[Math] Is KKT conditions necessary and sufficient for any convex problems

Question

In Boyd's Convex Optimization, pp. 243,

for any optimization problem … for which strong duality obtains, any pair of primal and dual optimal points must satisfy the KKT conditions

i.e. $\mathrm{strong ~ duality} \implies \mathrm{KKT ~ is ~ necessary ~ condition ~ for ~ optimal ~ solution}$

and in pp. 244,

(When the primal problem is convex) if $\tilde{x}, \tilde{\lambda}, \tilde{\mu}$ are any points that satisfy the KKT conditions, then $\tilde{x}$ and $(\tilde{\lambda}, \tilde{\mu})$ are primal and dual optimal, with zero duality gap.

If duality gap = 0, the problem satisfies strong duality, and in the 3rd paragraph:

If a convex optimization problem … satisfies Slater’s condition, then the KKT conditions provide necessary and sufficient conditions for optimality

For me it means: (for any convex problems KKT is already sufficient for optimal)

$$\mathrm{KKT} \implies \mathrm{optimal ~ with ~ zero ~ duality ~ gap} \implies \mathrm{strong ~ duality} \implies \mathrm{KKT ~ is ~ also ~ necessary}$$

so KKT is necessary and sufficient for any convex problems? (Because Slater's condition can be automatically satisfied for the zero duality gap)

Answer 1

The KKT conditions are not necessary for optimality even for convex problems. Consider $$ \min x $$ subject to $$ x^2\le 0. $$ The constraint is convex. The only feasible point, thus the global minimum, is given by $x=0$. The gradient of the objective is $1$ at $x=0$, while the gradient of the constraint is zero. Thus, the KKT system cannot be satisfied.