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

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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.

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Notes

  1. Note, that if \(\lambda _k = y_k / \rho _k\) and \(\rho _k = 1\) the step \((\varDelta x_k, \varDelta y_k, \varDelta z_k)\) equals the step \((\widetilde{\varDelta x_k}, \widetilde{\varDelta \lambda _k}, \widetilde{\varDelta y_k})\) of Chen and Goldfarb (2009).

  2. Further information and download of WORHP on www.worhp.de.

  3. See www.netlib.org/blas.

References

  • Armand P, Omheni R (2017a) A globally and quadratically convergent primal–dual augmented Lagrangian algorithm for equality constrained optimization. Optim Methods Softw 32(1):1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Armand P, Omheni R (2017b) A mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization. J Optim Theory Appl 173:1–25

  • Armand P, Benoist J, Omheni R, Pateloup V (2014) Study of a primal–dual algorithm for equality constrained minimization. Comput Optim Appl 59(3):405–433

    Article  MathSciNet  MATH  Google Scholar 

  • Benson H, Vanderbei R, Shanno D (2002) Interior-point methods for nonconvex nonlinear programming: filter methods and merit functions. Comput Optim Appl 23(2):257–272

    Article  MathSciNet  MATH  Google Scholar 

  • 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 95–127

    Chapter  Google Scholar 

  • Benson HY, Sen A, Shanno DF (2007) Convergence analysis of an interior-point method for nonconvex nonlinear programming, Technical report

  • Boman EG (1999) Infeasibility and negative curvature in optimization, Ph.D. thesis. Stanford University

  • 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 85–110

    Chapter  Google Scholar 

  • Byrd RH, Curtis FE, Nocedal J (2010) Infeasibility detection and SQP methods for nonlinear optimization. SIAM J Optim 20(5):2281–2299

    Article  MathSciNet  MATH  Google Scholar 

  • Chen L, Goldfarb D (2006a) Interior-point \(\ell 2\)-penalty methods for nonlinear programming with strong global convergence properties. Math Program 108(1):1–36

    Article  MathSciNet  MATH  Google Scholar 

  • 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 York

  • Chen L, Goldfarb D (2009) An interior-point piecewise linear penalty method for nonlinear programming. Math Program 128(1):73–122

    MathSciNet  MATH  Google Scholar 

  • Conn A, Gould N, Toint P (2000) Trust-region methods. SIAM, Philadelphia

  • Conn A, Gould G, Toint P (2013) Lancelot: a Fortran package for large-scale nonlinear optimization (Release A), Springer Series in Computational Mathematics. Springer, Berlin

    Google Scholar 

  • Curtis FE (2012) A penalty-interior-point algorithm for nonlinear constrained optimization. Math Program Comput 4(2):181–209

    Article  MathSciNet  MATH  Google Scholar 

  • Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201–213

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher R (1983) Penalty functions. Springer, Berlin, pp 87–114

    MATH  Google Scholar 

  • 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 26–40

    Google Scholar 

  • Fletcher R, Leyffer S (2002) Nonlinear programming without a penalty function. Math Program 91(2):239–269

    Article  MathSciNet  MATH  Google Scholar 

  • Forsgren A, Gill PE (1998) Primal–dual interior methods for nonconvex nonlinear programming. SIAM J Optim 8(4):1132–1152

    Article  MathSciNet  MATH  Google Scholar 

  • Forsgren A, Gill PE, Wright MH (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):525–597

    Article  MathSciNet  MATH  Google Scholar 

  • Gertz EM, Gill PE (2004) A primal–dual trust region algorithm for nonlinear optimization. Math Program 100(1):49–94

    MathSciNet  MATH  Google Scholar 

  • Gill PE, Robinson DP (2010) A primal–dual augmented Lagrangian. Comput Optim Appl 51(1):1–25

    Article  MathSciNet  MATH  Google Scholar 

  • Gill PE, Robinson DP (2013) A globally convergent stabilized SQP method. SIAM J Optim 23(4):1983–2010

    Article  MathSciNet  MATH  Google Scholar 

  • Gill PE, Kungurtsev V, Robinson DP (2017a) A stabilized SQP method: global convergence. IMA J Numer Anal 37(1):407–443

