Wireless Network Design: Optimization Models and Solution ...

4 An Introduction to Integer and Large-Scale Linear Optimization 95
B −1 is a bottleneck operation in the simplex method. (It is worth noting that modern
simplex implementations do not compute B −1 directly, but instead use numerical
linear algebra methods that improve numerical stability and reduce computational
effort by exploiting the sparsity of the A matrix.) Therefore, if special structures
exist in the constraint matrix, it may be possible to compute B −1 by specialized
algorithms rather than by using general methods from linear algebra. Basis partitioning
algorithms compute B −1 by decomposing B into specially structured blocks,
and then employing a specialized algorithm to find corresponding blocks to its inverse.
Only a small submatrix of B −1 needs to be computed under these schemes,
which reduces both computational and memory requirements for solving problems
by the simplex method. Foundational work in this area was performed by Schrage
[25], and has been applied to network flow problems in several studies; see [18, 19]
for multicommodity flow applications.
4.3.4.2 Interior Point Methods
As mentioned previously, interior-point methods were originally developed to solve
nonlinear optimization problems. These algorithms had a dramatic implication in
the field of linear programming in the early 1980s by yielding the first polynomialtime
algorithm for linear programming problems. Interior point algorithms are more
sophisticated than simplex-based algorithms. Due to worst-case time-complexity
functions that are polynomial in the input size of the linear program, they often
outperform simplex approaches for very-large-scale problems (roughly 10,000 variables
or more), depending on problem-specific details such as the presence of special
constraint structures. See, e.g., [12, 21, 31] for discussion on these methods on
linear and nonlinear optimization problems. There is currently much development in
computation and theory for interior-point algorithms, some of which is specifically
focused on interior-point approaches for large-scale problems (e.g., [9]).
4.3.4.3 Heuristics
Exact optimization techniques such as the ones described above can become impractical
because of the size and/or complexity of real-world instances. Rather than
eschewing optimization as an option, one can turn to heuristic methods for solving
problems. Heuristics are designed to quickly provide good-quality feasible solutions,
but without a guarantee of optimality, or in some cases, without a guarantee
of feasibility.
Naturally, heuristic techniques are as diverse in complexity and effectiveness as
are exact optimization techniques. Many heuristic algorithms are tailored specifically
for the problem at hand, employing specific properties of the problem to obtain
fast and effective heuristic techniques. There are too many such approaches to name,
but one of the most famous of these is the Lin-Kernighan algorithm [16] for the Euclidean
traveling salesman problem. A common trend over the last few decades in