I just have learned a nice necessary and sufficient condition for convex optimization KKT form but i can't find exercises or examples for this. Can someone help me? ( some examples or exercies or book that have exercises about this)
Consider convex optimization problem (P):
minimize $f(x)$
subject to $g_i(x) \leq 0$ for $i=1,..,m$
$x \in \Omega$
Where $f: \mathbb{R}^n \to \mathbb{R}$ is convex function and $\Omega$ is a nonempty, convex set.
Slater's condition constraint qualification (CQ) holds in (P) if there exists $u \in \Omega$ such that $g_i(u)<0$ for all $i=1,..,m$.
Theorem: [Karush-Kuhn-Tucker Optimality Conditions in Convex Programming] Suppose that Slater's CQ holds in (P). Then $x^*$ is an optimal solution to (P) if and only if there exist nonnegative Lagrange multipliers $(\lambda_1,..,\lambda_m) \in \mathbb{R}^m$ such that
$$0 \in \partial f(x^*) + \sum_{i=1}^m \lambda_i \partial g_i(x^*) + N(x^*,\Omega)$$
See: http://people.scs.carleton.ca/~bertossi/dmbi/material/Convex%20Analysis.pdf (page 138)
There are examples and exercises for KKT conditions and Slater's constraint qualification in chapter 5 of Boyd and Vandenberghe "Convex Optimization", which is available for free at the author's website https://web.stanford.edu/~boyd/cvxbook/ .