LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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 We define the objective function as: and constraints of the problem as: . . . The Data 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: Conclusion: 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 A link to Matlab codes for this problem in the Github repository: https://github.com/asalimil/LMI-for-Linear-Programming Related LMIs LMI for Feasibility Problem External Links - LMI in Control Systems Analysis, Design and Applications <noinclude> Return to Main Page LMIs in Control/Tools