Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 93
max
π
z IP(π). (4.31)
Observe that because z IP(π) is determined by means of the inner minimization problem
(4.30), problem (4.31) is a maximin problem (maximization of a minimum objective).
For linear programming problems, such as (4.25), there will exist a π = π ⋆
such that z IP(π ⋆ ) = zP (assuming again that zP has an optimal solution). We say that
there is no duality gap for these problems. For integer and/or nonlinear programming
problems, duality gaps may indeed exist.
In some situations, (4.31) can be solved directly. When direct solution of this Lagrangian
dual problem is not possible, a very common technique for solving (4.31)
is via subgradient optimization, which exploits the fact that IP(π) is a piecewiselinear
concave function of π. (In fact, when maximizing a concave function, one
correctly refers to “supergradients” rather than subgradients, although both terms
are used in the literature in this case.) We refer the reader to [6, 8] for excellent discussions
of this technique, along with the classical text [28]. However, we provide
a sketch of the subgradient optimization method here for completeness.
Step 0. Set iteration counter i = 0, and choose an initial estimate, π 0 , of the dual
values π. (One often starts with π 0 ≡ 0.) Select a positive integer R to be a maximum
number of iterations, and ε > 0 to be an acceptable subgradient norm for algorithm
termination. Initialize the best lower bound computed thus far as LB = −∞. Continue
to Step 1.
Step 1. Solve IP(π i ), and obtain an optimal solution ( ˆx 1,i ,..., ˆx |K|,i ). (Note that in
(4.30), the x k -vectors can be optimized in |K| separable problems.) If z IP(π i ) > LB,
then set LB = z IP(π i ) . (Note that because subgradients are being used in this procedure,
we are not guaranteed that z IP(π 0 ) , z IP(π 1 ) ,... will be a nondecreasing sequence
of values.) Continue to Step 2.
Step 2. One subgradient of IP(π i ) is given by s i = (b − ∑k∈K A k ˆx k,i ). If the (Euclidean)
norm of s i is sufficiently small, then the current dual estimate is sufficiently
close to optimal, and we stop. Thus, we terminate if ||s i || < ε. Also, to control the
maximum computational effort that we exert, we can also terminate if i = R. Otherwise,
continue to Step 3.
Step 3. Based on the subgradient computed in Step 2, we adjust our dual estimate by
moving in the (normalized) direction of the subgradient multiplied by a step length
parameter α i determined based on the iteration. That is,
π i+1 i si
= πi + α
||si . (4.32)
||
Note that the step-length parameter α i is a critical consideration in guaranteeing key
convergence properties of the subgradient optimization algorithm; see [6, 8, 28]. Set
i = i + 1, and return to Step 1.
Remark 4.6. There are many computational and theoretical issues that arise in subgradient
optimization that are beyond the scope of an introductory chapter such as
this. Some of the finer points that are particularly important regard the dual updat-