (ZeTeM)University of BremenBremenGermany

Personalised recommendations