X - UWSpace - University of Waterloo

l, and Theorem 3. lb mles out the possibiiity that the second subproblem cm be unbounded
for k=l. Theorem 3.1~ ensures that the unboundedness (violation of the Assumption) is
detected at k=l.
Theorem 3.la For k=l. subproblem - SP: is either infeasible or hm a bounded optimal
solution.
Prool: When k=l, SP: - hûs no h variable. By the Assumptinn, and the negative objective
coefficients of the artificial variables, spi - can't have an unbounded solution. Therefore, SP~ -
is either infeasible or has a finite optimai solution.
-
Theorem 3.1b For k=l. subproblem is either infeasible or has a bounded optimal
solrrriotl.
Proof: When k=l. the subproblem SPI has no cuts or 0 variable. The possibility of
unboundedness is ruled out by the Assumption. and the large negative objective coefficients
of the utificid variables. Therefore, is either infeasible or has a bounded optimal
solution.
Suppose a modeller unintentionally submits a mode1 that violates the Assumption and
has an unbounded optimd value. Then, the algorithm detects the violation of the Assumption
and stops.