Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 7Static optimization

1 INTRODUCTION

Static optimization theory is concerned with finding those points (if any) at which a real‐valued function ϕ, defined on a subset S of ℝⁿ, has a minimum or a maximum. Two types of problems will be investigated in this chapter:

1. Unconstrained optimization (Sections 7.2–7.10) is concerned with the problem
   
   where the point at which the extremum occurs is an interior point of S.
2. Optimization subject to constraints (Sections 7.11–7.16) is concerned with the problem of optimizing ϕ subject to m nonlinear equality constraints, say g₁(x) = 0, …, g_m(x) = 0. Letting g = (g₁, g₂, …, g_m)′ and
   
   the problem can be written as
   
   or, equivalently, as
   

We shall not deal with inequality constraints.

2 UNCONSTRAINED OPTIMIZATION

In Sections 7.2–7.10, we wish to show how the one‐dimensional theory of maxima and minima of differentiable functions generalizes to functions of more than one variable. We start with some definitions.

Let ϕ : S → ℝ be a real‐valued function defined on a set S in ℝⁿ, and let c be a point of S. We say that ϕ has a local minimum at c if ...