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Nonlinear Optimization and Related Topics pp 15–49Cite as

A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints

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Part of the book series: Applied Optimization ((APOP,volume 36))

Abstract

A new primal-dual algorithm is proposed for the minimization of non-convex objective functions subject to simple bounds and linear equality constraints. The method alternates between a classical primal-dual step and a Newton-like modified barrier step in order to ensure descent on a suitable merit function. Convergence of a well-defined subsequence of iterates is proved from arbitrary starting points. Preliminary numerical results are presented.

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References

  • CPLEX 4.0. High-performance linear, integer and quadratic programming soft- ware. CPLEX Optimization, Inc., Incline Village, Nevada, USA, 1995.

    Google Scholar 

  • E. D. Andersen, J. Gondzio, C. Mészdros, and X. Xu. Implementation of interior point methods for large scale linear programming. In T. Terlaky, ed., `Interior Point Methods in Mathematical Programming, pp. 189 - 252, Dordrecht, The Netherlands, 1996. Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • I. Bongartz, A. R. Conn, N.I.M. Gould, and Ph. L. Toint. CUTE: Constrained and Unconstrained Testing Environment. ACM Transactions on Mathematical Software, 21 (1), 123 - 160, 1995.

    Article  MATH  Google Scholar 

  • J. F. Bonnans and C Pola. A trust-region interior point algorithm for linearly constrained optimization. Technical Report 1948, INRIA, Roquencourt, France, 1993.

    Google Scholar 

  • T. M. Carpenter, I. J. Lustig, J. M. Mulvey, and D. F. Shanno. Higher-order predictor-corrector interior point methods with application to quadratic objectives. SIAM Journal on Optimization, 3 (4), 696 - 733, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • A. R. Conn, N. I. M. Gould, and Ph. L. Toint. Global convergence of a class of trust region algorithms for optimization with simple bounds. SIAM Journal on Numerical Analysis, 25, 433-460, 1988. See also same journal 26: 764 - 767, 1989.

    Google Scholar 

  • A. R. Conn, N. I. M. Gould, and Ph. L. Toint. LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A). Number 17 in. LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A). Number 17 in `Springer Series in Computational Mathematics. Springer Verlag, Heidelberg, Berlin, New York, 1992.

    Google Scholar 

  • A. R. Conn, N. I. M. Gould, and Ph. L. Toint. Numerical experiments with the lancelot package (Release A) for large-scale nonlinear optimization. Mathematical Programming, Series A, 73 (1), 73 - 110, 1996a.

    MathSciNet  MATH  Google Scholar 

  • A. R. Conn, N. I. M. Gould, D. Orban, and Ph. L. Toint. A primal-dual trust-region algorithm for minimizing a non-convex function subject to bound and linear equality constraints. Technical Report (in preparation), Department of Mathematics, FUNDP, 1999.

    Google Scholar 

  • A. R. Conn, Nick Gould, A. Sartenaer, and Ph. L. Toint. Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints. SIAM Journal on Optimization, 6 (3), 674 - 703, 1996b.

    Article  MathSciNet  MATH  Google Scholar 

  • International Business Machine Corporation. Optimization Subroutine Library: Guide and Reference,second edn, 1990.

    Google Scholar 

  • I. S. Duff and J. K. Reid. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Transactions on Mathematical Software, 9(3), 302325, 1983.

    Google Scholar 

  • I. S. Duff, N. I. M. Gould, J. K. Reid, J. A. Scott, and K. Turner. The factorization of sparse symmetric indefinite matrices. IMA Journal of Numerical Analysis, 11, 181 - 204, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Forsgren, P. E. Gill, and J. R. Shinnerl. Stability of symmetric ill-conditioned systems arising in interior point methods for constrained optimization. SIAM Journal on Matrix Analysis and Applications, 17 (1), 187 - 211, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  • A. L. Forsgren and P. E. Gill. Primal-dual interior methods for nonconvex nonlinear programming. Technical Report TRITA-MAT-1996-0S4, Royal Institute of Technology, Stockholm, Sweden, 1996.

    Google Scholar 

  • A. L. Forsgren and W. Murray. Newton methods for large-scale linear equality-constrained minimization. SIAM Journal on Matrix Analysis and Applications, 14 (2), 560 - 587, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Fourr and S. Mehrotra. Solving symmetrical indefinite systems for an interior-point method for linear programming. Mathematical Programming, Series A, 62 (1), 15 - 39, 1993.

    MathSciNet  Google Scholar 

  • P. E. Gill, W. Murray, D. B. Ponceléon, and M. A. Saunders. Solving reduced KKT systems in barrier methods for linear and quadratic programming. Technical Report SOL 91-7, Dept. of Operations Research Stanford University, Stanford, USA, 1991.

