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The Sequential Quadratic Programming Method

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1989)

Abstract

Sequential (or Successive) Quadratic Programming (SQP) is a technique for the solution of Nonlinear Programming (NLP)problems. It is, as we shall see, an idealized concept, permitting and indeed necessitating many variations and modifications before becoming available as part of a reliable andefficient production computer code. In this monograph we trace the evolution of the SQP method through some important special cases of nonlinear programming, up to the most general form of problem. To fully understandthese developments it is important to have a thorough grasp of the underlying theoretical concepts, particularly in regard to optimality conditions. In this monograph we include a simple yet rigorous presentation of optimality conditions, which yet covers most cases of interest.

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References

  1. E. M. L. Beale, Numerical Methods In: Nonlinear Programming, J. Abadie, ed., North- Holland, Amsterdam, 1967.

    Google Scholar 

  2. I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and Unconstrained Testing Environment ACM Trans. Math. Software, 21, 1995, pp. 123–160.

    CrossRef  MATH  Google Scholar 

  3. R. H. Byrd, N. I. M. Gould, J. Nocedal and R. A. Waltz, An Active-Set Algorithm for Nonlinear Programming Using Linear Programming and Equality Constrained Subproblems, Math. Programming B, 100, 2004 pp. 27–48.

    MathSciNet  MATH  Google Scholar 

  4. R. H. Byrd, J. Nocedal and R. B. Schnabel, Representations of quasi-Newton matrices and their use in limited memory methods, Math. Programming, 63, 1994, pp. 129–156.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. C. M. Chin, A new trust region based SLP filter algorithm which uses EQP active set strategy, PhD thesis, Dept. of Mathematics, Univ. of Dundee, 2001.

    Google Scholar 

  6. C. M. Chin and R. Fletcher, On the global convergence of an SLP-filter algorithm that takes EQP steps, Math. Programming, 96, 2003, pp. 161–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. R. M. Chamberlain, C. Lemarechal, H. C. Pedersen and M. J. D. Powell, The watchdog technique for forcing convergence in algorithms forconstrained optimization, In: Algorithms for Constrained Minimization of Smooth Nonlinear Functions, A. G. Buckley and J.-L. Goffin, eds., Math. Programming Studies, 16, 1982, pp. 1–17.

    Google Scholar 

  8. A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust Region Methods, MPS-SIAM Series on Optimization, SIAM Publications, Philadelphia, 2000.

    CrossRef  MATH  Google Scholar 

  9. R. Courant, Variational methods for the solution of problems of equilibrium and vibration, Bull. Amer. Math. Soc., 49, 1943, pp. 1–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. J. E. Dennis and J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28, 1974, pp. 549–560.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. R. Fletcher, A model algorithm for composite nondifferentiable optimization problems, In: Nondifferential and Variational Techniques in Optimization, D. C. Sorensen and R. J.-B. Wets eds., Math.Programming Studies, 17, 1982, pp. 67–76.

    Google Scholar 

  12. R. Fletcher, Second order corrections for nondifferentiable optimization, In: Numerical Analysis, Dundee 1981, G. A. Watson ed., Lecture Notes in Mathematics 912, Springer-Verlag, Berlin, 1982.

    Google Scholar 

  13. R. Fletcher, Practical Methods of Optimization, 1987, Wiley, Chichester.

    Google Scholar 

  14. R. Fletcher, Dense Factors of Sparse Matrices, In: Approximation Theory and Optimization, M. D. Buhmann and A. Iserles, eds, C.U.P., Cambridge, 1997.

    Google Scholar 

  15. R. Fletcher, A New Low Rank Quasi-Newton Update Scheme for Nonlinear Programming, In: System Modelling and Optimization, H. Futura, K. Marti and L. Pandolfi, eds., Springer IFIP series in Computer Science, 199, 2006, pp. 275–293, Springer, Boston.

    Google Scholar 

  16. R. Fletcher, N. I. M. Gould, S. Leyffer, Ph. L. Toint, and A. Wächter, Global convergence of trust-region SQP-filter algorithms for general nonlinear programming, SIAM J. Optimization, 13, 2002, pp. 635–659.

    CrossRef  MATH  Google Scholar 

  17. R. Fletcher, A. Grothey and S. Leyffer, Computing sparse Hessian and Jacobian approximations with optimal hereditary properties, In: Large-Scale Optimization with Applications, Part II: Optimal Design and Control, L. T. Biegler, T. F. Coleman, A. R. Conn and F. N. Santosa, Springer, 1997.

    Google Scholar 

  18. R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Programming, 91, 2002, pp. 239–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. R. Fletcher and S. Leyffer, Filter-type algorithms for solving systems of algebraic equations and inequalities, In: G. di Pillo and A. Murli, eds, High Performance Algorithms and Software for Nonlinear Optimization, Kluwer, 2003.

