Kuhn-tucker conditions matlab

I am solving a maximisation problem subject the the restriction that xi<=delta*TR(x1,x2) for i=1,2. There are two lagrangians associated with my problem:

    L1=U1(x1,x2)+lambda1*(delta*TR(x1,x2)-x1)

    L2=U2(x1,x2)+lambda2*(delta*TR(x1,x2)-x2)

I need to solve for the vector (x1,x2)

I am investigating two cases:

Case 1: The restriction is binding in both lagrangians i.e. xi=delta*TR(x1,x2) for i=1,2. The Kuhn-Tucker conditions for this require that lambda1>0 and lambda2>0. Thus, I have four FOCS's:

 (i) dL1/dx1 
 (ii) dL1/dlambda1 
 (iii) dL2/dx2 
 (iv) dL2/dlambda2

I used FSOLVE with a system of 4 equations and got the result that I was looking for. So this case works fine.

Case (ii)

The restriction xi<=delta*TR(x1,x2) is only binding in the first lagrangian (L1). The Kuhn-Tucker conditions for this require that lambda1>0 and lambda2=0.

My question is: How do I incorporate lambda1>0 and lambda2=0 into my system of equations? Do i need to drop one of the first order conditions? Or do I need to drop the lambda2(delta*TR(x1,x2)-x2) term in the second lagrangian?

I realise that this is more of a maths question than Matlab question, but I would greatly appreciate any help!

Thanks, Kai

Answer 1

[Matt J]

You can use LSQNONLIN, which is very similar to FSOLVE, but allows you to incorporate bounds like lambda1>=0.

On the other hand, even if this is homework, if you are allowed to use solvers like FSOLVE, I wonder why you wouldn't be allowed to just use FMINCON to solve the whole problem for you, instead of manually attacking the KKT conditions.

[Kai]

When using FSOLVE I have to put the FOC in a function file and then I use the following code in my script file.

      clear
        b=1;
        n=4.5;
        c=1;
        e=0;
        m=6;
        f=5;

        d=0.05:0.05:0.5;
        xyl=zeros(4,max(size(d)));
         v0=[1;1;10;4];
        options =optimset('Display','iter','MaxFunEvals', 1000,'TolFun',1e-5);
        %'TolFun',1e-5,'MaxFunEvals',2000,'MaxIter',2000);
        %'FunValCheck', 'on', 'PlotFcns', @optimplotfval
        for i=1:max(size(d))

[xylf,FVAL,EXITFLAG,OUTPUT,JACOB] = FSOLVE(@(v)      myfunc3(v,d(i),f,m,n,b,c,e),v0,options);
         xyl(:,i) = xylf ;
        end

How can I change this to use LSQNONLIN? And where do I insert code for the restrictions that I need on x1 and x2.

Cheers, Kai

[Matt J]

It is very similar to FSOLVE and should be obvious from the documentation for LSQNONLIN

 >> doc lsqnonlin