## 2Linear Programming

In `lp_prob` there are 54 linear programming test problems with sizes to nearly 700 variables and 500 constraints. In order to define the 22'nd problem and solve it execute the following in Matlab:
```  n      = 22;
Prob   = probInit('lp_prob',n);
Result = tomRun('',Prob);
```
The general formulation in TOMLAB for a linear programming problem is:
 min x
f(x) = cT x

s/t
 xL ≤ x ≤ xU bL ≤ A x ≤ bU
(1)
where c, x, xL, xU Rn, A Rm1 × n, and bL,bU Rm1. Equality constraints are defined by setting the lower bound to the upper bound.

An example of a problem of this class, (that is also found in the TOMLAB Quickguide) is lpQG:

 min x1,x2
f(x1,x2) = −7x1−5x2

s/t
 x1+2x2 ≤ 6 4x1+x2 ≤ 12 x1,x2 ≥ 0
(2)

File: tomlab/quickguide/lpQG.m
```% lpQG is a small example problem for defining and solving
% linear programming problems using the TOMLAB format.

Name  = 'lpQG';     % Problem name, not required.
c     = [-7 -5]';   % Coefficients in linear objective function
A     = [ 1  2
4  1 ];   % Matrix defining linear constraints
b_U   = [ 6 12 ]';  % Upper bounds on the linear inequalities
x_L   = [ 0  0 ]';  % Lower bounds on x

% x_min and x_max are only needed if doing plots
x_min = [ 0  0 ]';
x_max = [10 10 ]';

% b_L, x_U and x_0 have default values and need not be defined.
% It is possible to call lpAssign with empty [] arguments instead
b_L   = [-inf -inf]';
x_U   = [];
x_0   = [];

% Assign routine for defining an LP problem. This allows the user
% to try any solver, including general nonlinear solvers.
Prob = lpAssign(c, A, b_L, b_U, x_L, x_U, x_0, Name,...
[], [], [], x_min, x_max, [], []);

% Calling driver routine tomRun to run the solver.
% The 1 sets the print level after optimization.

% Result.x_k contains the optimal decision variables.
% Result.f_k is the optimal value.

Result = tomRun('pdco', Prob, 1);

%Result = tomRun('lpSimplex', Prob, 1);
%Result = tomRun('minos', Prob, 1);
%Result = tomRun('snopt', Prob, 1);
%Result = tomRun('conopt', Prob, 1);
%Result = tomRun('knitro', Prob, 1);
%Result = tomRun('cplex', Prob, 1);
%Result = tomRun('xpress-mp', Prob, 1);```