By P.L. Hammer, E.L. Johnson and B.H. Korte (Eds.)

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**Extra resources for Discrete Optimization II, Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium co-sponsored by IBM Canada and SIAM Banff, Aha. and **

**Example text**

C. D. 3) with cija 0 for all i, 1= 2,3, . . , n - 1 may be minimized with a network flow algorithm applied to a network with n vertices. Capacities cb = cij are given to the arcs joining intermediary vertices, capacities c:j= max (Zy:; cb- cjj, 0) to the arcs from the source to the intermediary vertices and capacities c;,, = max (cjj-Cy=;' c:~,0) to the arcs from the intermediary vertices to the sink; if C ~ S c ~ ;~ ~an ,arc ~ joining the source to the sink, with capacity c;, = C ~ - C ; = ctj ~ is added; otherwise C;=l ctj-c0 must be subtracted from the value of the maximum flow.

Borwein, A strong duality theory for the minimum of a family of convex programs, Dalhousie University, Halifax (June 1978). A. Burdet, Enumerative inequalities in integer programming, Math. Programming 2 (1972) 32-64. A. Burdet, Polaroids: A new tool in nonconvex and integer programming, Naval Res. Logist. Quart. 20 (1973) 13-24. R. Fulkerson, Blocking Polyhedra, in: B. , Graph Theory and Its Applications (Academic Press, New York, 1972) 93-112. R. Fulkerson, Anti-blocking polyhedra, J. Combinatorial Theory 12 (1972) 50-71.

Balm 28 basic variables, there are also integrality constraints on some of the nonbasic variables. In [14] we first proved this principle for arbitrary cuts, by using subadditive functions, then applied it to cuts from disjunctions. Here we prove the principle directly for the latter case, without recourse to concepts outside the framework of disjunctive programming. 1), and assume in addition that some components of x are integer-constrained. In order for the principle that we are going to discuss to be applicable, it is necessary for each Ahx, h E Q, to have a lower bound, say b,h.