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.
We define the objective function as:
and constraints of the problem as:
Suppose that , , and are given parameters where and . Moreover, is an vector of positive variables.
The Optimization Problem
The optimization problem is to minimize the objective function, when the aforementioned linear constraints are satisfied.
The LMI: LMI for linear programming
The mathematical description of the optimization problem can be readily written in the following LMI formulation:
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.
A link to Matlab codes for this problem in the Github repository:
-  - LMI in Control Systems Analysis, Design and Applications