Abstract
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.
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Communicated by A. V. Fiacco
The research for this paper was supported by NIH Grants Nos. HL-04664, HL-18968, and RR-7, and by NCI Contract No. CB-84235. Partial support for the second author was provided by a grant from Waters Instruments, Rochester, Minnesota. While preparing the manuscript for this paper, the authors benefited from an extraordinary referee's report on Ref. 3 written by Professor J. L. Goffin, McGill University, Montreal, Canada. The authors thank Ms. B. Dane for typing the manuscript and preparing the drawings.
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Censor, Y., Lent, A. An iterative row-action method for interval convex programming. J Optim Theory Appl 34, 321–353 (1981). https://doi.org/10.1007/BF00934676
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DOI: https://doi.org/10.1007/BF00934676