A penalty-interior-point algorithm for nonlinear constrained optimization | Mathematical Programming Computation Skip to main content
Springer Nature Link
Log in

A penalty-interior-point algorithm for nonlinear constrained optimization

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization due to their Newton-like qualities, both in terms of their scalability and convergence behavior. Each of these two strategies, however, have certain disadvantages that make their use either impractical or inefficient for certain classes of problems. The goal of this paper is to present a penalty-interior-point method that possesses the advantages of penalty and interior-point techniques, but does not suffer from their disadvantages. Numerous attempts have been made along these lines in recent years, each with varying degrees of success. The novel feature of the algorithm in this paper is that our focus is not only on the formulation of the penalty-interior-point subproblem itself, but on the design of updates for the penalty and interior-point parameters. The updates we propose are designed so that rapid convergence to a solution of the nonlinear optimization problem or an infeasible stationary point is attained. We motivate the convergence properties of our algorithm and illustrate its practical performance on large sets of problems, including sets of problems that exhibit degeneracy or are infeasible.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benson, H.Y., Sen, A., Shanno, D.F.: Convergence analysis of an interior-point method for nonconvex nonlinear programming. Optimiz. Methods Softw. submitted (2010)

  2. Borchers B., Mitchell J.E.: An improved branch and bound algorithm for mixed integer nonlinear programming. Comput. Operat. Res. 21(4), 359–367 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Breitfeld, M.G., Shanno, D.F.: A globally convergent penalty-barrier algorithm for nonlinear programming and its computational performance. RUTCOR Research Report, RRR 12-94, Rutgers University, New Brunswick, NJ, USA. Technical report (1994)

  4. Breitfeld M.G., Shanno D.F.: Computational experience with penalty-barrier methods for nonlinear programming. Ann. Operat. Res. 62, 439–463 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byrd R.H., Curtis F.E., Nocedal J.: Infeasibility detection and SQP methods for nonlinear optimization. SIAM J. Optimiz. 20(5), 2281–2299 (2008)

    Article  MathSciNet  Google Scholar 

  6. Byrd R.H., Gilbert J.-Ch., Nocedal J.: Trust region method based on interior point techniques for nonlinear programming. Math. Programm. 89(1), 149–185 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Byrd R.H., Gould N.I.M., Nocedal J., Waltz R.A.: An algorithm for nonlinear optimization using linear programming and equality constrained subproblems. Math. Program. 100(1), 27–48 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Byrd R.H., Gould N.I.M., Nocedal J., Waltz R.A.: On the convergence of successive linear-quadratic programming algorithms. SIAM J. Optimiz. 16(2), 471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byrd R.H., Nocedal J., Waltz R.A.: Steering exact penalty methods for nonlinear programming. Optimiz. Methods Softw. 23(2), 197–213 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen L., Goldfarb D.: Interior-point L2 penalty methods for nonlinear programming with strong global convergence properties. Math. Program. 108(1), 1–36 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, L., Goldfarb, D.: On the fast local convergence of interior-point l2 penalty methods for nonlinear programming, technical report. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, USA. Technical Report x (2006)

  12. Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser A 91, 201–213 (2002)

    Article  MATH  Google Scholar 

  13. Fiacco, A.V., McCormick, G.P.: Nonlinear programming: sequential unconstrained minimization techniques. Classics in Applied Mathematics. SIAM, Philadelphia, PA, USA (1990)

  14. Forsgren A., Gill P.E., Wright M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44(4), 525–597 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fourer R., Gay D.M., Kernighan B.W.: AMPL: a modeling language for mathematical programming. Brooks/Cole, Pacific Grove (2002)

    Google Scholar 

  16. Goldfarb D., Polyak R.A., Scheinberg K., Yuzefovich I.: A modified barrier-augmented lagrangian method for constrained minimization. Comput. Optimiz. Appl. 14, 55–74 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gould N.I.M., Bongartz I., Conn A.R., Toint Ph.L.: CUTE: constrained and unconstrained testing environment. ACM Transact. Math. Softw. 21(1), 123–160 (1995)

    Article  MATH  Google Scholar 

  18. Gould, N.I.M., Orban, D., Toint, Ph.L.: An interior-point l1-penalty method for nonlinear optimization, technical report. Rutherford Appleton Laboratory, Chilton, Oxfordshire, England. Technical report (2003)

  19. Gould N.I.M., Orban D., Toint Ph.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Transact. Math. Softw. 29(4), 373–394 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gould, N.I.M., Robinson, D.P.: A second derivative SQP method: global convergence. SIAM J. Optimiz. submitted (2010)

  21. Grossmann I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optimiz. Eng. 3(3), 227–252 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hock W., Schittkowski K.: Test examples for nonlinear programming codes. J. Optimiz. Theory Appl. 30(1), 127–129 (1980)

    Article  MATH  Google Scholar 

  23. Hu X., Ralph D.: Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optimiz. Theory Appl. 123(2), 365–390 (2004)

    Article  MathSciNet  Google Scholar 

  24. Jittorntrum K., Osborne M.: A modified barrier function method with improved rate of convergence for degenerate problems. J. Aus. Math. Soc. Ser. B 21, 305–329 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Leyffer S., Lopez-Calva G., Nocedal J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optimiz. 17(1), 52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maratos, N.: Exact Penalty Function Algorithms for Finite-Dimensional and Control Optimization Problems. PhD thesis, Department of Computing and Control, University of London (1978)

  27. Morales J.L.: A numerical study of limited memory BFGS methods. Appl. Math. Lett. 15(4), 481–488 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nocedal J., Wächter A., Waltz R.A.: Adaptive barrier update strategies for nonlinear interior methods. SIAM J. Optimiz. 19(4), 1674–1693 (2009)

    Article  MATH  Google Scholar 

  29. Nocedal J., Wright S.J.: Numerical Optimization, Springer Series in Operations Research, 2nd edn. Springer, New York (2006)

    Google Scholar 

  30. Polyak, R.A.: Smooth optimization methods for solving nonlinear extremal and equilibrium problems with constraints. In: Eleventh international symposium on mathematical programming, Bonn, Germany (1982)

  31. Polyak R.A.: Modified barrier functions (theory and methods). Math. Program. 54(1–3), 177–222 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Polyak R.A.: Primal-dual exterior point method for convex optimization. Optimiz. Methods Softw. 23(1), 141–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Quesada I., Grossmann I.E.: An LP/NLP based branch and bound algorithm for convex minlp optimization problems. Comput. Chem. Eng. 16(10–11), 937–947 (1992)

    Article  Google Scholar 

  34. Scheel H., Scholtes S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Operat. Res. 25(1), 1–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vanderbei R.J., Shanno D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optimiz. Appl. 13, 231–252 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wächter A., Biegler L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optimiz. 16, 1–31 (2005)

    Article  MATH  Google Scholar 

  37. Wächter A., Biegler L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, 18018, USA

    Frank E. Curtis

Authors
  1. Frank E. Curtis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank E. Curtis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Curtis, F.E. A penalty-interior-point algorithm for nonlinear constrained optimization. Math. Prog. Comp. 4, 181–209 (2012). https://doi.org/10.1007/s12532-012-0041-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-012-0041-4

Keywords

Mathematics Subject Classification

Search

Navigation