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1.1 Mathematical optimization 3
framework, are most likely).
An amazing variety of practical problems involving decision making (or system
design, analysis, and operation) can be cast in the form of a mathematical opti-
mization problem, or some variation such as a multicriterion optimization problem.
Indeed, mathematical optimization has become an important tool in many areas.
It is widel y used in engineering, in electronic design automation, automatic con-
trol systems, and optimal design problems arising in civil, chemical, mechanical,
and aerospace engineering. Optimization is used for problems arising in network
design and operation, finance, supply chain management, scheduling, and many
other areas. The list of applications is still steadily expanding.
For most of these applications, mathematical optimization is used as an aid to
a human deci si on maker, system designer, or system operator, who supervises the
process, checks the results, and modifi es the problem (or the solution approach)
when necessary. This human decision maker also carries out any actions suggested
by the optimization problem, e.g., buying or selling asset s to achieve the opt imal
portfolio.
A r el atively recent phenomenon opens the possibility of many other applications
for mathematical optimization. With the proliferation of computers embedded in
products, we have seen a rapid growth in embedded optimization. In these em-
bedded applications, optimization is used to automatically make real-time choices,
and even carry out the associated actions, with no (or little) human intervention or
oversight. In some application areas, this blending of tradition al automatic control
systems and embedded optimization is well under way; in others, it is just start-
ing. Embedded real-time optimization raises some new challenges: in particular,
it requires solution methods that are extremely reliable, and solve problems in a
predictable amount of time (and memory).
1.1.2 Solving optimization problems
A solution method for a class of optimization problems is an algor ith m that com-
putes a solution of the problem (to some given accuracy), given a particular problem
from the class, i.e., an instance of the problem. Since the late 1940s, a large effort
has gone into developing algorithms for solving various classes of optimization prob-
lems, analyzing their properties, and developing good software implementations.
The effectiveness of these algorithms, i.e., our ability to solve the optimization prob-
lem (1.1), varies considerably, and depends on factors such as the particular forms
of the objective and constraint functions, how many variables and constraints there
are, and special structure, such as sparsity. (A problem is sparse if each constr aint
function depends on only a small number of the variables).
Even when the objective and constraint functions are smooth (for example,
polynomials) the general optimization problem (1.1) is surprisingly difficult to solve.
Approaches to the general problem therefore involve some kind of compromise, such
as very long computation time, or the possibility of not finding the solution. Some
of these methods are discussed in §1.4.
There are, however, some important exceptions to the general rule that most
optimization problems are difficul t to solve. For a few problem classes we have