Parametric Linear Programming: Some Extensions*

Abstract

AbstractThis paper considers a special version of the general problem of imputing regions in the space of parameters to optimal solutions and/or to optimal values Z* of the problem: Optimize the scalar-valued function subject to the constraints Inthe special version, the constraints are linear arid the objective function is i.e., a “mixture” of linear objective functions with undetermined “weights” λi. It is shown that the set of values of λi which allow finite optima, as well as those subsets of it for which basic feasible solutions are also optimal feasible, are convex polyhedra. The set of finite optima associated with each feasible basis is convex. Even when the values of some of the parameters are held fixed, the set of those values of the others whichmaintain a given optimum is a convex polyhedron. Furthermore, a convex combination of basic feasible solutions produces the same combination of optima only for those (vectors of) parameter values which are common to the polyhedra in the parameter space....