Abstract
Methods related to Wolfe's recursive method for the resolution of degeneracy in linear programming are discussed, and a nonrecursive variant which works with probability one suggested. Numerical results for both nondegenerate problems and problems constructed to have degenerate optima are reported. These are obtained using a careful implementation of the projected gradient algorithm [11]. These are compared with results obtained using a steepest descent approach which can be implemented by means of a closely related projected gradient method, and which is not affected by degeneracy in principle. However, the problem of correctly identifying degenerate active sets occurs with both algorithms. The numerical results favour the more standard projected gradient procedure which resolves the degeneracy explicitly. Extension of both methods to general polyhedral convex function minimization problems is sketched.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.W. Bassett and R.W. Koenker, An empirical quantile function for linear models with iid errors, J. Am. Stat. Assoc. 77 (1982) 407–415.
D.I. Clark and M.R. Osborne, On linear restricted and interval least squares problems, IMA J. Numer. Anal. 8 (1988) 23–36.
A. Dax, Linear programming via least squares, Lin. Alg. Appl. 111 (1988) 313–324.
A. Dax, Thel 1 solution of linear equations subject to linear constraints, SISSC 10 (1989) 328–340.
R. Fletcher, Degeneracy in the presence of roundoff errors, Lin. Alg. Appl. 106 (1988) 149–183.
R. Fletcher, Resolving degeneracy in quadratic programming, Report NA/135, Mathematics Department, University of Dundee (1991).
R. Fourer, A simplex algorithm for piecewise linear programming, Math. Progr. 33 (1985) 204–233.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, A practical anti-cycling procedure for linearly constrained optimization, Math. Progr. 45 (1989) 437–474.
M.J. Hopper and M.J.D. Powell, A technique that gains speed and accuracy in the minimax solution of over-determined linear equations, in:Mathematical Software III, ed. J.R. Rice (Academic Press, 1977) pp. 15–34.
J.A. Krommes, The WEB system of structured software design and documentation for Fortran, Ratfor, and C, Technical Report, Princeton University; available by anonymous ftp from ss01.ppp1.gov in directory/pub/fweb (1989).
M.R. Osborne,Finite Algorithms in Optimization and Data Analysis (Wiley, 1985).
M.R. Osborne, The reduced gradient algorithm, in:Statistical Data Analysis, ed. Y. Dodge (North-Holland, 1987) pp. 95–108.
R.T. Rockafellar,Network Flows and Monodromic Optimization (Wiley, 1984).
D.M. Ryan and M.R. Osborne, On the solution of highly degenerate linear programmes, Math. Progr. 41 (1988) 385–392.
P. Wolfe, A technique for resolving degeneracy in linear programming, SIAM J. 11 (1963) 205–211.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
George, K., Osborne, M.R. On degeneracy in linear programming and related problems. Ann Oper Res 46, 343–359 (1993). https://doi.org/10.1007/BF02023104
Issue Date:
DOI: https://doi.org/10.1007/BF02023104