Approximation and Fitting
Last edited: February 2, 2026china ece
Last edited: February 2, 2026China Economy Index
Last edited: February 2, 2026Complementary Slackness
Last edited: February 2, 2026Consider:
\begin{align} f_{0}\qty(x^{*}) &= \text{inf}_{x} \qty(f_{0}\qty(x) + \sum_{i=1}^{m} \lambda_{i}f_{i}\qty(x) + \sum_{i=1}^{p} v_{i}h_{i}\qty(x)) \\ &\leq f_{0}\qty(x) + \dots \\ &\leq f_{0}\qty(x) \end{align}
So the inequality holds strictly
conjugate function
Last edited: February 2, 2026The conjugate of a function \(f\) is \(f^{*}\qty(y) = \text{sup}_{x \in \text{dom }f} \qty(y^{T}x - f\qty(x))\). \(f^{*}\) is convex, even if \(f\) is not.
(fyi \(\text{sup}_{x} = \max_{x}\))