Mathematical optimization

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework.

Since the following is valid:

A is some subset of the Euclidean space A of f is variously called an objective function, criterion function, loss function, cost function (minimization), utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution.

In mathematics, conventional optimization problems are usually stated in terms of minimization.

A local minimum ∀𝐱∈Awhere‖𝐱−𝐱∗‖≤δ,

the expression minx∈ℝ(x2+1)

This denotes the minimum value of the objective function 2</sup> + 1"}},"i":0}}]}'/>x + 1, when choosing x = 0.

Similarly, the notation

x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

Optimal input arguments

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argminx∈(−∞,−1]x2+1,

or equivalently

(−∞,−1] that minimizes (or minimize) the objective function 2</sup> + 1"}},"i":0}}]}'/>x + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answer is argmaxx∈[−5,5],y∈ℝxcos⁡y,

or equivalently

x cos y, with the added constraint that {5, 2kπ}, where arg max are sometimes also written as