Abstract
In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numcrical tests. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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References
J. Bracken, G.P. McCormick, Selected applications of nonlinear programming, Wiley, New York, 1968.
J.W. Burke, An exact penalization viewpoint of constrained optimization, SIAM J. Control Optim. 29 (1991) 968–998.
R.H. Byrd, J. Nocedal, An analysis of reduced Hessian methods for constrained optimization, Math. Programming 49 (1991) 285–323.
R.H. Byrd, J. Nocedal, R.B. Schnabel, Representations of quasi-Newton-matrices and their use in limited memory methods, Math. Programming 63 (1994) 129–156.
Th.F. Coleman, A.R. Conn, Nonlinear programming via an exact penalty function: Global analysis, Math. Programming 24 (1982) 137–161.
Th.F. Coleman, A.R. Conn, Nonlinear programming via an exact penalty function: Asymptotic analysis, Math. Programming 24 (1982) 123–136.
Th.F. Coleman, A.R. Conn, On the local convergence of a quasi-Newton-method for the nonlinear programming problem, SIAM J. Numer. Anal. 21 (1984) 755–769.
Th.F. Coleman, P.A. Fenyes, Partitioned quasi-Newton methods for nonlinear equality constrained optimization, Math. Programming 53 (1992) 17–44.
A.R. Conn, T. Pietrzykowski, A penalty function method converging directly to a constrained optimum, SIAM J. Numer. Anal. 14 (1977) 348–374.
K. Crusius, Ein global konvergentes Verfahren der projizierten Richtungen mit nicht notwendig zulàssigen Iterationspunkten, Ph.D. Thesis, Mainz University, Mainz, Germany, 1983.
R.S. Dembo, A set of geometric programming test problems and their solutions, Math. Programming 10 (1976) 192–213.
J.C. Dodu, P. Huard, Utilisation de mises à jour doubles dans les méthodes de quasi-Newton, Comptes Rendus de l' Academie de Sciences Paris Série I, 313 (1991) 329–334.
I.S. Duff, A.M. Erisman, J.K. Reid, Direct method for sparse matrices, Oxford Univ. Press, Oxford, 1986.
R. Fletcher, Practical methods of optimization, 2nd ed., Wiley, Chicester, 1987.
D. Gabay, Reduced quasi-Newton methods with feasibility improvement for nonlinearly constrained optimization, Math. Programming Stud. 16 (1982) 18–44.
Ph.E. Gill, W. Murray, Numerically stable methods for quadratic programming, Math. Programming 14 (1978) 349–372.
Ph.E. Gill, W. Murray, M. Saunders, M.H. Wright, Some theoretical properties of an augmented Lagrangian merit function, in: P.M. Pardalos (Ed.), Advances in Optimization and Parallel Computing, North Holland, Amsterdam, 1992, pp. 101–128.
Ph.E. Gill, W. Murray, M. Saunders, M.H. Wright, Sparse matrix methods in optimization, SIAM J. Sci. Comp. 5 (1984) 562–589.
Ph.E. Gill, S.J. Hammarling, W. Murray, M. Saunders, M.H. Wright, Users Guide for NOPSOL (ver. 4.0), Department O R, Stanford University, Report SOL 86-2, 1986.
Ch. Gurwitz, Local convergence of a two-piece update of a projected Hessian matrix, SIAM J. Optim. 4 (1994) 461–485.
Ch. Gurwitz, M. Overton, Sequential quadratic programming methods based on approximating a projected Hessian matrix, SIAM J. Sci. Comp. 10 (1989) 631–653.
D.M. Himmelblau, Applied nonlinear programming, McGraw-Hill, New York, 1972.
J. Heinz, P. Spellucci, A successful implementation of the Pantoja-Mayne SQP method, Optim. Meth. Software 4 (1994) 1–28.
W. Hock, K. Schittkowski, Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems 187, Springer, Berlin, 1981.
D.Q. Mayne, E. Polak, A superlinearly convergent algorithm for constrained optimization problems, Math. Programming Stud. 16 (1982) 45–61.
J.F.A. Pantoja, D.Q. Mayne, Exact penalty function algorithm with simple updating of the penalty parameter, J. Optim. Theory Appl. 69 (1991) 441–467.
W. Murray, J.P. Prieto, A sequential quadratic programming algorithm using an incomplete solution of the subproblem, SIAM J. Optim. 5 (1995) 590–640.
J. Nocedal, M. Overton, Projected Hessian updating algorithms for nonlinearly constrained optimization, SIAM J. Numer. Anal. 22 (1985) 821–850.
H.K. Overley, Structured Secant Updates for Nonlinear Constrained Optimization, Ph.D. Thesis, Rice University, Rice, Texas, 1991.
K. Schittkowski, The nonlinear programming method of Wilson, Han and Powell with an augmented Lagrangian type line search function I, II, Numer. Math. 38 (1981) 83–128.
K. Schittkowski, More test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems 282, Springer, Berlin, 1987.
P. Spellucci, Numerische Verfahren der nichtlinearen Optimierung, Birkhäuser, Basel, 1993.
P. Spellucci, donlp2: do nonlinear programming, code obtainable via anonymous ftp from netlib as netlib/opt/donlp2.
P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds.), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.
P. Spellucci, Sequential quadratic programming: Theory, implementation, problems, in: M.J. Beckmann, K.W. Gaede, K. Ritter, H. Schneeweiss (Eds.), Methods of Operations Research, vol. 53, Anton Hain, Meisenheim, 1985, pp. 183–213.
P. Spellucci, A new technique for inconsistent QP-problems in the SQP-method, Technical University at Darmstadt, Department of Mathematics, preprint 1561, Darmstadt (1993), to appear in Mathematical Methods of Operations Research, vol. 48 (1998).
G.W. Stewart, The effects of rounding error on an algorithm for downdating a Cholesky factorization, J.I.M.A. 23 (1979) 203–213.
Y. Xie, Reduced Hessian algorithm for solving large scale equality constrained optimization problems, Ph.D. Thesis, University of Colorado at Boulder, Boulder, Colorado, 1991.
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Spellucci, P. An SQP method for general nonlinear programs using only equality constrained subproblems. Mathematical Programming 82, 413–448 (1998). https://doi.org/10.1007/BF01580078
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DOI: https://doi.org/10.1007/BF01580078