Abstract
An iterative linear programming algorithm for the solution of the convex programming problem is proposed. The algorithm partially solves a sequence of linear programming subproblems whose solution is shown to converge quadratically, superlinearly, or linearly to the solution of the convex program, depending on the accuracy to which the subproblems are solved. The given algorithm is related to inexact Newton methods for the nonlinear complementarity problem. Preliminary results for an implementation of the algorithm are given.
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References
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill, New York, New York, 1969.
Shiau, T. H.,Iterative Linear Programming for Linear Complementarity and Related Problems, University of Wisconsin, Madison, Wisconsin, Computer Sciences Department, Technical Report No. 507, 1983.
Pang, J. S.,Inexact Newton Methods for the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 36, pp. 54–71, 1986.
Josephy, N. H.,Newton's Method for Generalized Equations, University of Wisconsin, Madison, Wisconsin, Mathematics Research Center, Technical Summary Report No. 1965, 1979.
Josephy, N. H.,Quasi-Newton Methods for Generalized Equations, University of Wisconsin, Madison, Wisconsin, Mathematics Research Center, Technical Summary Report No. 1966, 1979.
Robinson, S. M.,Strongly Regular Generalized Equations, Mathematics of Operations Research, Vol. 5, pp. 43–62, 1980.
Garcia Palomares, U. M., andMangasarian, O. L.,Superlinearly Convergent Quasi-Newton Algorithms for Nonlinearly Constrained Optimization Problems, Mathematical Programming, Vol. 11, pp. 1–13, 1976.
Han, S. P.,Superlinearly Convergent Variable-Metric Algorithms for General Nonlinear Programming Problems, Mathematical Programming, Vol. 11, pp. 263–282, 1976.
Powell, M. J. D.,The Convergence of Variable-Metric Methods for Nonlinearly Constrained Optimization Calculations, Nonlinear Programming, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, London, England, Vol. 3, pp. 27–63, 1978.
Rosenbrock, H. H.,Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal, Vol. 3, pp. 175–184, 1960.
Himmelblau, D. M.,Applied Nonlinear Programming, McGraw-Hill, New York, New York, 1972.
Murtagh, B. A., andSaunders, M. A.,Minos 5.0 User's Guide, Stanford University, Technical Report No. SOL-83.20, 1983.
Colville, A. E.,A Comparative Study on Nonlinear Programming Codes, IBM New York Scientific Center, Technical Report No. 320–2949, 1968.
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Communicated by O. L. Mangasarian
This material is based on research supported by the National Science Foundation, Grants DCR-8521228 and CCR-8723091, and by the Air Force Office of Scientific Research, Grant AFOSR-86-0172. The author would like to thank Professor O. L. Mangasarian for stimulating discussions during the preparation of this paper.
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Ferris, M.C. Iterative linear programming solution of convex programs. J Optim Theory Appl 65, 53–65 (1990). https://doi.org/10.1007/BF00941159
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DOI: https://doi.org/10.1007/BF00941159