Wireless Network Design: Optimization Models and Solution ...

72 J. Cole Smith and Sibel B. Sonuc
a dual constraint, and each dual variable is associated with a primal constraint. If
some constraints (4.2b) are of the ≤ sense (or ≥ sense), then their associated dual
(π) variables are restricted to be nonnegative (or nonpositive). Also, if (4.2c) does
not enforce nonnegativity of the x-variables (or enforces nonpositivity), then their
associated constraints (4.7b) are of the = sense (or of the ≤ sense).
There are vital relationships that exist between the primal and dual. First, suppose
that there exists a feasible primal solution x ′ and a feasible dual solution π ′ . Then
(π ′ ) T Ax ′ = (π ′ ) T b due to primal feasibility, and (π ′ ) T Ax ′ ≥ c T x ′ due to dual feasibility
and nonnegativity of x. Therefore, combining these expressions, (π ′ ) T b ≥ c T x ′ ,
i.e., the dual objective function value for any feasible solution π ′ is greater than or
equal to the primal objective function value for any primal feasible solution x ′ . A
more general statement is given by the following Weak Duality Theorem.
Theorem 4.3. Consider a set of primal and dual linear programming problems,
where the primal is a maximization problem and the dual is a minimization problem.
Let zD(π ′ ) be the dual objective function value corresponding to any dual feasible
solution π ′ , and let zP(x ′ ) be the primal objective function value corresponding to
any primal feasible solution x ′ . Then zD(π ′ ) ≥ xP(x ′ ).
Corollary 4.1. If the primal (dual) is unbounded, then the dual (primal) is infeasible.
Theorem 4.3 states that any feasible solution to a maximization problem yields a
lower bound on the optimal objective function value for its dual minimization problem,
and any feasible solution to a minimization problem yields an upper bound
on the optimal objective function value for its dual maximization problem. It thus
follows that if one problem is unbounded, there cannot exist a feasible solution to
its associated dual as stated by Corollary 4.1. (It is possible for both the primal and
dual to be infeasible.)
The Strong Duality Theorem below is vital in large-scale optimization.
Theorem 4.4. Consider a set of primal and dual linear programming problems, and
suppose that both have feasible solutions. Then both problems have optimal solu-
be the primal and dual optimal objective function values,
tions, and letting z⋆ P and z⋆D respectively, we have that z⋆ P = z⋆D .
For instance, the dual of problem (4.1) is given by
min 9π1 + 10π2 + π3
(4.8a)
s.t. 3π1 + π2 + π3 ≥ 1 (4.8b)
π1 + 2π2 − π3 ≥ 1 (4.8c)
π1,π2,π3 ≥ 0. (4.8d)
Recalling that the primal problem (4.1) has an optimal objective function value of
29/5, observe that, as stated by the Weak Duality Theorem, all feasible solutions to
problem (4.8) have objective function value at least 29/5. For instance, π 1 = (1,0,0)
and π 2 = (1/3,1/2,0) are both feasible, and have objective function values 9 and