# LMIs in Control/pages/LMI for convex minimization problem

LMIs in Control/pages/LMI for convex minimization problem

## The System

Let ${\displaystyle A(x)}$  and ${\displaystyle B(x)}$  are matrix functions of appropriate dimensions.

Moreove, the matrix functions ${\displaystyle A(x)}$  and ${\displaystyle B(x)}$  are linear in vector ${\displaystyle x}$  where ${\displaystyle x}$  is an ${\displaystyle m\times 1}$  vector.

## The Data

A convex function ${\displaystyle f(x)}$  is given and ${\displaystyle x}$  is the vector of decision variables.

## The Optimization Problem

The optimization problem is to minimize the convex function ${\displaystyle f(x)}$  such that a contraint between two convex functions are satisfied for the decision variables.

## The LMI: LMI for convex minimization problem

{\displaystyle {\begin{aligned}&{\text{min}}\;\quad f(x)&\\&A(x)

## Implementation

A link to Matlab codes for this problem in the Github repository: