Robert B. Schnabel
Abstract
We describe a new package, UNCMIN, for finding a local minimizer of a real valued function of more than one variable. The novel feature of UNCMIN is that it is a modular system of algorithms, containing three different step selection strategies (line search, dogleg, and optimal step) that may be combined with either analytic or finite difference gradient evaluation and with either analytic, finite difference, or BFGS Hessian approximation. We present the results of a comparison of the three step selection strategies on the problems in More, Garbow, and Hillstrom in two separate cases: using finite difference gradients and Hessians, and using finite difference gradients with BFGS Hessian approximations. We also describe a second package, REVMIN, that uses optimization algorithms identical to UNCMIN but obtains values of user-supplied functions by reverse communication.
- 1 DENNIS, J. E., JR., GAY, D. M., AND WELSCH, R. E. An adaptive nonlinear least-square algorithm. ACM Trans. Math. Softw. 7 (1981), 348-368. Google Scholar
- 2 DENNIS, J. E., JR., AND MEI, H. H.W. Two new unconstrained optimization algorithms which use function and gradient values. J. Optimization Theory and Its Applications 28 (1979), 453-482.Google Scholar
- 3 DENNIS, J. E., JR., AND SCHNABEL, R. B. Numerical Methods for Nonlinear Equations and Unconstrained Optimization. Prentice-Hall, Englewood Cliffs, N. J., 1983.Google Scholar
- 4 FLETCHER, R. Practical Methods of Optimization, Vol. 1. Unconstrained Optimization. Wiley, New York, 1980. Google Scholar
- 5 GAY, D.M. Some tips for writing portable software. Tech. Rept. TR-14, Center for Computational Research in Economics and Management Science, Massachusetts Institute of Technology, Cambridge, Mass., 1980.Google Scholar
- 6 GAY, D.M. Computing optimal locally constrained steps. SIAM J. Sci. Stat. Comput. 2, (1981), 186-197.Google Scholar
- 7 GAY, D.M. Subroutines for unconstrained minimization using a model/trust-region approach. ACM Trans. Math. Softw. 9 (1983), 503-524. Google Scholar
- 8 GILL, P. E., AND MURRAY, W. Newton-type methods for unconstrained and linearly constrained optimization. Math. Program. 28 (1974), 311-350.Google Scholar
- 9 GILL, P. E., MURRAY, W., PICKEN, S. M., AND WRIGHT, M.H. The design and structure of a Fortran program library for optimization. ACM Trans. Math. Softw. 5 (1979), 259-283. Google Scholar
- 10 GILL, P. E., MURRAY, W., AND WRIGHT, M.H. Praztical Optimization. Academic Press, New York, 1981.Google Scholar
- 11 GOLDFARB, D. Factorized variable metric methods for unconstrained optimization. Math. Comput. 30 (1976), 796-811.Google Scholar
- 12 HAMMING, R.W. Introduction to Applied Numerical Analysis. McGraw-Hill, New York, 1971. Google Scholar
- 13 Harwell Subroutine Library, A Catalogue of Subroutines (M. J. Hopper, Ed.), Computer Science and Systems Division, A.E.R.E. Harwell, Oxon., England.Google Scholar
- 14 IMSL Library Reference Manual, International Mathematical and Statistical Libraries, Houston, Tex.Google Scholar
- 15 KROGH, F. T. VODQ/SVDQ/DVDQwVariable order integrators for numerical solution of ordinary differential equations. Subroutine Write-Up, Section 314, Jet Propulsion Laboratory, Pasadena, Calif.Google Scholar
- 16 LAwsoN, C., BLOCK, N., AND GARRETT, R. Fortran IV subroutines for contour plotting. Technical Memo 106, Section 314, Jet Propulsion Laboratory, Pasadena, Calif.Google Scholar
- 17 MINPACK Documentation, Applied Mathematics Division, Argonne National Laboratory, Argonne, Ill.Google Scholar
- 18 MOR~, J.J. The Levenberg-Marquardt algorithm: Implementation and theory. In G. A. Watson, Numerical Analysis, Dundee 1977, Lecture Notes in Mathematics 630. Ed., Springer-Verlag, Berlin, pp. 105-116.Google Scholar
- 19 MOR~, J. J. On the design of optimization software. In S. Incerti and G. Treccani, Eds., Otimazzazione Nonlineare e Applicazioni. Pitagora Editrice, Bologna, Italy, 1980.Google Scholar
- 20 MOR~:, J.J. Notes on optimization software. In M. J. D. Powell, Ed., Nonlinear Optimization 1981, Academic Press, New York, 1982, pp. 339-352.Google Scholar
- 21 MOR~, J. J., GARBOW, B. S., AND HILLSTROM, Ko E. Testing unconstrained optimization software. ACM Trans. Math. Softw. 7 (1981), 17-41. Google Scholar
- 22 MOR~:, J. J., AND SORENSEN, D.C. Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, (1983), 553-572.Google Scholar
- 23 NAG Fortran Library Manual, The Numerical Algorithms Group (USA), Downers Grove, Ill.Google Scholar
- 24 OSTERWEIL, L. J., AND FOSDICK, L.D. DAVE--A validation error detection and documentation system for Fortran programs. Softw. Pract. Exp. 6, (1976), 473-486.Google Scholar
- 25 RYDER, B.G. The PFORT verifier. Softw. Pract. Exp. 4, (1974), 359-377.Google Scholar
- 26 SHANNO, D. F., AND PHUA, K.H. Minimization of unconstrained multivariate functions. ACM Trans. Math. Softw. 2, {1976), 87-94. Google Scholar
- 27 SHULTZ, G. A., SCHNABEL, R. B., AND BYRD, R.H. A family of trust region based algorithms for unconstrained minimization with strong global convergence properties. SIAM J. Numer. Anal. 22, (1985), 47-67.Google Scholar
- 28 SORENSEN, D.C. Newton's method with a model trust region modification. SIAM Jo Numer. Anal. 19, {1982), 409-426.Google Scholar
- 29 WAGENER, J.L. Status of work towards revision of programming language Fortran. SIGNUM Newsletter 19, 3 (July 1984). Google Scholar
- 30 WEISS, B.E. A modular software package for solving unconstrained nonlinear optimization problems. M. S. thesis, Department of Computer Science, University of Colorado at Boulder, 1980.Google Scholar
Index Terms
- A modular system of algorithms for unconstrained minimization
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Reviews
Henry W. Mosteller
This paper describes UNCMIN, a modular system of FORTRAN subroutines for solving :7S x f( x), :9F:Y where x is an n-dimensional vector and f is a scalar function of x. The first and second derivatives of f should be continuous. This is an important restriction. Many real problems have discontinuous function values as well as first and second derivatives. A nice feature of the package is its modular approach. It offers the following options: (1)Three different step selection methods—line search, dogleg, and hookstep. (2)Analytic or finite difference gradient evaluation. (3)Analytic, finite difference, or BFGS Hessian calculation. The Newton minimization algorithm is used to find the minimum. A large number of standard test functions were minimized using various choices from the above options. Some conclusions are: the choice of step selection method does not change the results much; and the BFGS Hessian approximation gives a smaller number of function evaluations compared to the finite difference Hessian. The ideas in this paper are explained in more detail in [1]. The reader who is interested in this approach to unconstrained minimization should read this book.
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