# Conjugate function

The convex conjugate of $LaTeX: f$ on $LaTeX: X$, denoted $LaTeX: f^*$ on $LaTeX: X^*$, is the greatest convex approximation from below:

$LaTeX: f^*(x^*) = \sup \{ x^T x^* - f(x): x \in X\}$,

and $LaTeX: X^* = \{x^*: f^*(x^*) < \infty\}$, i.e. $LaTeX: X^*$ is the effective domain of $LaTeX: f^*$). The concave conjugate of $LaTeX: f$ on $LaTeX: X$, denoted $LaTeX: f^{\wedge}$ on $LaTeX: X^{\wedge}$, is the least concave approximation from above:

$LaTeX: f^{\wedge}(x^*) = \inf\{ x^T x* - f(x): x \in X\}$,

and $LaTeX: X^{\wedge} = \{x^*: f^{\wedge}(x^*) > -\infty}$. This is a foundation for Lagrangian duality, viewed in response space.