    Google Scholar 

  20. R. Fletcher, S. Leyffer, and Ph. L. Toint, On the global convergence of a filter-SQP algorithm, SIAM J. Optimization, 13, 2002, pp. 44–59.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. R. Fletcher, S. Leyffer, and Ph. L. Toint, A Brief History of Filter Methods, Preprint ANL/MCS-P1372-0906, Argonne National Laboratory, Mathematics and Computer Science Division, September 2006.

    Google Scholar 

  22. R. Fletcher and E. Sainz de la Maza, Nonlinear programming and nonsmooth optimization by successive linear programming, Math. Programming, 43, 1989, pp. 235–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, 2nd Edn., Duxbury Press, 2002.

    Google Scholar 

  24. K. R. Frisch, The logarithmic potential method of convex programming, Oslo Univ. Inst. of Economics Memorandum, May 1955.

    Google Scholar 

  25. P. E. Gill, W. Murray and M. A. Saunders, SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization, SIAM Review, 47, 2005, pp. 99–131.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. C. C. Gonzaga, E. Karas, and M. Vanti, A globally convergent filter method for nonlinear programming, SIAM J. Optimization, 14, 2003, pp. 646–669.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton’s method, SIAM J. Num. Anal., 23, pp. 707–716.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. N. I. M. Gould and Ph. L. Toint, Global Convergence of a Non-monotone Trust-Region SQP-Filter Algorithm for Nonlinear Programming, In: Multiscale Optimization Methods and Applications, W. W. Hager, S. J. Huang, P. M. Pardalos and O. A. Prokopyev, eds., Springer Series on Nonconvex Optimization and Its Applications, Vol. 82, Springer Verlag, 2006.

    Google Scholar 

  29. S. P. Han, A globally convergent method for nonlinear programming, J. Opt. Theo. Applns., 22, 1976, pp. 297–309.

    CrossRef  Google Scholar 

  30. M. R. Hestenes, Multiplier and gradient methods, J. Opt. Theo. Applns, 4, 1969, pp. 303–320.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. W. Karush, Minima of functions of several variables with ineqalities as side conditions, Master’s Thesis, Dept. of Mathematics, Univ. of Chicago, 1939.

    Google Scholar 

  32. H. W. Kuhn and A. W. Tucker, Nonlinear Programming, In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman, ed., University of California Press, 1951.

    Google Scholar 

  33. O. L. Mangasarian and S. Fromowitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints J. Math. Analysis and Applications, 17, 1967, pp. 37–47.

    CrossRef  MATH  Google Scholar 

  34. N. Maratos, Exact penalty function algorithms for finite dimensional and control optimization problems, Ph.D. Thesis, Univ. of London, 1978.

    Google Scholar 

  35. B. A. Murtagh and M. A. Saunders, A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints, Math. Programming Studies, 16, 1982, pp. 84–117.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comp., 35, 1980, pp. 773–782.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. J. Nocedal and M. L. Overton, Projected Hessian updating algorithms for nonlinearly constrained optimization, SIAM J. Num. Anal., 22, 1985, pp. 821–850.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. T. Pietrzykowski, An exact potential method for constrained maxima, SIAM J. Num. Anal., 6, 1969, pp. 217–238.

    CrossRef  MathSciNet  Google Scholar 

  39. R. Polyak, Modified barrier functions (theory and methods), Math. Programming, 54, 1992, pp. 177–222.

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. M. J. D. Powell, A method for nonlinear constraints in minimization problems, In: Optimization, R. Fletcher ed., Academic Press, London, 1969.

    Google Scholar 

  41. M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, In: Numerical Analysis, Dundee 1977, G. A. Watson, ed., Lecture Notes in Mathematics 630, Springer Verlag, Berlin, 1978.

    Google Scholar 

  42. A. A. Ribeiro, E. W. Karas, and C. C. Gonzaga, Global convergence of filter methods for nonlinear programming, Technical report, Dept. of Mathematics, Federal University of Paraná, Brazil, 2006.

    Google Scholar 

  43. S. M. Robinson, A quadratically convergent method for general nonlinear programming problems, Math. Programming, 3, 1972, pp. 145–156.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in non-convex programming, SIAM J. Control, 12, 1974, pp. 268–285.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. Ph. L. Toint, On sparse and symmetric updating subject to a linear equation, Math. Comp., 31, 1977, pp. 954–961.

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optimization, 16, 2005, pp. 1–31.

    CrossRef  MATH  Google Scholar 

  47. A. Wächter and L. T. Biegler, Line search filter methods for nonlinear programming: Local convergence, SIAM J. Optimization, 16, 2005, pp. 32–48.

    CrossRef  MATH  Google Scholar 

  48. R. B. Wilson, A simplicial algorithm for concave programming, Ph.D. dissertation, Harvard Univ. Graduate School of Business Administration, 1960.

    Google Scholar 

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Fletcher, R. (2010). The Sequential Quadratic Programming Method. In: Di Pillo, G., Schoen, F. (eds) Nonlinear Optimization. Lecture Notes in Mathematics(), vol 1989. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11339-0_3

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