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ound constraints. The two parts are linked through the pnmal (or dual) linking consaaints
containing the matnx La. The primal (or dual) linking variables are yr (or y) which appear in
linking constmints of both parts. Each linking pnmal consaint has a corresponding duai
linking variable and similarly. there is 3 linking dual constra.int corresponding CO each linking
primal variable. Each pnmal artificid variable induces a corresponding upper bound
consmint to a dual linking variable. Similarly. there is a dual axtificial variable corresponding
to each upper bound constraint for a pnmal linking variable. These mificial variables cm be
adjusted to satisfy the linking constraints. thus dlowing each part to act independently.
However. because of the high cost of the artificial variables, it may k desirable. in the
optimal solution. for the pans to coopente. Thus. the iutificial variables (primai and dual) are
an important aspect of the notion of "pans" of a linear program. The pnmal mificial variables
may in fact represent 3 red aspect of the situation such as unfillable demand or emergency
purchase in inventory consuaints: wherher real or mificial. these variables ensure the
feasibility of linhng constraints.
The two-stage (or block-triangular) structure is a special case for which part one has
no linking consminu and y2 does not exist. i.e. BI. LII, Li:. Dz and L2 are dl zero. This
structure *ses. eg.. in a two period model, in which the linking consnaints represent the
influence of decisions in the fint period on those in the second.
convergence.
An assumption is made in order to simplify the dgorïthm, and to guarantee
Assum~tion: The set of nonlinliing constraints in each part. together with the upper bound
consrnints and nonneptivity constraints. define bounded feasible regions for the x,, y, vectors.
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