Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 67
of these products (if demand is price elastic). For transportation networks, finding
a best route for travelers is a common problem. Similarly, in communication networks,
one may be interested in finding a collection of routes for data transferred
through the system. However, it may be unclear how to best define decision variables
in some problems. For instance, while it may be straightforward to define some variable
xa to represent the amount of production, and xp to denote its price, defining
variables corresponding to data routes can be done in several different ways. The
choice of variable definition plays a large role in how the optimization problem will
ultimately be solved.
In optimization models, one seeks a “best” set of decision variable values, according
to some specified metric value. A function that maps a set of variable
values to this metric is called an objective function, which is either minimized or
maximized depending on the context of the problem. Some of the most common
objectives minimize cost for an operation or maximize a profit function. A solution
that achieves the best objective function value is called an optimal solution.
Decision variables are usually bounded by some restrictions, called constraints,
related to the nature of the problem. (If not, the problem is called unconstrained;
however, problems considered in this chapter are virtually all subject to constraints.)
Constraints can be used for many purposes, and are most commonly employed to
state resource limitations on a problem, enforce logical restrictions, or assist in constructing
components of an objective function. For the purposes of this chapter, we
will assume that constraints are either of the form g(x) ≤ 0, g(x) = 0, or g(x) ≥ 0,
where x is a vector of decision variables. For example, if multiple data streams are
to be sent on a route, the bandwidth of links on the network may limit the maximum
sum of traffic that passes through the link at any one time, and needs to be accounted
for in the optimization process via constraints. We may also state logical constraints
which stipulate that each communication pair must send and receive its desired traffic.
Another common set of logical constraints are nonnegativity constraints on the
variables, which prevent negative flows, production values, etc., when they do not
make sense for a given problem. An example of using constraints to assist in constructing
an objective function arises when one wishes to minimize the maximum
of several functions, say, f1(x),..., fk(x), where x is a vector of decision variables.
One can define a decision variable z that represents maxi=1,...,k{ fi(x)}, and enforce
constraints z ≥ fi(x) for each i = 1,...,k. Collectively, these constraints actually
state that z ≥ maxi=1,...,k{ fi(x)}, but when an objective of minimizing z is given,
z takes its smallest possible value, and hence z = maxi=1,...,k{ fi(x)} at optimality.
A mathematical optimization model that seeks to optimize an objective function,
possibly over a set of constraints is called a mathematical program.
Any solution that satisfies all constraints is called a feasible solution; those that
violate at least one constraint are infeasible. The set of all feasible solutions defines
the feasible region of the problem. Hence, optimization problems seek a feasible
solution that has the best objective function value. When there exist multiple feasible
solutions that achieve the same optimal objective function value, we say that
alternative optimal solutions exist.