    Google Scholar 

  • P. E. Gill, W. Murray, D. B. Ponceléon, and M. A. Saunders. Primal-dual methods for linear programming. Mathematical Programming, Series A, 70 (3), 251 - 278, 1995.

    MATH  Google Scholar 

  • P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright. A Schur complement method for sparse quadratic programming. In M. G. Cox and S. Hammar-ling, eds, `Reliable Numerical Computation, pp. 113 - 138, Oxford, 1990. Clarendon Press.

    Google Scholar 

  • N. I. M. Gould. On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem. Mathematical Programming, 32, 90 - 99, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  • N. I. M. Gould. An algorithm for large-scale quadratic programming. IMA Journal of Numerical Analysis, 11 (3), 299 - 324, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • N. I. M. Gould. Constructing appropriate models for large-scale, linearly-constrained, nonconvex, nonlinear, optimization algorithms. Technical Report RAL-TR 95-037, Rutherford Appleton Laboratory, Chilton, England, 1995.

    Google Scholar 

  • Harwell Subroutine Library. A catalogue of subroutines (release 12). Advanced Computing Department, Harwell Laboratory, Harwell, UK, 1995.

    Google Scholar 

  • N. J. Higham and S. Cheng. Modifying the inertia of matrices arising in opti- mization. Linear Algebra and its Applications, 275-276, 261 - 279, 1998.

    Article  MathSciNet  Google Scholar 

  • B. A. Murtagh and M. A. Saunders. Large-scale linearly constrained optimization. Mathematical Programming, 14, 41 - 72, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  • B. A. Murtagh and M. A. Saunders. MINOS 5.4 USERS GUIDE. Technical Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, USA, 1993. Revised 1995.

    Google Scholar 

  • D. B. Ponceleón. Barrier methods for large-scale quadratic programming. PhD thesis, Department of Computer Science, Stanford University, Stanford, California 94305, USA, 1990.

    Google Scholar 

  • S. M. Robinson. A quadratically convergent algorithm for general nonlinear programming problems. Mathematical Programming, 3, 145 - 156, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  • J. B. Rosen and J. Kreuser. A gradient projection algorithm for nonlinear constraints. In F. Lootsma, ed., `Numerical Methods for Nonlinear Optimization, London, 1972. Academic Press.

    Google Scholar 

  • D. F. Shanno. Algorithms for linear programming. In E. Spedicato, ed., `Algorithms for continuous optimization: the state of the art, Dordrecht, The Netherlands, 1994. Kluwer Academic Publishers.

    Google Scholar 

  • E. M. Simantiraki and D. F. Shanno. An infeasible-interior-point method for linear complementarity problems. Technical Report RRR 7-95, RUTCOR, Rutgers University, New Jersey, USA, 1995.

    Google Scholar 

  • R. J. Vanderbei and T. J. Carpenter. Symmetrical indefinite systems for interior point methods. Mathematical Programming, 58 (1), 1 - 32, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • J. H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, UK, 1965.

    MATH  Google Scholar 

  • S. J. Wright. An interior point algorithm for linearly constrained optimization. SIAM Journal on Optimization, 2 (3), 450 - 473, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Zhang and D. Zhang. Superlinear convergence of infeasible-interior-point methods for linear programming. Mathematical Programming, 66 (3), 361 - 377, 1994.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY, 10598, USA

    Andrew R. Conn

  2. Rutherford Appleton Laboratory, Computational Science and Engineering Department, Chilton, Oxfordshire, OX11 0QX, England

    Nicholas I. M. Gould

  3. Department of Mathematics, Facultés Universitaires ND de la Paix, 61, rue de Bruxelles, B-5000, Namur, Belgium, EU

    Philippe L. Toint

Authors
  1. Andrew R. Conn
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  2. Nicholas I. M. Gould
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  3. Philippe L. Toint
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Editor information

Editors and Affiliations

  1. University Di Roma La Sapienza, Italy

    Gianni Di Pillo

  2. University of Pisa, Italy

    Franco Giannessi

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© 2000 Springer Science+Business Media Dordrecht

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Conn, A.R., Gould, N.I.M., Toint, P.L. (2000). A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints. In: Pillo, G.D., Giannessi, F. (eds) Nonlinear Optimization and Related Topics. Applied Optimization, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3226-9_2

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  • DOI: https://doi.org/10.1007/978-1-4757-3226-9_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4823-6

  • Online ISBN: 978-1-4757-3226-9

  • eBook Packages: Springer Book Archive

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