Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 71
a basis of R m . The method begins by choosing a basis of A that corresponds to an
extreme-point solution, and moves to an adjacent basis (i.e. a set of columns that differs
by only one column from the current basis) at each iteration. Each move should
improve some extended metric of the objective function (see the lexicographic rule),
which then guarantees that the algorithm terminates either with an optimal solution,
or in a direction of unboundedness. This method can easily accommodate the case
in which a starting basis corresponding to a basic feasible solution is not readily
available (using the two-phase or big-M methods [5]), and identifies when a problem
is infeasible as well. There are also alternative interior point approaches to
solving LPs, inspired by nonlinear optimization ideas generated in the 1960s. These
algorithms are discussed very briefly in Section 4.3.4.
4.2.3 Duality
Duality is an important complement to linear programming theory. Consider the
general linear programming form given by (4.2). A relaxation of the problem is
given as:
z(π) = max c T x + π T (b − Ax) (4.5a)
s.t. x ≥ 0, (4.5b)
where π ∈ R m is a set of dual values that are specified as inputs for now, and where
z(π) = ∞ if (4.5) is unbounded. Assume for the following discussion that an optimal
solution exists to (4.2), and let z ⋆ be the optimal objective function value to (4.2).
Then z(π) ≥ z ⋆ . To see this, note that any feasible solution to problem (4.2) is also
feasible to (4.5), and this solution yields the same objective function value in both
formulations. However, the reverse is not necessarily true. Hence, for each choice
of π, we obtain an upper bound on z ⋆ . To achieve the tightest (i.e., lowest) bound
possible, we wish to optimize the following problem:
minz(π),
where z(π) = max
π x≥0 (cT − π T A)x + π T b. (4.6)
Because x is bounded only by nonnegativity restrictions, z(π) will be infinite if any
component of (c T − π T A) is positive. On the other hand, if (c T − π T A) ≤ 0, then
choosing x = 0 solves problem (4.5), and z(π) = π T b. Therefore, (4.6) is equivalent
to
min π T b (4.7a)
s.t. A T π ≥ c. (4.7b)
The formulation (4.7) is called the dual of (4.2), which we refer to as the primal
problem. (Note that “primal” and “dual” are relative terms, and either can refer to a
minimization or a maximization problem.) Each primal variable is associated with