Wireless Network Design: Optimization Models and Solution ...

82 J. Cole Smith and Sibel B. Sonuc
tains complicating variables or complicating constraints, respectively. (Examples
of these situations are shown in Sections 4.3.1 and 4.3.2, respectively.)
In this case, we can attempt to exploit the special structure of these mathematical
programming problems by using large-scale optimization methods. In fact,
“large-scale optimization” is a broad term and refers to many different types of
advanced techniques for solving large optimization problems. For this chapter, we
focus primarily on decomposition techniques due to their common use in solving
telecommunication-related problems.
The rest of this section is organized as follows. We discuss the Benders decomposition
technique in Section 4.3.1, and illustrate it using a capacitated evacuation
network example. Next, we discuss the Dantzig-Wolfe decomposition method in
Section 4.3.2, illustrated using a multicommodity flow example. We then describe
in Section 4.3.3 key principles behind Lagrangian optimization. Finally, we briefly
discuss several other large-scale optimization strategies in Section 4.3.4.
4.3.1 Benders Decomposition
To motivate the use of Benders decomposition [7], we begin by introducing a
capacitated network design problem in which arcs can fail. Consider a network
G = (V,A) with node set V and candidate arc set A, and let di denote the supply/demand
value of node i ∈ V (where di > 0 indicates that there is a supply of
di units at node i, and di < 0 indicates that a demand of |di| is requested at node
i). Define the forward star of node j ∈ V (i.e., the nodes adjacent from node j) as
FS( j) = {i ∈ V : ( j,i) ∈ A}, and the reverse star of node j ∈ V (those nodes adjacent
to node j) as RS( j) = {i ∈ V : (i, j) ∈ A}.
Arc (i, j) can be constructed at a cost of ri j ≥ 0. If constructed, the arc will have
a capacity of qi j > 0, and a per-unit flow cost of ci j ≥ 0. (Note that all flows are
assumed to be simultaneous, meaning that the total flow across arc (i, j) is no more
than qi j.)
A network design problem would hence seek to minimize the cost of building
arcs plus the cost of routing flows across the established arcs. However, we also
consider in this problem the situation that some sets of arcs can possibly fail. For
ease of exposition, we consider here the case in which any single arc can fail, although
the discussion that follows easily extends to the case in which combinations
of arcs can fail. In fact, we adopt the terminology failure scenario from [11], where
scenario s ∈ A refers to the event that some arc s ∈ A fails. Let pi j ≥ 0 be the probability
that (only) arc (i, j) fails. Another interpretation here is that pi j represents
the proportion of time that arc (i, j) will fail over a long-term network flow situa-
tion. (Note that we use a single index for arcs where convenient, e.g., p s and p
are equivalently used. Also, for simplicity, we assume that ∑(i, j)∈A p i j = 1, which is
without loss of generality by adding a dummy arc if necessary.)
Now, the objective becomes the minimization of arc construction costs, plus the
expected flow costs that will take place after any arc fails. Note that the cost for
i j