fmincon nonlinear equality constraint

Hi,

I am trying to implement the following equality constraint for using it in fmincon:

where

and TargetVariance is a fixed numeric value. The size of w is 1xN.

This is a minimal example of my code.

%Input
Mu = [.1, .15, .12];
Cov = [.3, .1, .2;
    .1, .5, .1;
    .2, .1, .7];

% Prepare vector of ones.
N = numel(Mu);
one = ones(1,N);

% Objective function.
NegPFMu = @(w) -Mu * w';

% Set equal weights as initial point.
w0 = one * (1/N);

% Sum of weights constraint.
Aeq = one;
beq = 1;

% Shortselling constraint.
lb = zeros(1,N);

% Minimize.
wopt = fmincon(NegPFMu, w0, [], [], Aeq, beq, lb);

Can you help me with the constraint?

I was thinking about something like the following but I don't how to call that as my nonlcon then.

function [c, ceq] = mycon(w, Cov, TargetVariance)
PortfolioVariance = @(w) w*Cov*w'
ceq = PortfolioVariance(w) - Targetvariance;
c = [];
end

Could you help me please?

Answer 1

[Matt J]

wopt = fmincon(NegPFMu, w0, [], [], Aeq, beq, lb,[], @(w) mycon(w, Cov, TargetVariance));

function [c, ceq] = mycon(w, Cov, TargetVariance)
 ceq = w*Cov*w'- Targetvariance;
 c = [];
end

[Clemens Gersch]

works perfectly, thanks. Would there also be a possibility to solve that with an anonymous function?

[Matt J]

@(w) mycon(w, Cov, TargetVariance) is an anonymous function.

However, if you mean you would like to avoid creating the local function mycon, then you could do this,

wopt = fmincon(NegPFMu, w0, [], [], Aeq, beq, lb,[],...
              @(w) deal([], w*Cov*w'- Targetvariance) );

[Clemens Gersch]

That was exactly what I was looking for. Thank you!