It is not often that optimization problems have unconstrained solutions. Typically, some or all of the constraints will matter. Over this chapter and the next, we examine necessary conditions for optima in such a context.
If the constraints do bite at an optimum x, it is imperative, in order to characterize the behavior of the objective function f around x, to have some knowledge of what the constraint set looks like in a neighborhood of x. Thus, a first step in the analysis of constrained optimization problems is to require some additional structure of the constraint set D, beyond just that it be some subset of ℝ n . The structure that we shall require, and the order in which our analysis will proceed, is the subject of the following section.
Constrained Optimization Problems
It is assumed in the sequel that the constraint set D has the form
where U ⊂ ℝ n is open, g: ℝ n → ℝ k , and h: ℝ n → ℝ l . We will refer to the functions g = (g 1, …, g k ) as equality constraints, and to the functions h = (h 1,…, h l ) as inequality constraints.
This specification for the constraint set D is very general, much more so than might appear at first sight. Many problems of interest in economic theory can be written in this form, including all of the examples outlined in Section 2.3.
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