# LMIs in Control/pages/LMI for Feasibility Problem

**LMI for Feasibility Problem in Optimization**

The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.

**The System**[edit | edit source]

Assume that we have two matrices as follows:

which are matrix functions of variables .

**The Data**[edit | edit source]

Suppose that the matrices and are given.

**The Optimization Problem**[edit | edit source]

The optimization problem is to find variables such that the following constraint is satisfied:

**The LMI:** LMI for Feasibility Problem[edit | edit source]

This optimization problem can be converted to a standard LMI problem by adding a slack variable, .

The mathematical description for this problem is to minimize in the following form of the LMI formulation:

**Conclusion:**[edit | edit source]

In this problem, and are decision variables of the LMI problem.

As a result, these variables are determined in the optimization problem such that the minimum value of is found while the inequality constraint is satisfied.

**Implementation**[edit | edit source]

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

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

**Related LMIs**[edit | edit source]

**External Links**[edit | edit source]

- [1] - LMI in Control Systems Analysis, Design and Applications