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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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On secant updates for use in general constrained optimization
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by Richard Tapia PDF
Math. Comp. 51 (1988), 181-202 Request permission


In this paper we present two new classes of successive quadratic programming (SQP) secant methods for the equality-constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation. We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or q-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.
  • Mordecai Avriel, Nonlinear programming, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Analysis and methods. MR 0489892
  • M. C. Bartholomew-Biggs, Matrix Updating in Nonlinear Programming Calculation, TR No. 156, The Hatfield Polytechnic Numerical Optimization Center, Hatfield, Hertfordshire, England, 1985. P. T. Boggs & J. W. Tolle, An Efficient Strategy for Utilizing a Merit Function in Nonlinear Programming Algorithms, T-R 85-5, Curriculum in Operations Research and Systems Analysis, University of North Carolina, Chapel Hill.
  • Paul T. Boggs, Jon W. Tolle, and Pyng Wang, On the local convergence of quasi-Newton methods for constrained optimization, SIAM J. Control Optim. 20 (1982), no. 2, 161–171. MR 646946, DOI 10.1137/0320014
  • C. G. Broyden, J. E. Dennis Jr., and Jorge J. Moré, On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl. 12 (1973), 223–245. MR 341853
  • Thomas F. Coleman and Andrew R. Conn, On the local convergence of a quasi-Newton method for the nonlinear programming problem, SIAM J. Numer. Anal. 21 (1984), no. 4, 755–769. MR 749369, DOI 10.1137/0721051
  • J. E. Dennis Jr. and Jorge J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977), no. 1, 46–89. MR 445812, DOI 10.1137/1019005
  • John E. Dennis Jr. and Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. MR 702023
  • J. E. Dennis Jr. and Homer F. Walker, Convergence theorems for least-change secant update methods, SIAM J. Numer. Anal. 18 (1981), no. 6, 949–987. MR 638993, DOI 10.1137/0718067
  • R. Fletcher, Practical methods of optimization. Vol. 1, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1980. Unconstrained optimization. MR 585160
  • R. Fontecilla, The Lack of Positive Definiteness in the Hessian in Constrained Optimization, TR 1334, Department of Computer Science, University of Maryland, College Park, Maryland 20742, 1983.
  • Rodrigo Fontecilla, Trond Steihaug, and Richard A. Tapia, A convergence theory for a class of quasi-Newton methods for constrained optimization, SIAM J. Numer. Anal. 24 (1987), no. 5, 1133–1151. MR 909070, DOI 10.1137/0724075
  • Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical optimization, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. MR 634376
  • S. T. Glad, Properties of updating methods for the multipliers in augmented Lagrangians, J. Optim. Theory Appl. 28 (1979), no. 2, 135–156. MR 536718, DOI 10.1007/BF00933239
  • Shih Ping Han, Superlinearly convergent variable metric algorithms for general nonlinear programming problems, Math. Programming 11 (1976/77), no. 3, 263–282. MR 483440, DOI 10.1007/BF01580395
  • Shih Ping Han, Dual variable metric algorithms for constrained optimization, SIAM J. Control Optim. 15 (1977), no. 4, 546–565. MR 459635, DOI 10.1137/0315037
  • Jorge Nocedal and Michael L. Overton, Projected Hessian updating algorithms for nonlinearly constrained optimization, SIAM J. Numer. Anal. 22 (1985), no. 5, 821–850. MR 799115, DOI 10.1137/0722050
  • M. J. D. Powell, The convergence of variable metric methods for nonlinearly constrained optimization calculations, Nonlinear programming, 3 (Proc. Sympos., Special Interest Group Math. Programming, Univ. Wisconsin, Madison, Wis., 1977) Academic Press, New York-London, 1978, pp. 27–63. MR 507858
  • R. A. Tapia, Diagonalized multiplier methods and quasi-Newton methods for constrained optimization, J. Optim. Theory Appl. 22 (1977), no. 2, 135–194. MR 459641, DOI 10.1007/BF00933161
  • R. A. Tapia, Quasi-Newton methods for equality constrained optimization: equivalence of existing methods and a new implementation, Nonlinear programming, 3 (Proc. Sympos., Special Interest Group Math. Programming, Univ. Wisconsin, Madison, Wis., 1977) Academic Press, New York-London, 1978, pp. 125–164. MR 507861
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Math. Comp. 51 (1988), 181-202
  • MSC: Primary 90C30; Secondary 49D37, 90C20
  • DOI:
  • MathSciNet review: 942149