Lagrangian saddle point

Theorem 6.21 (Kuhn-Tucker Saddle Point Condition) Assume an optimization problem of the form (6.37), where $f:R^{m}\rightarrow R$ and $c:R^{m}\rightarrow R$ for $i\in[n]$ are arbitrary functions, and a Lagrangian

$L(x,\alpha) = f(x) + \sum_{i=1}^{n}\alpha_{i}c_{i}(x)$ where $\alpha_{i}\geq 0$.

If a pair of variables $(x^{*},\alpha^{*})$ with $x^{*}$ and $\alpha^{*}\geq 0$, for all $i = \in [n]$ exists such that for all $x\in R^m$ and $\alpha\in[0,\infty)^{n}$,

$L(x^{*},\alpha) \leq L(x^{*},\alpha^{*}) \leq L(x,\alpha^{*})$ (Saddle Point)

then $x^{*}$ is a solution to (6.37).

 

Book: Chapter 6, Page 166, Learning from Kernels.