Optimization problem: Difference between revisions

Revision as of 01:41, 28 January 2018

Revision as of 01:53, 28 January 2018

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&\operatorname{subject\;to}

& &g_i(x) \leq 0, \quad i = 1,\dots,m \\

&&&h_ih_j(x) = 0, \quad ij = 1, \dots,p

\end{align}</math>

where

* <math>f(x): \mathbb{R}^n \to \mathbb{R}</math> is the '''[[Loss function|objective function]]''' to be minimized over the ''n''-variable vector <math>x</math>,

* <math>g_i(x) \leq 0</math> are called '''inequality [[Constraint (mathematics)|constraints]]''', and

* <math>h_ih_j(x) = 0</math> are called '''equality constraints'''., and

* <math>m \geq 0\ and\ p \geq 0</math>.

By convention, the standard form defines a '''minimization problem'''. A '''maximization problem''' can be treated by [[Additive inverse|negating]] the objective function.

If <math>m</math> and <math>p</math> equal 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a '''minimization problem'''. A '''maximization problem''' can be treated by [[Additive inverse|negating]] the objective function.

==Combinatorial optimization problem==