Originally posted by: Mark L. Stone

I am trying to understand the convexity requirements on the objective function and constraints for the MATLAB toolbox function cplexqcp in CPLEX 12.6.0.1.  Specifically, what are the requirements for the matrix H for the quadratic objective term and (each) matrix Q for the quadratic constraint term(s) as to whether they need be positive semi-definite?  I thought that at least H need not be positive semi-definite if solutiontarget is set to 2, but this does not appear to be the case.

It appears to be the case that if there is a quadratic constraint (Q is not empty), then H needs to be positive semi-definite, regardless of the value of solutiontarget.  Is this correct?  Is there some other parameter or way of formulating the problem which allows a non positive semi-definite H (presuming there is a non-empty Q) and even perhaps allows non positive semi-definite Q?

Consider the following toy problem with non-convex H and convex Q

>> cplex_options = cplexoptimset('solutiontarget',2);

>> [x,fval,exitflag,output]=cplexqcp([1 0;0 -1],[2 ;3],[],[],[],[],zeros(2,1),[1 0;0 1],1,[-1;-1],[1;;1],[0;0],cplex_options)
Error using cplexqcp (line 649)
CPLEX Error  5002: Q in %s is not positive semi-definite.

Now change H to be positive definite, and the problem is solved (no surprise here)

>> [x,fval,exitflag,output]=cplexqcp([1 0;0 1],[2 ;3],[],[],[],[],zeros(2,1),[1 0;0 1],1,[-1;-1],[1;;1],[0;0],cplex_options)
x =
   -0.5547
   -0.8321
fval =
   -3.1056
exitflag =
     1
output =
          cplexstatus: 1
    cplexstatusstring: 'optimal'
           iterations: 0
            algorithm: 4
                 time: 0
              message: 'Function converged to a solution x.'

Or go back to the original problem with non-convex H and eliminate Q (so it now is a QP) and is solved (without resorting to cplexqp)

>> [x,fval,exitflag,output]=cplexqcp([1 0;0 -1],[2 ;3],[],[],[],[],[],[],[],[-1;-1],[1;1],[0;0],cplex_options)
x =
    -1
    -1
fval =
    -5
exitflag =
     1
output =
          cplexstatus: 1
    cplexstatusstring: 'optimal'
           iterations: 0
            algorithm: 4
                 time: 0
              message: 'Function converged to a solution x.'

Even eliminating the quadratic term in the objective doesn't allow non-convex constraint:

>> [x,fval,exitflag,output]=cplexqcp([],[2 ;3],[],[],[],[],zeros(2,1),[1 0;0 -1],1,[-1;-1],[1;;1],[0;0],cplex_options)
Error using cplexqcp (line 649)
CPLEX Error  5002: Q in %s is not positive semi-definite.

I suggest the documentation be beefed up to make these matters clearer.  Thanks.

Originally posted by: Mark L. Stone

Bump.  Thanks.

The solution target parameter applies only to QP and MIQP, see the reference documentation of CPX_PARAM_SOLUTIONTARGET.

CPLEX currently does not implement an algorithm that can solve problems that contain quadratic constraints with non positive semi-definite Q matrices.

Originally posted by: Mark L. Stone

Thanks.  And to add to/clarify your point, cplexqcp does not allow non-positive semi-definite H even if Q is positive semi-definite.