Numerical decomposition of a convex function
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Given then×p orthogonal matrixA and the convex functionf:Rn→R, we find two orthogonal matricesP andQ such thatf is almost constant on the convex hull of ± the columns ofP, f is sufficiently nonconstant on the column space ofQ, and the column spaces ofP andQ provide an orthogonal direct sum decomposition of the column space ofA. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a pointx, of a convex function. Applications to convex programming are discussed.
Key WordsConvex functions convex programming cone of directions of constancy numerical stability
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- 1.Ben-Israel, A., Ben-Tal, A., andZlobec, S.,Optimality in Nonlinear Programming: A Feasible Directions Approach, Wiley, New York, New York, 1981.Google Scholar
- 2.Wolkowicz, H.,Geometry of Optimality Conditions and Constraint Qualifications: The Convex Case, Mathematical Programming, Vol. 19, pp. 32–60, 1980.Google Scholar
- 3.Wolkowicz, H.,The Method of Reduction in Convex Programming, Journal of Optimization Theory and Applications, Vol. 40, pp. 349–378, 1983.Google Scholar
- 4.Wolkowicz, H.,Calculating the Cone of Directions of Constancy, Journal of Optimization Theory and Applications, Vol. 25, pp. 451–457, 1978.Google Scholar
- 5.Baart, M. L.,The Use of Autocorrelation for Pseudo-Rank Determination in Noisy Ill-Conditioned Linear Least-Squares Problems, IMA Journal of Numerical Analysis, Vol. 2, pp. 241–247, 1982.Google Scholar
- 6.Golub, G., Klema, V., andStewart, G. W.,Rank Degeneracy and Least Squares Problems, Stanford University, Research Report, 1976.Google Scholar
- 7.Rockafellar, R. T.,Some Convex Programs Where Duals Are Linearly Constrained, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, pp. 293–322, 1970.Google Scholar
- 8.Coleman, T. F., andSorensen, D. C.,A Note on the Computation of an Orthonormal Basis for the Null Space of a Matrix, Cornell University, Department of Computer Science, Technical Report No. 82–510, 1982.Google Scholar
- 9.Parlett, B. N.,Analysis of Algorithms for Reflections in Bisectors, SIAM Review, Vol. 13, pp. 197–208, 1971.Google Scholar
- 10.Parlett, B. N.,The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey, 1980.Google Scholar
- 11.Wilkinson, J. H.,Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, New Jersey, 1963.Google Scholar