Minimizing a function in matlab

I would like to minimize w'Hw, with respect to w, where w is a vector, and H is matrix.

And with the following constraint, |w1|+|w开发者_StackOverflow中文版2|+|w3| < 3, ie. the l1 norm of the weights vector is less that 3.

How can I do this in matlab?

Thanks

You're trying to solve a quadratic minimization problem with linear constraints (also known as quadratic programming).

Do you know anything about your matrix H -- in particular, is it positive semidefinite? I would really expect this to be the case, since this is usual for the problem domains in which quadratic programming problems usually crop up.

If H really is positive semidefinite, and your only constraint is |w1|+|w2|+|w3| < 3, then, as Richie Cotton has already pointed out, the minimum is trivially at w=0. Maybe you have some additional constraints?

If you do have additional constraints, but H is still positive semidefinite, there are existing efficient solvers for this class of problem. In MATLAB, take a look at quadprog.

You'll have to reformulate your single nonlinear constraint |w1|+|w2|+|w3| < 3 as a series of linear constraints.

In the one-dimensional case, the constraint |w1| < 1 turns into two linear constraints:

 w1 < 1
-w1 < 1.

In the two-dimensional case, the constraint |w1| + |w2| < 1 turns into four linear constraints:

 w1+w2 < 1
 w1-w2 < 1
-w1+w2 < 1
-w1-w2 < 1

I'll leave the extension to three dimensions to you.

you need to use the optimization toolbox, specifically fmincon:

use fun to establish w'Hw, and you want c(eq) = (|w1|+|w2|+|w3| - 3) <0 which you set with nonlcon (in the documentation).

I'd suggest you look at the fminsearch function in the matlab documentation.

Rasman, below is the fmincon code I am using:

    function PortfolioWeights = GMVPC1Type2(SCM)
    w0 = [0.1 0.1 0.1 0.1 0.1];
    n = length(w0);

    options = optimset('Display','final-detailed');
    PortfolioWeights = fmincon(@myobj2,w0,[],[],ones(1,n),1,[],[],@myconstraint1,options)

        function f = myobj2(w)
        w = [w(1);w(2);w(3);w(4);w(5)];
        f = w'*SCM*w;
        end
    end
    -----------------------------------------------------------------------------
    function [c ceq] = myconstraint1(w)

    c = abs(w(1))+abs(w(2))+abs(w(3))+abs(w(4))+abs(w(5))-1
    ceq = [];

    end
    ------------------------------------------------------------------------------

I added in options = optimset('Display','final-detailed'); as you suggested. I get the following message:

Optimization stopped because the predicted change in the objective function,
6.115031e-009, is less than options.TolFun = 1.000000e-006, and the maximum constraint
violation, 0.000000e+000, is less than options.TolCon = 1.000000e-006.

Optimization Metric                                                 Options
abs(steplength*directional derivative) =  6.12e-009        TolFun =  1e-006 (default)
max(constraint violation) =  0.00e+000                     TolCon =  1e-006 (default)

Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
                                 1

PortfolioWeights =

    0.2000    0.2000    0.2000    0.2000    0.2000

The matrix I am using is:

0.000257165759136336    8.48196702102889e-05    9.27141501220362e-05    0.000111360154790061    0.000155196440517440
8.48196702102889e-05    0.000277377166669392    0.000101880007672550    0.000107375764193076    0.000117042329431538
9.27141501220362e-05    0.000101880007672550    0.000300697293925817    0.000112004860252653    0.000134354417344316
0.000111360154790061    0.000107375764193076    0.000112004860252653    0.000311028738698100    0.000147296211557256
0.000155196440517440    0.000117042329431538    0.000134354417344316    0.000147296211557256    0.000376418027192374