Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 85
all extreme rays to the dual feasible region for scenario s, we have that SP s ( ˆx) is
feasible if and only if
(g s − T s ˆx) T d ≤ 0 ∀d ∈ Γ s . (4.18)
Based on (4.17) and (4.18), we can reformulate (4.14) as the following equivalent
problem, where now z s is a decision variable that takes on the optimal objective
function value of SP s (x).
min c T x + ∑ z
s∈S
s
(4.19a)
s.t. Ax = b (4.19b)
z s ≥ (g s − T s x) T π ∀s ∈ S, π ∈ Ω s
(4.19c)
(g s − T s x) T d ≤ 0 ∀s ∈ S, d ∈ Γ s
(4.19d)
x ≥ 0. (4.19e)
Note that (4.19c) is equivalent to (4.17) since optimality forces zs to the maximum
value of (gs −T sx) T π over all extreme points π in Ω s . Inequalities (4.19c) are sometimes
called “optimality inequalities” (or “optimality cuts” if they are used to remove
a targeted solution that reports an infeasible candidate solution with respect to
the relationship between the x- and z-variables), while (4.19d) are sometimes called
“feasibility inequalities” (or “feasibility cuts” if they are used to remove a solution
x to which no feasible solution corresponds in (4.15) for some s ∈ S).
Observe, however, that problem (4.19) is likely to be far more difficult to solve
than (4.14) itself because the sets Ω s and Γ s may both contain an exponential number
of elements. Hence, we adapt a cutting-plane strategy in which inequalities of
the form (4.19c) and (4.19d) are dynamically added to the problem only as needed.
Initially, we start with only a subset (usually empty) Ω s ⊆ Ω s of extreme points to
each dual feasible region, and also an (empty) subset Γ s ⊆ Γ s of extreme rays to
each dual feasible solution. Then, the relaxed master problem is the same as (4.19),
but with Ω s replaced by Ω s , and with Γ s replaced by Γ s . (The problem is called
“relaxed” because many or all of the constraints (4.19c) and (4.19d) are initially
missing from the formulation. Most studies refer simply to a master problem instead,
with “relaxed” being implied.) The Benders decomposition algorithm then
proceeds as follows.
Step 0. Initialize by setting Ω s = Γ s = /0 for each s ∈ S, and continue to Step 1.
Step 1. Solve the relaxed master problem. If the relaxed master problem is infeasible,
then the original problem is infeasible as well, and the process terminates.
Otherwise, obtain optimal solution ˆx, ˆz s , for each s ∈ S. Observe that the problem
may be unbounded, with some ˆz s values taking on values of −∞. Continue to Step
2.
Step 2. Solve dual subproblem DSPs ( ˆx), ∀s ∈ S, given the value of ˆx computed in
Step 1. If a dual subproblem is unbounded for some s ∈ S, then an extreme direction
dˆ ∈ Γ s exists for which (gs − T s ˆx) T d ˆ > 0. Therefore, we add dˆ to Γ s (thus creating
a new “feasibility cut” of the form (4.19d)) to avoid regenerating an ˆx that makes