We start with an overview of methods for optimization with inequality constraints which are important in economics, finance, engineering and science applications. The fundamental mathematics results behind these methods are associated with the Karush-Kuhn-Tucker conditions. Depending on the properties of the objective function and of the constraints the methods fall in the category of solvers for linear and nonlinear programming and these include well-known methods such as the simplex method and interior point methods. We will discuss a particular class of problems known as a linear complementarity problem and a class of techniques known as semi-smooth Newton methods.
Next we describe a technique for 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. In Part II we will describe a particular nonlinear evolution problem under constraints where we can take advantage of inequality-constrained optimization methods.