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Proof : The Assumption guarantees that the optimal value of P is bounded. Then. the
boundedness of each subproblem is proven as follows: by the Assumption. the non-artificial
variables x, and y, are bounded, and the artificial variables. vr, cannot cause unboundedness
because the artificial variables have nonnegativity and large negahve objective coefficients. Also.
- 1 k-1the
A,:-' variables are bounded because of nonnegativity and the constra.int e Ait -1. The 8,
variables in upper bound subproblerns are also bounded by the similar argument to the proof of
Theorem 3.2b in the previous chapter. Therefore. the optimal value of each subproblem is
bounded.
Theorem 4.8 states chat Step O of the algorithm accurately detects the feasibility of the
whole problem P.
Theorem 48 Problern P is infeasibie ifand onfy if Step O of tlze algonthm reports infeasibility
of P.
Prwf : (The "iF' pan) if a subproblern is found to be infeasible at step O. then at least one
nonlinhng constraint and upper bound consvaint for a subproblem is infeasible because the
linking consuaints cm aiways be satisfied for some choice of the mificial variables. Then,
infeasibility of the nonlinking constraint and upper bound constra.int implies that P is infeasible.
(The "only if' put) if problem P is infesible, then at leut one part's set of nonlinking
constnints and upper bound constraints is infeasible because the linking consvaints cm'? be
viohted. This infesibility will be detected at step O.