Wireless Network Design: Optimization Models and Solution ...

92 J. Cole Smith and Sibel B. Sonuc
4.3.3 Lagrangian Optimization
Another common (and more general) technique for addressing problems having
complicating constraints is by Lagrangian optimization. The idea here is to penalize
violations to the complicating constraints within the objective function, and omit
those complicating constraints from the constraint set. Indeed, we have already seen
a basic application of this concept in Section 4.2.3, where constraints Ax = b were
omitted from the constraint set, and a penalty term π T (b −Ax) was added to the objective
function. (This strategy led us to develop the dual formulation for linear programs.)
The multipliers that are used to penalize violations to these constraints are
called Lagrangian duals, and the process of moving constraints out of a constraint
set and into the objective function is often referred to as “dualizing” the constraints.
Whereas all constraints were dualized in Section 4.2.3, it is more common to
dualize only the complicating constraints, leaving behind a separable problem (or
otherwise easily solvable problem, as mentioned in Remark 4.4). In the context
of the multicommodity flow problem, we would dualize the capacity constraints
(4.24b).
We again consider the generic formulation (4.25). Assume for simplicity that
X k = {x ≥ 0 : H k x k = g k } is nonempty and bounded. (If X k is empty, then the
problem is infeasible. If X k is unbounded, we can bound it by placing a constraint
e T x k ≤ M for a very large value M, and note that any solution lying on this constraint
indicates a direction of unboundedness.) A relaxation of (4.25) is given as follows
(to which we will later refer as an “inner problem,” and often called the “Lagrangian
function” or “Lagrangian subproblem”), for any given vector π:
IP(π) : min ∑ (c
k∈K
k ) T x k + π T (b − ∑ A
k∈K
k x k ) (4.30a)
s.t. H k x k = g k
∀k ∈ K (4.30b)
x k ≥ 0 ∀k ∈ K. (4.30c)
Problem IP(π) is a relaxation of (4.25) because any solution to (4.25) is a feasible
solution to IP(π) with the same objective function value (regardless of the value of
π). That is, let zP be the optimal objective function value to (4.25) (assuming that
(4.25) has an optimal solution), and let zIP(π) be the optimal objective function value
to IP(π), where zIP(π) = ∞ if IP(π) is unbounded. Then zIP(π) ≤ zP for any vector π.
We can therefore use problem IP(π) to yield lower bounds to the optimal objective
function value of (4.25).
Note here that if constraint (4.25b) were written as the inequality ∑k∈K Akxk ≥ b,
then we would formulate (4.30) in the same way, but with the stipulation that π
should be nonnegative. This will ensure that the objective penalty term, πT (b −
∑k∈K Akxk ), remains nonpositive when x is feasible to (4.25b), and that zIP(π) ≤ zP
as desired. If (4.25b) were written in the ≤ sense, we would insist that π ≤ 0.
In order to provide the best possible lower bound for zP, we can seek a value of
π that maximizes zIP(π). Thus, the Lagrangian dual problem is: