Solving Cardinality Constrained Portfolio Optimization Problem in Convex.jl

[Ayush Pandey]

Hello,

The normal Portfolio Optimization problem is when we are given a portfolio of assets and the only restriction is that the sum of weights assigned to assets is equal to 1(and of course 0<=wi<=1).

But when we put the limit on number of assets to be invested, this problem becomes a Cardinality Constrained Portfolio Optimization Problem(for e.g. I have portfolio of 10 assets but at a time I can only invest in 5 assets).

In such case we introduce a binary variable yi = {0,1}

yi = 0 when no investment is made in asset i and yi = 1 when the investment is made in asset i.

Now, this problem becomes a Mixed Integer Quadratic Programming Problem.

I was trying to solve it using Convex.jl but couldn't succeed. I would be great if someone could help :)

The problem is in the following lines of the code

#Defining variables

w = Variable(10);

y = Variable(10, :Bin);

#Defining constraints

c1 = sum(w) == 1;

c2 = sum(y) = 5;

c3 = w_lower <= w*y'; 

Constraint:

<= constraint

lhs: 0

rhs: AbstractExpr with

head: *

size: (5, 5)

sign: Convex.NoSign()

vexity: Convex.NotDcp()

vexity: Convex.NotDcp()

c4 = w*y' <= w_upper;

Yours Sincerely,

Ayush Pandey        
LinkedIn Profile   
GitHub

[Miles Lubin]

If you double-post, please give a link (https://groups.google.com/d/msg/julia-users/B88L_byCcy0/zM-KyWsqBgAJ) to avoid duplicate effort on the part of those answering.

[Ayush Pandey]

Thank you Miles ! I would be taking this into consideration from next time onwards . 

Yours Sincerely,

Ayush Pandey        
LinkedIn Profile   
GitHub

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[Madeleine Udell]

You've expressed the last constraint by multiplying two variables, which is not a DCP compliant operation. In fact, I'm not entirely sure why you want an outer product constraint; I'd expect you to want to constrain the sum of the weights w

w_lower <= sum(w)

and also enforce that no weight can be positive unless the corresponding element of y is 1, which you can enforce using a big-M relaxation:

w[i] <= M*y[i]

where M is some suitably large constant.

And then you can do the same for the upper bounds.