Problem using "linprog" command.

For a linear problem:

min C^(T) x

such that : A_1 x <= b_1

A_2 x = b_2

x >=0

The dual is given by:

max - Lambda^(T) b_2 + mu^(T) b_1

such that A_2^(T) * Lambda + A_1 ^(T) * A_1 <= C^(T)

mu >=0

------------------------------------------

Question:

If I call the "linprog" command to solve the primal , the lagrange multipliers obtained are not satisfying the inequality constraint of the dual.

Also, the lagrange multipliers of Primal ane not same as the optimal values of the dual.

Please find attached the example with which I am working and finding the above mentioned problem.

%---------Primal------------------


f = [98,97,85,97,93,93,0,2,5,10,0,2,5,10,0,2,5,10,0,2,5,10,0,2,5,10,0,2,5,10,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,2,5,10,12,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15,5,10,12,15];
A_eq = 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B_eq = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1];

A_ineq = 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B_ineq = [74;72;82;86;88;64;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0];

lb = zeros(1, length(f));



[values,rmp,exitflag,OUTPUT, lambda] = linprog(f,A_ineq,B_ineq,A_eq,B_eq,lb);
     

%---------------------Dual-----------------------
f_D = [B_ineq' -B_eq'];

A_D_ineq = [ -A_ineq' A_eq'];

B_D_ineq = f';

[values_D,rmp_D,exitflagD,OUTPUTD, lambda_D] = linprog(f_D,A_D_ineq,B_D_ineq,[],[],zeros(90,1));
 

Answer 1

[Shreshth]

Hello Devyani,

I can understand that you are facing issues with the primal and dual formulations for your optimisation process.

On analysing your primal and dual statements the following issues can cause the discrepancy in the results that you are getting:

1. Dual Constraint Typo: In your dual problem statement, you have A_1^T * A_1 which seems to be a typo. Based on the standard form of a dual problem, it should be A_2^T \lambda + A_1^T \mu \leq C.

2. Variable Bounds: For the dual problem, you correctly set the bounds for the dual variables to be non-negative. However, make sure that the bounds are applied to the correct variables. The dual variables corresponding to the equality constraints ((\lambda)) do not necessarily have to be non-negative, but the dual variables corresponding to the inequality constraints ((\mu)) should be.

3. Implementation in ‘linprog’: Your implementation in ‘linprog’ for both primal and dual seems to have correctly identified the bounds for the primal variables and the dual variables associated with inequality constraints. However, ensure that the coefficients and constraints are correctly placed in the ‘linprog’ function calls.

By implementing the above corrections you should get the correct optimization results.

For further clarification on the use of ‘linprog’, you can refer to the below MathWorks documentation:

www.mathworks.com/help/optim/ug/linprog.html

Hope it helps.