Testing a class of methods for solving minimization problems with simple bounds on the variables

Conn, Andrew R.; Gould, Nicholas I. M.; Toint, Philippe L.

doi:10.1090/S0025-5718-1988-0929544-3

Testing a class of methods for solving minimization problems with simple bounds on the variables
HTML articles powered by AMS MathViewer

by Andrew R. Conn, Nicholas I. M. Gould and Philippe L. Toint PDF

Math. Comp. 50 (1988), 399-430 Request permission

Abstract:

We describe the results of a series of tests for a class of new methods of trust region type for solving the simple bound constrained minimization problem. The results are encouraging and lead us to believe that the methods will prove useful in solving large-scale problems.

References

Dimitri P. Bertsekas, Constrained optimization and Lagrange multiplier methods, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 690767
R. K. Brayton and J. Cullum, An algorithm for minimizing a differentiable function subject to box constraints and errors, J. Optim. Theory Appl. 29 (1979), no. 4, 521–558. MR 552105, DOI 10.1007/BF00934451
A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Global convergence of a class of trust region algorithms for optimization with simple bounds, SIAM J. Numer. Anal. 25 (1988), no. 2, 433–460. MR 933734, DOI 10.1137/0725029
E. E. Cragg and A. V. Levy, Study on a supermemory gradient method for the minimization of functions, J. Optim. Theory Appl. 4 (1969), 191–205. MR 248967, DOI 10.1007/BF00930579
J. Cullum and R. K. Brayton, Some remarks on the symmetric rank-one update, J. Optim. Theory Appl. 29 (1979), no. 4, 493–519. MR 552104, DOI 10.1007/BF00934450
Ron S. Dembo, Stanley C. Eisenstat, and Trond Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), no. 2, 400–408. MR 650059, DOI 10.1137/0719025

On the Minimization of Quadratic Functions Subject to Box Constraints

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
R. Fletcher, Practical methods of optimization. Vol. 1, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1980. Unconstrained optimization. MR 585160
R. Fletcher and M. P. Jackson, Minimization of a quadratic function of many variables subject only to lower and upper bounds, J. Inst. Math. Appl. 14 (1974), 159–174. MR 384156
U. M. García-Palomares, Superlinearly convergent algorithms for linearly constrained optimization problems, Nonlinear programming, 2 (Proc. Special Interest Group on Math. Programming Sympos., Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York, 1974, pp. 101–119. MR 0406522

Minimization Subject to Bounds on the Variables

Some Theoretical Properties of an Augmented Lagrangian Function

Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical optimization, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. MR 634376
A. Griewank and Ph. L. Toint, On the unconstrained optimization of partially separable functions, Nonlinear optimization, 1981 (Cambridge, 1981) NATO Conf. Ser. II: Systems Sci., Academic Press, London, 1982, pp. 301–312. MR 775354
A. Griewank and Ph. L. Toint, Partitioned variable metric updates for large structured optimization problems, Numer. Math. 39 (1982), no. 1, 119–137. MR 664541, DOI 10.1007/BF01399316
Willi Hock and Klaus Schittkowski, Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems, vol. 187, Springer-Verlag, Berlin-New York, 1981. MR 611512, DOI 10.1007/BF00934594
J. J. Moré, Recent developments in algorithms and software for trust region methods, Mathematical programming: the state of the art (Bonn, 1982) Springer, Berlin, 1983, pp. 258–287. MR 717404
Jorge J. Moré, Burton S. Garbow, and Kenneth E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software 7 (1981), no. 1, 17–41. MR 607350, DOI 10.1145/355934.355936
J. L. Nazareth, Analogues of Dixon’s and Powell’s theorems for unconstrained minimization with inexact line searches, SIAM J. Numer. Anal. 23 (1986), no. 1, 170–177. MR 821913, DOI 10.1137/0723012
Dianne Prost O’Leary, A generalized conjugate gradient algorithm for solving a class of quadratic programming problems, Linear Algebra Appl. 34 (1980), 371–399. MR 591439, DOI 10.1016/0024-3795(80)90173-1
M. J. D. Powell, A new algorithm for unconstrained optimization, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 31–65. MR 0272162
M. J. D. Powell, Some global convergence properties of a variable metric algorithm for minimization without exact line searches, Nonlinear programming (Proc. Sympos., New York, 1975) SIAM-AMS Proc., Vol. IX, Amer. Math. Soc., Providence, R.I., 1976, pp. 53–72. MR 0426428
M. J. D. Powell, How bad are the BFGS and DFP methods when the objective function is quadratic?, Math. Programming 34 (1986), no. 1, 34–47. MR 819873, DOI 10.1007/BF01582161
D. F. Shanno and Kang Hoh Phua, Matrix conditioning and nonlinear optimization, Math. Programming 14 (1978), no. 2, 149–160. MR 474819, DOI 10.1007/BF01588962

SIGNUM Newsletter

Trond Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal. 20 (1983), no. 3, 626–637. MR 701102, DOI 10.1137/0720042
Ph. L. Toint, Some numerical results using a sparse matrix updating formula in unconstrained optimization, Math. Comp. 32 (1978), no. 143, 839–851. MR 483452, DOI 10.1090/S0025-5718-1978-0483452-7

Sparse Matrices and Their Uses