Nonlinear One-Parametric Bottleneck Linear Programming

Schubert, I. S. and Zimmermann, Uwe (1985) Nonlinear One-Parametric Bottleneck Linear Programming.
Published in: Zeitschrift für Operations-Research : ZOR ; mathematical methods of operations research Vol. 29 (5). pp. 187-201.


We consider the one-parametric linear bottleneck problem min{c(x,t)| x in P(t)} where the bottleneck objective c(x,t):= max{cj(t) | xj>0} is minimized subject to linear constraints, i.e. P(t):={x| A(t)x=b(t), x>=0}. All coefficients are assumed to be continuous functions of one real parameter t which varies in a real interval S. A method is developed for constructing a partition of S into subintervals on which either a basis stays optimal or the problem stays infeasible. Finiteness of the partition is due to certain finiteness assumptions on the zeros of particular combinations of the coefficient functions. Using a lexicographic refinement of the objective function a characterization of the optimality interval of a fixed basis is derived which is independent on explicit information about other bases.

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