# Convex optimization problem

## 1 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:

\begin{split} & f_0(x) \to \min\limits_{x \in \mathbb{R}^n}\\ \text{s.t. } & f_i(x) \leq 0, \; i = 1,\ldots,m\\ & Ax = b, \end{split} \tag{COP}

where all the functions f_0(x), f_1(x), \ldots, f_m(x) are convex and all the equality constraints are affine. It sounds a bit strange, but not all convex problems are convex optimization problems.

\tag{CP} f_0(x) \to \min\limits_{x \in S},

where f_0(x) is a convex function, defined on the convex set S. The necessity of affine equality constraint is essential.

Example

This problem is not a convex optimization problem (but implies minimizing the convex function over the convex set):

\begin{split} & x_1^2 + x_2^2 \to \min\limits_{x \in \mathbb{R}^n}\\ \text{s.t. } & \dfrac{x_1}{1 + x_2^2} \leq 0\\ & (x_1 + x_2)^2 = 0, \end{split} \tag{CP}

while the following equivalent problem is a convex optimization problem

\begin{split} & x_1^2 + x_2^2 \to \min\limits_{x \in \mathbb{R}^n}\\ \text{s.t. } & \dfrac{x_1}{1 + x_2^2} \leq 0\\ & x_1 + x_2 = 0, \end{split} \tag{COP}

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