# LMIs in Control/pages/LMI for Linear Programming

**LMI for Linear Programming**

Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.

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

We define the objective function as:

and constraints of the problem as:

.

.

.

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

Suppose that , , and are given parameters where and . Moreover, is an vector of positive variables.

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

The optimization problem is to minimize the objective function, when the aforementioned linear constraints are satisfied.

**The LMI:** LMI for linear programming[edit | edit source]

The mathematical description of the optimization problem can be readily written in the following LMI formulation:

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

Solving this problem results in the values of variables which minimize the objective function. It is also worthwhile to note that if , the computational cost for solving this problem would be .

There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.

**Implementation**[edit | edit source]

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

https://github.com/asalimil/LMI-for-Linear-Programming

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

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

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