Let \(H\) be a Hilbert space, \(f: H\to H\) a continuously Gâteaux differentiable function and \(g: H\to H\) a multivalued mapping. For the numerical solution of the problem \(f(u)+ g(u)\ni 0\), the generalized Newton method \(u_{n+1}= (f'(u_ n)+ g)^{-1}(f'(u_ n)[u_ n]- f(u_ n))\) is considered. For the case when \(g\) is the subdifferential mapping of a lower semicontinuous proper convex function, a quadratic convergence result is proved. A problem of this type appears when formulating the usual optimality condition for the minimization of the sum of a twice-differentiable convex function and a lower semicontinuous proper convex function. A numerical example illustrating the method is also presented.