An iterative row-action method for interval convex programming
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
The iterative primal-dual method of Bregman for solving linearly constrained convex programming problems, which utilizes nonorthogonal projections onto hyperplanes, is represented in a compact form, and a complete proof of convergence is given for an almost cyclic control of the method. Based on this, a new algorithm for solving interval convex programming problems, i.e., problems of the form minf(x), subject to γ≤Ax≤δ, is proposed. For a certain family of functionsf(x), which includes the norm ∥x∥ and thex logx entropy function, convergence is proved. The present row-action method is particularly suitable for handling problems in which the matrixA is large (or huge) and sparse.
- Robers, P. D., andBen-Israel, A.,An Interval Programming Algorithm for Discrete Linear L 1-Approximation Problems, Journal of Approximation Theory, Vol. 2, pp. 323–336, 1969.
- Robers, P. D., andBen-Israel, A.,A Suboptimal Method for Interval Linear Programming, Linear Algebra and Its Applications, Vol. 3, pp. 383–405, 1970.
- Herman, G. T., andLent, A.,A Family of Iterative Quadratic Optimization Algorithms for Pairs of Inequalities, with Application in Diagnostic Radiology, Mathematical Programming Study, Vol. 9, pp. 15–29, 1978.
- Herman, G. T., andLent, A.,Iterative Reconstruction Algorithms, Computers in Biology and Medicine, Vol. 6, pp. 273–294, 1976.
- Herman, G. T., Lent, A., andLutz, P. H.,Relaxation Methods For Image Reconstruction, Communications of the Association for Computing Machinery, Vol. 21, pp. 152–158, 1978.
- Bregman, L. M.,The Relaxation Method of Finding the Common Point of Convex Sets and Its Application to the Solution of Problems in Convex Programming, USSR Computational Mathematics and Mathematical Physics, Vol. 7, pp. 200–217, 1967.
- Hildreth, C.,A Quadratic Programming Procedure, Naval Research Logistics Quarterly, Vol. 4, pp. 79–85, 1975; see alsoErratum, Naval Research Logistic Quarterly, Vol. 4, p. 361, 1975.
- Lent, A., andCensor, Y.,Extensions of Hildreth's Row-Action Method for Quadratic Programming, SIAM Journal on Control and Optimization, Vol. 18, pp. 444–454, 1980.
- D'Esopo, D. A.,A Convex Programming Procedure, Naval Research Logistics Quarterly, Vol. 6, pp. 33–42, 1959.
- Censor, Y., andHerman, G. T.,Row-Generation Methods for Feasibility and Optimization Problems Involving Sparse Matrices and Their Application, Sparse Matrix Proceedings-1978, Edited by I. S. Duff and G. W. Stewart, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, pp. 197–219, 1979.
- Censor, Y.,Row-Action Methods for Huge and Sparse Systems and Their Applications, SIAM Review, to appear.
- Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
- Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, Berlin, Germany, 1970.
- Ponstein, J.,Seven Kinds of Convexity, SIAM Review, Vol. 9, pp. 115–119, 1967.
- Ben-Israel, A.,Linear Equations and Inequalities on Finite Dimensional, Real or Complex, Vector Spaces: A Unified Theory, Journal of Mathematical Analysis and Applications, Vol. 27, pp. 367–389, 1969.
- Lent, A.,A Convergent Algorithm for Maximum Entropy Image Restoration, with a Medical X-Ray Application, Image Analysis and Evaluation, Edited by R. Shaw, Society of Photographic Scientists and Engineers, Washington, DC, pp. 249–257, 1977.
- Daniel, J. W.,The Approximate Minimization of Functionals, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
- Censor, Y., Lakshminarayanan, A. V., andLent, A.,Relaxational Methods for Large-Scale Entropy Optimization Problems, with Application in Image Reconstruction, Information Linkage Between Applied Mathematics and Industry, Edited by P. C. C. Wang, Academic Press, New York, New York, pp. 539–546, 1979.
- An iterative row-action method for interval convex programming
Journal of Optimization Theory and Applications
Volume 34, Issue 3 , pp 321-353
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Interval convex programming
- entropy optimization
- large and sparse matrices
- nonorthogonal projections
- image reconstruction from projections
- Industry Sectors