Conjugate function

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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.


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