# Convex optimization problem

Note, that there is an agreement in notation of mathematical programming. The problems of the following type are called Convex optimization problem:

where all the functions $f(x), g_1(x), \ldots, g_m(x)$ are convex and all equality constraints are affine. It sounds a bit strange, but not all convex problems are convex optimization problems.

Where $f(x)$ is convex function, defined on the convex set $S$. The necessity of affine equality constraint is essential see Slater’s condition in Duality.

For example, this problem is not convex optimization problem (but implies minimizing convex function over the convex set):

while the following equivalent problem is convex optimization problem

Such confusion in notation is sometimes being avoided by naming problems of type $\text{(CP)}$ as abstract form convex optimization problem