How can I calculate the pseudo inverse of a matrix of a set of linear equations with nonlinear constraints

Hello, I would like to calculate the x values in a set of linear equations Cx=d subject to two constraints: 1. linear constraints for example 0<x1<10, .... 2. nonlinear constraints, (x1/x3)=(x2/x4),... Any help or suggestion ?? Thanks in advance

Answer 1

[Walter Roberson]

If there were bounds constraints or linear constraints but no nonlinear constraints the you could use http://www.mathworks.com/help/optim/ug/lsqlin.html

However, when you add in the nonlinear constraint, you need to switch to fmincon

[Ahmed Ragib Kafi]

Thanks alot Mr. Walter, I will try to formulate the problem using fmincon

Answer 2

[John D'Errico]

They are no longer a system of linear equations when that nonlinear equality constraint is added. The pseudo-inverse is not defined. For that matter, even the bound constraints make it no longer valid.

It appears that you have a problem that must be solved by a tool like fmincon, given the nonlinear constraint. Formulate the objective function as the sum of squares of the residuals of those linear equations.

[Ahmed Ragib Kafi]

Dear Mr. John, Thanks for your answer, but as I know there can be a linear system with nonlinear constraints. For example you can imagine any linear set of equations but it need that the ratio of x1/x3 is equal to the ration of x2/x4 and it is nonlinear contraints. Any way, I will try solving using fmincon as per your suggestion. Thanks again for your help.

[Walter Roberson]

Ratios can be coded by reducing by one variable. For example,

function r = myobjective(x)
   x1 = x(1);
   x2 = x(2);
   x3 = x(3);
   x13ratio = x1/x3;
   x4 = x2 / x13ratio;