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

We define the objective function as:

$c_{1}x_{1}+c_{2}x_{2}+...c_{n}x_{n}$

and constraints of the problem as:

$a_{11}x_{1}+a_{12}x_{2}+...+a_{1n}x_{n}<b_{1}$

$a_{21}x_{1}+a_{22}x_{2}+...+a_{2n}x_{n}<b_{2}$

.

.

.

$a_{m1}x_{1}+a_{m2}x_{2}+...+a_{mn}x_{n}<b_{m}$

The Data

Suppose that $c_{j}\in \mathbb {R} ^{n}$, $a_{ij}\in \mathbb {R} ^{n}$, and $b_{i}\in \mathbb {R} ^{m}$ are given parameters where $i=1,2,...,m$ and $j=1,2,...,n$. Moreover, $x=[x_{1}\quad x_{2}\quad ...\quad x_{n}]^{\text{T}}$ is an $n\times 1$ vector of positive variables.

The Optimization Problem

The optimization problem is to minimize the objective function, $c^{\text{T}}x$ 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:

${\begin{aligned}&{\text{min}}\quad c^{\text{T}}x&\\&{\text{s.t.}}\quad a_{i}^{\text{T}}x\leq b_{i}x\\\end{aligned}}$

Conclusion:

Solving this problem results in the values of variables $x$ which minimize the objective function. It is also worthwhile to note that if $m\geq n$, the computational cost for solving this problem would be $mn^{2}$.

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

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

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