Wireless Network Design: Optimization Models and Solution ...

84 J. Cole Smith and Sibel B. Sonuc
min c T x + ∑(h s∈S
s ) T y s
(4.14a)
s.t. Ax = b (4.14b)
T s x + G s y s = g s
∀s ∈ S (4.14c)
x ≥ 0, y s ≥ 0, ∀s ∈ S, (4.14d)
where A ∈ R m0 ×n and T s ∈ R m s ×n , ∀s ∈ S, G s ∈ R m s ×k , ∀s ∈ S, and all over vectors
have conforming dimensions. For the sake of simplicity, we suppose here that problem
(4.14) is not unbounded; the case in which (4.14) is unbounded requires only
slight modifications.
In Benders decomposition, we create a relaxed master problem, in which we
determine candidate x-solutions, and anticipate how those solutions affect the objective
function contribution terms corresponding to variables y s , ∀s ∈ S. Given the
values for x-variables, we can then formulate |S| subproblems, where subproblem
s ∈ S corresponds to the optimization over variables y s . We begin by developing the
subproblem formulation below, given a fixed set of values ˆx corresponding to x.
The dual of this problem is given as follows.
SP s ( ˆx) : min (h s ) T y s
(4.15a)
s.t. G s y s = g s − T s ˆx (4.15b)
y s ≥ 0 (4.15c)
DSP s ( ˆx) : max (g s − T s ˆx) T π (4.16a)
s.t. (G s ) T π ≤ h s
(4.16b)
First, let us assume that there exists an optimal solution to SPs ( ˆx), with objective
value zs . Then by the strong duality theorem, there is an optimal solution to DSPs ( ˆx)
with the same objective function value. Furthermore, let Ω s represent the set of all
extreme points to the dual feasible region for scenario s, and note that this dual
feasible region is invariant with respect to the given value of ˆx. We therefore have
that
z s = max
π∈Ω s(gs − T s ˆx) T π. (4.17)
Next, suppose that SPs ( ˆx) is infeasible, and therefore, that DSPs ( ˆx) is either infeasible
or unbounded. In this case, it is convenient to place certain “regularization”
restrictions on the subproblems to ensure that the dual is always feasible. (An easy
way of doing this is to add upper bounds on variables ys by adding the constraints
ys ≤ u, where u = (M,...,M) T and M is an arbitrarily large number. In this case,
each ys < M at optimality by assumption that the problem is bounded. Reversing
the constraints as −ys ≥ −u, and associating duals β with these primal constraints,
our dual constraints become (Gs ) T π − β ≤ hs , with β ≥ 0, and a feasible solution
exists in which β is set to be sufficiently large.) Now, if SPs ( ˆx) is infeasible, then
DSPs ( ˆx) must unbounded. There would therefore exist an extreme direction d to
the dual feasible solution such that (gs − T s ˆx) T d > 0. Letting Γ s denote the set of