For my thesis, I am using a certain theorem that states the equivalence of the dual optimization problem. My source for this is "Learning with Kernels" by Schoelkopf and Smola. The problem is that the theorem is not explicitly stated, but rather slowly developed starting from Theorem 6.26, on pages 170-171, culminating especially, but not only, in equation 6.64.
Did I summarize the facts correctly?
EDIT: I am looking for an answer that describes what exactly is wrong with the current wording, describes how to state the theorem correctly and gives a reference.
Consider the optimization problem \begin{equation*} \begin{aligned} & \underset{z \in \mathbb{R}^d}{\text{minimize}} & & f(z) \\ & \text{subject to} & & c_i(z) \leq 0 \text{ for all } i = 1, \ldots, n \end{aligned} \end{equation*} where $f, c_i\colon \mathbb{R}^d \to \mathbb{R}$ are convex and differentiable for all $i = 1, \ldots, n$. Define the Lagrangian to be \begin{align*} &L(z, \alpha) := f(z) + \sum_{i=1}^n \alpha_i c_i(z) \\ &\text{where } \alpha_i \geq 0 \text{ for all } i = 1, \ldots, n. \end{align*} Then the the above optimization problem is equivalent to \begin{equation*} \begin{aligned} & \underset{z \in \mathbb{R}^d, \alpha \in \mathbb{R}^n}{\text{maximize}} & & L(z, \alpha) \\ & \text{subject to} & & \alpha_i \geq 0 \text{ for all } i = 1, \ldots, n \\ & \text{and} & & \partial_z L(z, \alpha) = 0. \end{aligned} \end{equation*} Furthermore, if there exists $z \in \mathbb{R}^d$ such that for all $ i = 1, \ldots, n$, $c_i(z) < 0$, then the two optimization problems have a solution.
Additionally, if there is any solution $z^\ast$, $\alpha^\ast$, it will fulfill \begin{align*} \sum_{i=1}^n \alpha_i^\ast c_i(z^\ast) = 0 \end{align*} or, equivalently $\alpha_i^\ast c_i(z^\ast) = 0$ for all $i=1, \ldots, n$.