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:

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

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

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.

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

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:

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

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

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