Cutting-Plane Methods without Nested Constraint Sets

This paper gives general conditions for the convergence of a class of cutting-plane algorithms without requiring that the constraint sets for the sub-problems be sequentially nested. Conditions are given under which inactive constraints may be dropped after each subproblem. Procedures for generating cutting-planes include those of Kelley, Cheney and Goldstein, and a generalization of the one used by both Zoutendijk and Veinott. For algorithms with nested constraint sets, these conditions reduce to a special case of those of Zangwill for such problems and include as special cases the algorithms of Kelley, Cheney and Goldstein, and Veinott. Finally, the paper gives an arithmetic convergence rate.