Abstract
We consider sequential quadratic programming methods for solving constrained nonlinear programming problems. It is generally believed that these methods are sensitive to the accuracy by which partial derivatives are provided. One reason is that differences of gradients of the Lagrangian function are used for updating a quasi-Newton matrix, e.g., by the BFGS formula. The purpose of this paper is to show by numerical experimentation that the method can be stabilized substantially. The algorithm applies non-monotone line search and internal and external restarts in case of errors due to inaccurate derivatives while computing the search direction. Even in case of large random errors leading to partial derivatives with at most one correct digit, termination subject to an accuracy of 10−7 can be achieved in 90% of 306 problems of a standard test suite. On the other hand, the original version with monotone line search and without restarts solves only 30% of these problems under the same test environment. In addition, we show how initial and periodic scaled restarts improve the efficiency in situations with slow convergence.
Similar content being viewed by others
References
Armijo L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16, 1–3 (1966)
Boggs P.T., Tolle J.W.: Sequential quadratic programming. Acta Numer. 4, 1–51 (1995)
Bongartz I., Conn A.R., Gould N., Toint Ph.: CUTE: Constrained and unconstrained testing environment. Trans. Math. Softw. 21(1), 123–160 (1995)
Dai Y.H.: On the nonmonotone line search. J. Optim. Theory Appl. 112(2), 315–330 (2002)
Dai Y.H., Schittkowski K.: A sequential quadratic programming algorithm with non-monotone line search. Pac. J. Optim. 4, 335–351 (2008)
Gill P.E., Leonard M.W.: Limited-memory reduced-Hessian methods for unconstrained optimization. SIAM J. Optim. 14, 380–401 (2003)
Goldfarb D., Idnani A.: A numerically stable method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)
Grippo L., Lampariello F., Lucidi S.: A nonmonotone line search technique for Newtons’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)
Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 187. Springer (1981)
Liu D.C., Nocedal J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45, 503–528 (1989)
Maurer H., Mittelmann H.: Optimization techniques for solving elliptic control problems with control and state constraints: Part 1. Boundary control. Comput. Optim. Appl. 16, 29–55 (2000)
Maurer H., Mittelmann H.: Optimization techniques for solving elliptic control problems with control and state constraints: Part 2: Distributed control. Comput. Optim. Appl. 18, 141–160 (2001)
Nocedal J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)
Powell, M.J.D.: A fast algorithm for nonlinearly constraint optimization calculations. In: Watson, G.A. Numerical Analysis, Lecture Notes in Mathematics, vol. 630, Springer (1978)
Powell M.J.D.: The convergence of variable metric methods for nonlinearly constrained optimization calculations. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds) Nonlinear Programming 3, Academic Press, New York (1978)
Rockafellar T.: Augmented Lagrangian multiplier functions and duality in nonconvex programming. SIAM J. Control 12, 268–285 (1974)
Sachsenberg, B.: NLPIP: A Forrtan implementation of an SQP Interior Point algorithm for solving large scale nonlinear optimization problems: user’s guide, Report, Department of Computer Science, University of Bayreuth (2010)
Schittkowski, K.: Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, vol. 183. Springer (1980)
Schittkowski K.: On the convergence of a sequential quadratic programming method with an augmented Lagrangian search direction. Optimization 14, 197–216 (1983)
Schittkowski K.: NLPQL: A Fortran subroutine solving constrained nonlinear programming problems. Ann Oper Res 5, 485–500 (1985/1986)
Schittkowski K.: More Test Examples for Nonlinear Programming. Lecture Notes in Economics and Mathematical Systems, vol. 182. Springer, Berlin (1987)
Schittkowski K.: Solving nonlinear least squares problems by a general purpose SQP-method. In: Hoffmann, K.-H., Hiriart-Urruty, J.-B., Lemarechal, C., Zowe, J. (eds) Trends in Mathematical Optimization. International Series of Numerical Mathematics, vol. 84, pp. 295–309. Birkhäuser, Basel (1988)
Schittkowski K.: Numerical Data Fitting in Dynamical Systems. Kluwer Academic Publishers, Dordrecht (2002)
Schittkowski, K.: QL: A Fortran code for convex quadratic programming—user’s guide, Report, Department of Mathematics, University of Bayreuth (2003)
Schittkowski K.: An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints. Appl. Numer. Math. 59, 2999–3007 (2008)
Schittkowski, K.: An updated set of 306 test problems for nonlinear programming with validated optimal solutions: user’s guide, Report, Department of Computer Science, University of Bayreuth (2008)
Schittkowski, K.: NLPQLP: A Fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search—user’s guide, version 3.0, Report, Department of Computer Science, University of Bayreuth (2009)
Schittkowski K., Zillober C., Zotemantel R.: Numerical comparison of nonlinear programming algorithms for structural optimization. Struct. Optim. 7(1), 1–28 (1994)
Spellucci P.: DONLP2 users guide, Technical University at Darmstadt, Department of Mathematics, Darmstadt, Germany
Stoer J.: Foundations of recursive quadratic programming methods for solving nonlinear programs. In: Schittkowski, K. (eds) Computational Mathematical Programming, NATO ASI Series. Series F. Computer and Systems Sciences. vol. 15, Springer, Berlin (1985)
Toint P.L.: An assessment of nonmontone line search techniques for unconstrained optimization. SIAM J. Sci. Comput. 17, 725–739 (1996)
Toint P.L.: A nonmonotone trust-region algorithm for nonlinear optimization subject to convex constraints. Math. Program. 77, 69–94 (1997)
Wolfe P.: Convergence conditions for ascent methods. SIAM Rev. 11, 226–235 (1969)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schittkowski, K. A robust implementation of a sequential quadratic programming algorithm with successive error restoration. Optim Lett 5, 283–296 (2011). https://doi.org/10.1007/s11590-010-0207-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-010-0207-9