fmincon: nonliner constraint

I am using fmincon to find the solution (weights of each assets) that optimizing my investment portfolio.How can I make each of the weight less than 1, instead of less than or equal to 1, but the sum of weights equals 1? The codes are:

w = zeros (4,1)
cov = xlsread('gupiao.xls','equal weights','O13:R16')
sigmaP = w'*cov*w
delta_x = xlsread('gupiao.xls','sheet3','A503:D503')
delta_P = delta_x*w
fun = @(w)-[delta_x*w-0.5*[0.7*w'*cov*w+1/1000000*(0.1*36843+0.2*48772)]^2]
Aeq = ones(1,4)
beq = 1
A = []
b = []
lb = []
ub=ones(size(w))
w = fmincon(fun,w,A,b,Aeq,beq,lb,ub)

It gives me the results:

w =

      1.0000
      1.0000
      1.0000
     -2.0000

But, these results violate my constraint that each of the weight is less than 1. Could you please tell me how can I solve this problem?

Answer 1

[Walter Roberson]

A = ones(1,4);
b = 1-eps;
Aeq = [];
beq = [];

This forces the sum to be less than or equal to 1-eps, which should be equivalent to being less than 1.

[Xiangnan Liu]

Hi~ Thank you for replying me! I tried your codes, and it gives me w =

    1.0000
    1.0000
    1.0000
  -48.8791

I appologize for not stating my problem clearly, it should be: make each of the weight less than 1, instead of less than or equal to 1, but the sum of weights equals 1.

If so, could you please tell me what constraint should I use?

[Walter Roberson]

Ah, that is a different problem. To do that use your original setting but set

ub = ones(1,4)-eps;

[Xiangnan Liu]

I rewrote my codes applying your suggestions, it works! Thanks a lot！

Answer 2

[Stephan]

Hi,

i assume, negative values are not allowed. A share of -2 of the stock represented by variable 4 makes no sense.

If that is correct, use

lb=zeros(1,4)

Together with your Aeq and beq constraint this should work.

Best regards

Stephan

[Xiangnan Liu]

Hello! Thanks a lot for your reply! Actually, in my portfolio, negative weights are allowed, which means I am short selling. Could you please tell me what constraint should I use then?

Answer 3

[Matt J]

If the unconstrained minimum lies at w(i)=1, then there is no sense pursuing a constraint like w(i)<1. It would be like minimizing f(z)=(z-1)^2 with the constraint that z<1. The minimum cannot be attained.