Optimization Toolbox

Linear Programming with Equalities and Inequalities

The problem is

and you can load the matrices and vectors A, Aeq, b, beq, f, and the lower bounds lb into the MATLAB workspace with

• load sc50b
  

This problem in sc50b.mat has 48 variables, 30 inequalities, and 20 equalities.

You can use linprog to solve the problem:

• [x,fval,exitflag,output] = ...
      linprog(f,A,b,Aeq,beq,lb,[],[],optimset('Display','iter'));
  

Because the iterative display was set using optimset, the results displayed are

•   Residuals:   Primal      Dual      Duality     Total
                 Infeas     Infeas       Gap        Rel
                 A*x-b     A'*y+z-f     x'*z       Error
    ------------------------------------------------------
    Iter    0:  1.50e+003  2.19e+001  1.91e+004  1.00e+002
    Iter    1:  1.15e+002  2.94e-015  3.62e+003  9.90e-001
    Iter    2:  1.16e-012  2.21e-015  4.32e+002  9.48e-001
    Iter    3:  3.23e-012  5.16e-015  7.78e+001  6.88e-001
    Iter    4:  5.78e-011  7.61e-016  2.38e+001  2.69e-001
    Iter    5:  9.31e-011  1.84e-015  5.05e+000  6.89e-002
    Iter    6:  2.96e-011  1.62e-016  1.64e-001  2.34e-003
    Iter    7:  1.51e-011  2.74e-016  1.09e-005  1.55e-007
    Iter    8:  1.51e-012  2.37e-016  1.09e-011  1.51e-013
  
  Optimization terminated successfully.
  

For this problem, the large-scale linear programming algorithm quickly reduces the scaled residuals below the default tolerance of 1e-08.

The exitflag value is positive, telling you linprog converged. You can also get the final function value in fval and the number of iterations in output.iterations:

• exitflag =
       1
  fval =
   -70.0000
  output = 
        iterations: 8
      cgiterations: 0
         algorithm: 'lipsol'
  

Linear Least-Squares with Bound Constraints

Linear Programming with Dense Columns in the Equalities