CVX cannot solve:
maximize (X1*inv_pos(1+X2))
s.t. convex constraints <= 0
where X1 and X2 are positive variables.
With Hessian Matrix, we find that X1*inv_pos(1+X2) is joint concave with regarding to Y1 and Y2.
Below are the error messages in Matlab:
Error .* (line 173)
Disciplined convex programming error:
Cannot perform the operation: {real affine} .* {convex}Error * (line 36)
z = feval( oper, x, y );Error ObjFuc_aim (line 35)
maximize ( -Y1*inv_pos(1+Y2));
Thanks ahead!!!
Your CVX code models
X1/max(1+X2,0)
That is neither convex nor concave.
For X2 > -1, this simplifies to X1/(1+X2). Now restrict this to X1 and X2 are positive. Evaluating the Hessian of this at X1 = X2 = 1 produces one positive eigenvalue and one negative eigenvalue. So neither convex nor concave even on this restricted domain.
Thanks so much for the explanation, Sir ! 
What shall be do to convert or approximate X1/(1+X2) into a concave form, such that
maximize X1/(1+X2) if we restrict this to X1 and X2 are positive…
in the CVX optimization environment?
That function is not concave. The subject of what would be a good approximation for your purposes is outside the scope of this forum. If that is really the function you wish to maximize, you may be better off using a non-convex solver.