As flipper points out, the problem is that the hard maximum is not differentiable; that is, the differential of max (x, y) is undefined when x == y.  Most numerical approximation techniques require a smooth, differentiable function, so this is bad.<p>A related problem is that of bounding.  Suppose I want to minimize the parameters in a set of nonlinear equations; I can numerically differentiate the equations to get the gradient and the Hamiltonian and then I can minimize that.  But I may want to impose additional criteria, like x &#62; 0.  I could just say the function goes to some preposterously high number when x &#60;= 0, but then we have this hard corner problem again.  Instead if you use a continuous function like a logarithm to impose your bound, it affects the solution space minimally, makes for a solution that will not wander outside your bounds, and is differentiable.