Maximixing Sharpe Ratio Matlab using Cplex

Question

I am working on a problem where I need to find a rebalanced portfolio based on maximizing the Sharpe ratio using IBM Cplex tool in Matlab. From what I think, I have added all conditions and constraints, but when I try to solve it, I get integer infeasible.

Here is the code:

% mu and covariance matrix Q is given
% No of stocks
n = 20;
% Risk free rate
r = 0.045; 

% Optimization problem data
lb = zeros(n,1);
ub = inf*ones(n,1);
A = [(mu' - r) 0; ones(1,n) -1];
lhs = [1;0];
rhs = [1;0];
b  = 1;

% Define continuous and binary variables
variabtype = [char(ones([1 n])*('C')) char(ones([1 1])*('B'))];

% Compute minimum variance portfolio
cplex1 = Cplex('minYK');
% cplex1.Model.sense = 'minimize';
cplex1.addCols(zeros(n+1,1), [], [lb; 0], [ub; inf], variabtype);
cplex1.addRows(lhs, A, rhs);
cplex1.Param.qpmethod.Cur = 6; % concurrent algorithm
cplex1.Param.barrier.crossover.Cur = 1; % enable crossover
cplex1.Model.Q = [2*Q zeros(n,1)];

Answer 1

The Sharpe Ratio problem by itself is not a Quadratic Program, but it can be converted into a quadratic program (Link Here). Then follow the steps, make sure the dimensions of the mean returns is coherent with your program (if its row or column, trasnspose accordingly).

Q=[Q;zeros(1,n)];
Q=[Q zeros(n+1,1)];
%% CPLEX Solution
% Initialize the CPLEX object
cplex3 = Cplex('max_Sharpe');
cplex3.Model.sense = 'minimize';

% Optimization problem data
lb = zeros(n+1,1);
ub = inf*ones(n+1,1);
%
A = [[(  mu- r_rf/252) 0];[ones(1,n) -1]; [eye(n,n) -1*ones(n,1)]];
y = ones(n+1,1);
lhs = [1; 0; -inf*ones(n,1)];
rhs = [1; 0; zeros(n,1)];

y = ones(n+1,1);

% Add objective function and bounds on variables to CPLEX model
cplex3.addCols(zeros(n+1,1), [], lb, ub);
% Add constraints to CPLEX model
cplex3.addRows(lhs, A, rhs);

% Add quadratic part of objective function to CPLEX model
cplex3.Model.Q = 2*Q;
% Set CPLEX parameters
cplex3.Param.qpmethod.Cur = 6;
cplex3.Param.barrier.crossover.Cur = 1;% Concurrent algorithm
% Optimize the problem
cplex3.DisplayFunc = [];
cplex3.solve();

y = cplex3.Solution.x(1:n);
k = cplex3.Solution.x(n+1);

w_maxsharpe = y/k;