Wireless Network Design: Optimization Models and Solution ...

68 J. Cole Smith and Sibel B. Sonuc
4.2.2 Linear Programming Problems
A mathematical program is called a linear program (LP) if no nonlinear terms appear
in the objective function or in the constraints, and if no variables are restricted
to take on a discrete set of values (i.e., all variables are continuous and allowed to
take on fractions). LPs carry special attributes that enable them to be solved more
easily than general mathematical programs. For instance, the following problem is
an LP:
max x1 + x2
(4.1a)
s.t. 3x1 + x2 ≤ 9 (4.1b)
x1 + 2x2 ≤ 10 (4.1c)
x1 − x2 ≤ 1 (4.1d)
x1,x2 ≥ 0. (4.1e)
The feasible region of this LP is given in Figure 4.1.
The dotted line closest to the origin in Figure 4.2 represents the set of solutions
that have an objective function value x1 +x2 = 1, and the next dotted line represents
the set of solutions with objective function value 2. These objective contours are
parallel, and increase in the direction of the objective function gradient (which is
(1,1) in this case). To optimize a problem in two dimensions, one can slide the
objective contours as far as possible in the direction of the objective gradient (for
a max problem), which yields the optimal solution for this problem as depicted
by Figure 4.2. Note that because this solution lies at the intersection of the first
two constraints, the solution can be obtained by setting those two constraints as
equalities, and then by solving the simultaneous system of equations. This yields
x1 = 8/5 and x2 = 21/5, with objective function value 29/5.
Fig. 4.1 Feasible region of
the problem (4.1)
x2
Constraint (1b)
Feasible region
Constraint (1d)
Constraint (1c)
x1