Abstract
Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the simple technique of Pantoja and Mayne [36] and the highly successful, but expensive one of Tone [47]. Global convergence of a corresponding algorithm is shown under reasonable weak conditions. Numerical results are reported which show that this technique, combined with a special method for the case of regular subproblems, is quite competitive to highly appreciated established ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burke JV (1989) A sequential quadratic programming algorithm for potentially infeasible mathematical programs. J. Math. Anal. Applies 139:319–351
Burke JV (1992) A robust trust region algorithm for constrained nonlinear programming problems. SIOPT 2:325–347
Burke JV (1991) An exact penalization viewpoint of constrained optimization. SIAM J. Control and Optimization 29:968–998
Burke JV, Han SP (1989) A robust sequential quadratic programming method. Math. Prog. 43:277–303
Byrd RH, Tapia RA, Zhang Y (1992) An SQP augmented Lagrangian BFGS algorithm for constrained optimization. SIOPT 2:210–241
Byrd RH, Nocedal J, Schnabel B (1994) Representations of quasi-Newton-matrices and their use in limited memory methods. Math. Prog. 63:129–156
Chamberlain RM (1979) Some examples of cycling in variable metric methods for constrained minimization. Math. Prog. 16:378–383
Chamberlain RM, Powell MDJ, Lemarechal C, Petersen HC (1982) The watchdog technique for forcing convergence in algorithms for constrained optimization. Math. Prog. Study 16:1–17
Clarke FH (1987) Optimization and nonsmooth analysis, John Wiley, New York
Conn AR, Sinclair JW Quadratic programming via a non-differentiable penalty function. Univ. of Waterloo Dept. of Combinatorics and Optimization report CORR 75/15
Daniel JW (1973) Stability of definite quadratic programs. Math. Prog. 5:41–53
Dembo RS (1976) A set of geometric programming test problems and their solutions. Math. Prog. 10:192–213
El-Alem MM (1995) Global convergence without the assumption of linear independence for a trust-region algorithm for constrained optimization. J.O.T.A. 87:563–577
Felkel R (1996) An interior point method for large scale QP problems. THD Mathematics Department Report 1850
Fletcher R (1982) A model algorithm for composite nondifferentiable optimization problems. Math. Prog. 17:67–76
Fletcher R (1987) Practical methods of optimization. 2nd ed, Wiley, Chicester-New York
Fletcher R (1981) Second order correction for nondifferentiable optimization. 85-114 in Lect. Notes on Math. 912, Springer, Berlin Heidelberg New York
Fletcher R (1984) Al 1 penalty method for nonlinear constraints. In: Boggs PT, Byrd RH, Schnabel RB (eds.) Numerical optimization, SIAM Publications, Philadelphia
Friedlander A, Martinez JM, Santos SA (1994) On the resolution of linearly constrained convex minimization problems. SIOPT 4:331–339
Gill PhE, Murray W (1978) Numerically stable methods for quadratic programming. Math. Prog. 14:349–372
Gill PhE, Murray W, Wright MH (1980) Practical methods of optimization. Acad. Press, New York
Gill PhE, Murray W, Saunders M, Wright MH (1992) Some theoretical properties of an augmented Lagrangian merit function. In: Pardalos PM (ed.) Advances in optimization and parallel computing, North Holland, Amsterdam, pp. 101–128
Gill PhE, Hammarling S, Murray W, Saunders M, Wright, MH (1986) Users guide for NPSOL (Version 4.0). Dept. O.R. Stanford Univ. Rep. SOL 86-2
Goldfarb D, Idnani A (1983) A numerically stable dual method for solving strictly convex quadratic programs. Math. Prog. 27:1–33
Gould NIM (1991) An algorithm for large scale quadratic programming. I.M.A.J. Numer. Anal. 11:299–323
Himmelblau DM (1972) Applied nonlinear programming. McGraw-Hill, New York
Hock W, Schittkowski K (1981) Test examples for nonlinear programming codes. Lect. Not. in Econ. and Math. Syst. 187, Springer, Berlin Heidelberg New York
Heinz J, Spellucci P (1994) A successful implementation of the Pantoja-Mayne SQP- method. Optimization Methods and Software 4:1–28
Kanzow C (1994) An unconstrained optimization technique for large-scale linearly constrained convex minimization problems. Computing 53:101–117
Lucidi S (1988) New results on a class of exact augmented Lagrangians. J.O.T.A. 58:259–282
Luenberger DG (1984) Introduction to linear and nonlinear programming. 2nd ed, Addison-Wesley, Menlo Park
Mayne DQ, Sahba M (1985) An efficient algorithm for solving inequalities. J.O.T.A. 45:407–423
Murty KG (1988) Linear complementarity, linear and nonlinear programming. Heldermann, Berlin
Nocedal J, Overton M (1985) Projected Hessian updating alogorithms for nonlinearly constrained optimization. SINUM 22:821–850
Nowak I (1988) Ein quadratisches Optimierungsproblem mit Schlupfvariablen für die SQP- Methode zur Lösung des allgemeinen nichtlinearen Optimierungsproblems. Diplomarbeit, TH Dannstadt
Mayne DQ, Pantoja JFA (1991) Exact penalty function algorithm with simple updating of the penalty parameter J.O.T.A. 69:441–467
Panier ER, Tits AL (1991) Avoiding the Maratos effect by means of a nonmonotone line search. I. general constrained problems. SINUM 28:1183–1195
Powell MJD (1978) A fast algorithm for nonlinearly constrained optimization calculations. Lecture Notes on Mathematics 630, Springer, Berlin Heidelberg New York, pp. 144–157
Sahba M (1987) Globally convergent algorithm for nonlinearly constrained optimization problems. J.O.T.A. 52:291–309
Schittkowski K (1981) The nonlinear programming method of Wilson, Han and Powell with an augmented Lagrangian type line search function II. An efficient implementation using linear least squares sub-problems. Num. Math. 38:115–127
Schittkowski K (1983) On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function. Optimization 14:197–216
Schittkowski K (1987) More testexamples for nonlinear programming codes. Lect. Not. in Econ. and Math. Syst. 282, Springer, Berlin Heidelberg New York
Spellucci P (1993) Numerische Verfahren der nichtlinearen Optimierung. Birkhäuser, Basel
Spellucci P (1993) A SQP method for general nonlinear programs using only equality constrained sub-problems. THD FB4 Preprint 1562, Submitted for publication
Spellucci P (1985) Sequential quadratic programming: Theory, implementation, problems. In: Beckmann MJ, Gaede KW, Ritter K, Schneeweiss H (eds.) Methods of Operation Research 53, Anton Hain, Meisenheim, pp. 183–213
Spellucci P donlp2. Program and users guide available as donlp2.shar from netlib/opt/ donlp2
Tone K (1983) Revision of constraint approximations in the successive QP-method for nonlinear programming problems. Math. Prog. 26:144–152
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Spellucci, P. A new technique for inconsistent QP problems in the SQP method. Mathematical Methods of Operations Research 47, 355–400 (1998). https://doi.org/10.1007/BF01198402
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01198402