A wide field of study that deals with the theory, applications, and computational methods for optimization problems. An abstract formulation of such problems is to maximize a function f (known as an objective function) over a constraint set S, i.e.
where
Rn denotes the space of real
n-component vectors
x,
and
f is a real-valued function defined on
S. If
S consists only of vectors whose elements are integers, then the problem is one of
integer programming. Linear programming treats the case of
f as a linear function with
S defined by linear equations and/or constraints. Nonlinear objective functions with or without constraints (defined by systems of nonlinear equations) give rise to problems generally referred to as optimization problems.
Mathematical-programming problems arise in engineering, business, and the physical and social sciences.