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        
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