The following nonsmooth optimization problem is considered: \[ \min\{J(u)+\phi(u): u\in H^1_0(\Omega)\}, \] where \[ J(u)=\textstyle{{1\over 2}} a(u,u)-\ell(u),\;\phi(u)=\displaystyle{\int_\Omega}\Phi(u(x))dx, \] \(a(\cdot,\cdot)\) is a continuous, symmetric and \(H^1_0(\Omega)\)-elliptic bilinear form, \(\ell\in H^{-1}(\Omega)\) and \(\Phi\) is a piecewise quadratic convex function. Using piecewise linear finite elements the initial problem is reduced to a discrete elliptic variational inequality. To solve the discrete problem globally convergent multigrid methods are derived. Let \(2^{-j}\) be the order of diameter of triangles in the partition of \(\Omega\) and \(u_j\) the solution of the discrete variational inequality. The author gives estimates for the convergence of the multigrid approximations \(u^\nu_j\) to \(u_j\) of the form \[ |u^{\nu+1}_j-u_j|\leq(1-c(j+1)^{-3})|u^\nu_j-u_j|. \] Numerical experiments with the proposed methods are also discussed. For the quadratic obstacle problem \[ \min\{J(u): u\in H^1_0(\Omega); u(x)\leq\varphi(x), x\in\Omega\} \] similar results have been presented in the first part of the paper [Numer. Math. 69, No. 2, 167-184 (1994; Zbl 0817.65051)].