    Article  MathSciNet  MATH  Google Scholar 

  • Gill PE, Kungurtsev V, Robinson DP (2017b) A stabilized SQP method: superlinear convergence. Math Program 163(1):369–410

    Article  MathSciNet  MATH  Google Scholar 

  • Goldfarb D, Polyak R, Scheinberg K, Yuzefovich I (1999) A modified barrier-augmented Lagrangian method for constrained minimization. Comput Optim Appl 14(1):55–74

    Article  MathSciNet  MATH  Google Scholar 

  • Gould NIM, Toint PL (2010) Nonlinear programming without a penalty function or a filter. Math Program 122(1):155–196

    Article  MathSciNet  MATH  Google Scholar 

  • 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 117–150

  • Gould NIM, Loh Y, Robinson DP (2014) A filter method with unified step computation for nonlinear optimization. SIAM J Optim 24(1):175–209

    Article  MathSciNet  MATH  Google Scholar 

  • Gould NIM, Loh Y, Robinson DP (2015a) A nonmonotone filter SQP method: local convergence and numerical results. SIAM J Optim 25(3):1885–1911

    Article  MathSciNet  MATH  Google Scholar 

  • 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):545–557

    Article  MathSciNet  MATH  Google Scholar 

  • Greif C, Moulding E, Orban D (2014) Bounds on eigenvalues of matrices arising from interior-point methods. SIAM J Optim 24(1):49–83

    Article  MathSciNet  MATH  Google Scholar 

  • Hogg J, Scott JA (2012) New parallel sparse direct solvers for engineering applications, Technical report. Science and Technology Facilities Council

  • 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):231–257

    Article  MathSciNet  MATH  Google Scholar 

  • Izmailov AF, Solodov MV (2012) Stabilized SQP revisited. Math Program 133(1):93–120

    Article  MathSciNet  MATH  Google Scholar 

  • 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):545–571

    Article  MathSciNet  MATH  Google Scholar 

  • 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 167–183

  • Nocedal J, Wright S (2006) Numerical optimization. Springer, Berlin

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Parikh N, Boyd S (2014) Proximal algorithms. Found Trends Optim 1(3):127–239

    Article  Google Scholar 

  • Shen C, Zhang LH, Liu W (2016) A stabilized filter SQP algorithm for nonlinear programming. J Glob Optim 65(4):677–708

    Article  MathSciNet  MATH  Google Scholar 

  • 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):173–199

    Article  MathSciNet  MATH  Google Scholar 

  • Ulbrich M, Ulbrich S, Vicente NL (2003) A globally convergent primal–dual interior-point filter method for nonlinear programming. Math Program 100(2):379–410

    Article  MathSciNet  MATH  Google Scholar 

  • Vanderbei RJ (1999) LOQO: an interior point code for quadratic programming. Optim Methods Softw 11(1–4):451–484

    Article  MathSciNet  MATH  Google Scholar 

  • Vanderbei RJ, Shanno DF (1999) An interior-point algorithm for nonconvex nonlinear programming. Comput Optim Appl 13(1–3):231–252

    Article  MathSciNet  MATH  Google Scholar 

  • Wächter A, Biegler LT (2000) Failure of global convergence for a class of interior point methods for nonlinear programming. Math Program 88(3):565–574

    Article  MathSciNet  MATH  Google Scholar 

  • 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):25–57

    Article  MathSciNet  MATH  Google Scholar 

  • 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):391–408

    Article  MathSciNet  MATH  Google Scholar 

  • Yamashita H (1998) A globally convergent primal–dual interior point method for constrained optimization. Optim Methods Softw 10(2):443–469

    Article  MathSciNet  MATH  Google Scholar 

  • 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):479–499

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Optimization and Optimal Control, Center for Industrial Mathematics (ZeTeM), University of Bremen, Bibliothekstr. 5, 28359, Bremen, Germany

    Renke Kuhlmann & Christof Büskens

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  1. Renke Kuhlmann
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  2. Christof Büskens
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Correspondence to Renke Kuhlmann.

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Kuhlmann, R., Büskens, C. A primal–dual augmented Lagrangian penalty-interior-point filter line search algorithm. Math Meth Oper Res 87, 451–483 (2018). https://doi.org/10.1007/s00186-017-0625-x

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  • DOI: https://doi.org/10.1007/s00186-017-0625-x

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