Continuing from Part I, we discuss problems with inequality constraints which are important in economics, finance, engineering and science applications. We will discuss the linear complementarity problem and a class of techniques known as semi-smooth Newton methods applicable to solving differential equations with inequality constraints. For elliptic (or parabolic) PDEs the appropriate variational formulation through a minimization problem posed in Hilbert spaces leads to variational inequalities. We will show that their numerical solution in a finite dimensional space can be seen as the linear complementarity problem. Then we will describe a particular nonlinear evolution problem under constraints where we can take advantage of inequality-constrained optimization methods.