Wireless Network Design: Optimization Models and Solution ...

74 J. Cole Smith and Sibel B. Sonuc
4.2.4 Modeling with Nonlinear and Discrete Variables
Naturally, many real-world challenges involve nonlinear objective functions and/or
constraints, and/or variables that must take on a discrete set of values. These problems
are no longer considered to be linear programs, and cannot directly be solved
using the framework developed above. We briefly discuss in this subsection when
these problems arise, and what challenges we face in solving them.
For instance, consider a situation in which two wireless routers must be arranged
over a Euclidean space, and must be no more than d meters from each other. Letting
(xi,yi) be decision variables that specify where router i = 1,2 will be placed (on a
grid with one-meter spacing), we would have the constraint
�
(x1 − x2) 2 + (y1 − y2) 2 ≤ d.
Since this constraint is nonlinear, we say that the problem is a nonlinear program.
Nonlinearities may also arise in the objective function, for example, when minimizing
a nonlinear function of communication latency, which may be represented by a
nonlinear function of data transmission over a particular communication link. Nonlinear
programs can be extremely difficult to solve, even if there are few nonlinear
terms in the problem (see, e.g., [22]). The difficulty of, and methods used for, solving
these problems depend on the convexity or concavity of the objective function,
and whether or not the feasible region forms a convex set.
Several other applications may call for variables that take on only discrete values;
usually some subset of the integer values (and often just zero or one). These problems
are called integer programs (IP). An integer programming problem is often
called a binary or 0-1 program if all variables are restricted to equal 0 or 1. Problems
having a mixture of discrete and continuous variables are called mixed-integer
programs (MIP).
There are a vast array of uses for integer variables in modeling optimization
problems. The most obvious arises when decision variable values refer to quantities
that must logically be integer-valued, such as personnel or non-splittable units of
travel (e.g., cars on a transportation network). Logical entities may also be integervalued
(especially with 0-1 variables). For instance, variables may represent whether
or not to establish a link between two nodes in a network, or whether or not to
assign routing coverage to some point of access. Integer variables are also very
useful in constructing certain cost functions, when these functions do not satisfy
certain well-behaved properties that would allow linear programming formulations
to be used (e.g., for minimization problems, piecewise-linear convex functions can
be modeled as LPs, while piecewise-linear nonconvex functions generally require
the use of binary variables).