Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 69
In general, we can formulate any linear programming problem in standard form
as
max c T x (4.2a)
s.t. Ax = b (4.2b)
x ≥ 0, (4.2c)
where x is an n-dimensional (column) vector of decision variables, c is an ndimensional
vector of objective coefficient data, A is an m × n constraint coefficient
matrix (where without loss of generality, rank(A) = m), and b is an m-dimensional
right-hand-side vector that often states resource or requirement values of the optimization
model. Note that inequalities are readily accommodated by adding a
nonnegative slack variable or subtracting a nonnegative surplus variable from the
left-hand-side of less-than-or-equal-to or greater-than-or-equal-to constraints, respectively.
(The term “slack” is often used in conjunction with both less-thanor-equal-to
and greater-than-or-equal to constraints.) For instance, the constraint
7x1 + 2x2 + 8x3 ≤ 4 is equivalent to 7x1 + 2x2 + 8x3 + s = 4, along with the restriction
s ≥ 0 (where s serves as the slack variable). Minimization problems are
converted to maximization problems by multiplying the objective function by −1
and changing the optimization to maximization. Nonpositive variables xi are substituted
with nonnegative variables (replace xi with −xi), and unrestricted variables
x j are substituted with the difference of two nonnegative variables (replace x j with
x ′ j − x′′ j , with x′ j ≥ 0 and x′′ j ≥ 0).
Now, observe that the optimal solution for (4.2) is given at a “corner,” i.e., extreme
point of the feasible region. In fact, regardless of the orientation of the objective
function, it is easy to see that there will always exist an optimal solution at
one of these extreme points. More precisely, let X be the feasible region contained
Fig. 4.2 Optimization process
for solving the linear program
(4.1)
x2
Constraint (1b)
Optimal
x1 + x2 = 1 x1 + x2 = 2
Constraint (1d)
Constraint (1c)
x1