IBM ILOG CPLEX 12 with matlab, specify custom objective function

Question

I am very new to using IBM CPLEX , and am using CPLEX with Matlab. I was wondering how to compile a custom objective function in CPLEX using Matlab. The objective function is as follows: Optimization problem to replicate in CPLEX using MATLAB

Here aj is a column vector of size 36000 X 1 and A is a sparse matrix of size 36000 x 4503. wj is a column vector of size 4503 x 1 of optimization variables. Until now this is a simple cplexlsqnonneglin if we include the wj >=0 constraint. But I would also like to include the two other sum terms with 'beta' and lambda and the wjj = 0 constraint. Any help in recreating this optimization problem in CPLEX would be much appreciated.

Thanks in advance!

codehippo · Accepted Answer · 2014-02-19 19:49:56Z

When you add the other terms into the objective, your problem becomes a general quadratic program. Since wj >= 0 we have that ||wj||_1 = e'*wj. So we can write your problem as:

 minimize  0.5*(aj - A*wj)'*(aj - A*wj) + 0.5*beta*wj'*wj + lambda*e'*w
   wj
 subject to  wj >= 0, wj(j) = 0

After pushing the quadratic terms in the objective together we have the following QP:

minimize  0.5*aj'*aj -aj'*A*wj + 0.5*wj'*(A'*A + beta*I)*w + lambda*e'*w
 wj
subject to  wj >= 0, w(j) = 0

I can't help you with CPLEX. But you can solve this problem with Gurobi in MATLAB using the following code

m = 36000;
n = 4503;
A  = sprand(m, n, .01);
aj = rand(m, 1);
lambda = 0.1;
beta   = 0.4;

j = 300;

model.objcon = 0.5*aj'*aj;
model.obj    = -aj'*A + lambda*ones(1,n);
model.A      =  sparse(1, n);
model.sense  = '=';
model.rhs    = 0;
model.Q      = 0.5*(A'*A + beta*speye(n));
model.vtype  = 'C';
model.lb     = zeros(n, 1);
model.ub     = inf(n,1);
model.ub(j)  = 0; % set 0 <= wj(j) <= 0

params.outputflag = 1;
result = gurobi(model, params);

if strcmp(result.status, 'OPTIMAL')
    wj = result.x(1:n);
end

For more details see the documentation on Gurobi's MATLAB interface: http://www.gurobi.com/documentation/5.6/reference-manual/matlab_gurobi

Note you may want to create variables and constraints to avoid forming A'*A + beta*I in the objective. For example you could create a new variable r and a constraint r = A*wj. Then the objective wj'*(A'*A + beta*I) wj would become r'*r + beta*wj'*wj. This may help with the numerics.

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