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Considerations of numerical analysis in a sequential quadratic programming method

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Numerical Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1230))

Abstract

This paper describes some of the important issues of numerical analysis in implementing a sequential quadratic programming method for nonlinearly constrained optimization. We consider the separate treatment of linear constraints, design of a specialized quadratic programming algorithm, and control of ill-conditioning. The results of applying the method to two specific examples are analyzed in detail.

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References

  • Coleman, T. F. and Conn, A. R. (1982). Nonlinear programming via an exact penalty function: asymptotic analysis, Mathematical Programming 24, pp. 123–136.

    Article  MathSciNet  MATH  Google Scholar 

  • Coleman, T. F. and Conn, A. R. (1984). On the local convergence of a quasi-Newton method for the nonlinear programming problem, SIAM Journal on Numerical Analysis 21, pp. 755–769.

    Article  MathSciNet  MATH  Google Scholar 

  • Coleman, T. F. and Sorensen, D. C. (1984). A note on the computation of an orthogonal basis for the null space of a matrix, Mathematical Programming 29, pp. 234–242.

    Article  MathSciNet  MATH  Google Scholar 

  • Dembo, R. S. (1976). A set of geometric test problems and their solutions, Mathematical Programming 10, pp. 192–213.

    Article  MathSciNet  MATH  Google Scholar 

  • Dennis, J. E., Jr. and Moré, J. J. (1977). Quasi-Newton methods, motivation and theory, SIAM Review 19, pp. 46–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher, R. (1981). “Numerical experiments with an exact ℓ1 penalty function method”, in Nonlinear Programming 4 (O. L. Mangasarian, R. R. Meyer and S. M. Robinson, eds.), Academic Press, London and New York, pp. 99–129.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1983). On the representation of a basis for the null space, Report SOL 83-19, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1984a). User's guide to SOL/NPSOL (Version 2.1), Report SOL 84-7, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1984b). Procedures for optimization problems with a mixture of bounds and general linear constraints, ACM Transactions on Mathematical Software 10, pp. 282–298.

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1984c). User's guide to SOL/QPSOL (Version 3.2), Report SOL 84-6, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1985a). Model building and practical aspects of nonlinear programming, Report SOL 85-3, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1985b). The design and implementation of a quadratic programming algorithm, to appear.

    Google Scholar 

  • Gill, P. E., Murray, W., Saunders, M. A., Stewart, G. W. and Wright, M. H. (1985c). Properties of a representation of a basis for the null space, Report SOL 85-1, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Gill, P. E., Murray, W. and Wright, M. H. (1981). Practical Optimization, Academic Press, London and New York.

    MATH  Google Scholar 

  • Harris, P. M. J. (1973). Pivot selection methods of the Devex LP code, Mathematical Programming 5, pp. 1–28. [Reprinted in Mathematical Programming Study 4 (1975), pp. 30–57.]

    Article  MathSciNet  MATH  Google Scholar 

  • Murray, W. and Wright, M. H. (1978). Methods for nonlinearly constrained optimization based on the trajectories of penalty and barrier functions, Report SOL 78-23, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Nocedal, J. and Overton, M. L. (1983). Projected Hessian updating algorithms for nonlinearly constrained optimization, Report 95, Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, New York, New York.

    MATH  Google Scholar 

  • Ortega, J. M. and Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, London and New York.

    MATH  Google Scholar 

  • Peters, G. and Wilkinson, J. H. (1970). The least-squares problem and pseudo-inverses, Computer Journal 13, pp. 309–316.

    Article  MATH  Google Scholar 

  • Powell, M. J. D. (1974). “Introduction to constrained optimization”, in Numerical Methods for Constrained Optimization (P. E. Gill and W. Murray, eds.), pp. 1–28, Academic Press, London and New York.

    Google Scholar 

  • Powell, M. J. D. (1977a). A fast algorithm for nonlinearly constrained optimization calculations, Report DAMTP 77/NA 2, University of Cambridge, England.

    Google Scholar 

  • Powell, M. J. D. (1977b). Variable metric methods for constrained optimization, Report DAMTP 77/NA 5, University of Cambridge, England.

    Google Scholar 

  • Powell, M. J. D. (1983). “Variable metric methods for constrained optimization”, in Mathematical Programming: The State of the Art, (A. Bachem, M. Grötschel and B. Korte, eds.), pp. 288–311, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo.

    Chapter  Google Scholar 

  • Robinson, S. M. (1974). Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear programming algorithms, Math. Prog. 7, pp. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Schittkowski, K. (1981). The nonlinear programming method of Wilson, Han and Powell with an augemented Lagrangian type line search function, Numerische Mathematik 38, pp. 83–114.

    Article  MathSciNet  MATH  Google Scholar 

  • Schittkowski, K. (1982). On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function, Report SOL 82-4, Department of Operations Research, Stanford University, Stanford, California.

    Google Scholar 

  • Tone, K. (1983). Revisions of constraint approximations in the successive QP method for non-linear programming, Mathematical Programming 26, pp. 144–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilson, R. B. (1963). A Simplicial Algorithm for Concave Programming, Ph.D. Thesis, Harvard University.

    Google Scholar 

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Authors and Affiliations

  1. Systems Optimization Laboratory Department of Operations Research, Stanford University, 94305, Stanford, California, USA

    Philip E. Gill, Walter Murray, Michael A. Saunders & Margaret H. Wright

Authors
  1. Philip E. Gill
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  2. Walter Murray
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  3. Michael A. Saunders
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  4. Margaret H. Wright
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Editor information

Jean-Pierre Hennart

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© 1986 Springer-Verlag

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Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H. (1986). Considerations of numerical analysis in a sequential quadratic programming method. In: Hennart, JP. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072670

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  • DOI: https://doi.org/10.1007/BFb0072670

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17200-0

  • Online ISBN: 978-3-540-47379-4

  • eBook Packages: Springer Book Archive

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