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Roof : The Assumption guarantees that the optimal vaiue 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, v,, cannot cause unboundedness
because the artificial variables are nonnegative aid have large negative objective coefficients.
Also. the A variables are bounded because of nonnegativity and the sum to one constraint. The
8 variables are also bounded by an argument to similar to chat in the proof of Theomm 3.2b in
the previous chapter. Therefore. the optimai vaiue of each subproblem is bounded.
Theorem 4.2 States that Step O of the algorithm in each processor accurately detects
the feasibility of the whole problem P.
Theorem 4.2 Problern P is infeasible if and on- if the firsr step of the algorirlun in each
processor reports in feasibiliy 4 P.
Proof : (The "if' pan) If a subproblem is found to be infeasible m step 1 in any processor,
then the nonlinking constraints and upper bounds for the subproblem are infeasible because
the linking constnints can aiways be satisfied for some choice of the artificiai variables.
Infeasibility of the nonlinking conscraint and upper bound constraint implies that P is
infeasi ble.
(The "only if' part) If pmblem P is infeasible, then at least one piut's set of noniinking
consmints and upper bound constmints is infeasible because the Iinking constraints can't be
violated. This infeîsibility will be detected at step O.