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. maximize f(x), x ∈ S ⊆ Rn, where Rn denotes the space of real n-component vectors x, x = (x1,x2,…,xn)T 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.
maximize f(x), x ∈ S ⊆ Rn,
x = (x1,x2,…,xn)T
Mathematical-programming problems arise in engineering, business, and the physical and social sciences.