ccer05研究生数理经济学讲义2

Lecture Notes 1 & 2

Optimizing Theory

Chapter 1 Unconstrained Optimization

Theorem 1.1 Suppose :n f R R → is differentiable and n x R ∈ is a local maximizer or local minimizer of f , then ()0f x ∇=.

Theorem 1.2 Suppose :n f R R →is twice continuously differentiable and (')0f x ∇=.

1. If 'x is a local maximizer, then matrix 2(')D f x is negative semi-definite.

2. If 2(')D f x is negative definite, then x’ is a local maximizer.

Theorem 1.3 Suppose function :n f R R → is concave. If (')0f x ∇=, then x’ is a global maximizer of f .

Theorem 1.4 (Envelope Theorem) Consider unconstrained optimization problem:

n x R

Max ∈(,)f x a , where m a R ∈ is the vector of parameters. The function (,)f x a is continuous and differentiable. Suppose the solution point is **()x x a =. Denote ()((),)V a f x a a =. Then we must have

*()(,)()

j j V a f x a x x a a a ∂∂==∂∂ , 1,,j m =".

Chapter 2 Constrained optimization

2.1 The general structure

The variables of the problem will be considered to be in the form of a vector in n R . In addition to this vector, x , we have:

a. a feasible set K. Only x K ∈ is to be taken into account in the problem.

b. A continuous objective function, ()f x , whose value for x K ∈ is to be optimized.

Thus we can state a typical maximizing problem in formal terms as

Find *x K ∈ such that *()()f x f x ≥, for all x K ∈.

If such an *x exists, the problem has a weak global maximum-weak because it

satisfies the weak inequality, global because the inequality is satisfied for all x K ∈. A global optimum should not be confused with an unconstrained optimum. The latter implies that n K R =. We would have a strong maximum if we could find *x such that *()()f x f x >, for all x K ∈.

A weak optimum is equivalent to a non-unique optimum point since any x satisfying *()()f x f x = is also an optimum point. A strong global optimum implies a unique optimum.

If we reverse the inequalities we obtain a minimum, weak or strong as the case may be. A minimum for ()f x implies a maximum for [-()f x ]. The value *x is often called simply the solution of the optimum problem. To avoid confusion with other claimants for the same name in many economic models, we shall usually call it the optimal solution or optimum point.

Most calculus techniques cannot solve the problem as set out above, but can only solve a problem of the following kind:

Find *x K ∈ such that *()()f x f x ≥, for all ()x N K ∈∩,

where N is a neighborhood of *x .

Such a point is a weak local maximum. We can have a weak or strong local maximum, and weak, or strong local minimum.

It is obvious that a global optimum must also be a local optimum. Nevertheless, a local optimum is not necessarily global. Our interest is primarily in the global optimum. Thus, we are interested in conditions on the structure of the problem that will guarantee that a local optimum is also global. If such conditions are not satisfied, we need to adopt ad hoc procedures to locate the global optimum.

2.2 Constraints and the Feasible Set

The feasible set, over which the variables are permitted to range, may be defined in any suitable way. In the case of discrete variables, the feasible set may even be described by enumeration. Typically, however, the feasible set will be defined by equalities or inequalities involving relationships between the variables.

The boundaries of the feasible set are crucial in optimizing problems. In all our discussions of optimizing, it will be assumed that the constraints are such that as to give a closed feasible set. Otherwise the problem is usually without a solution. This is normally guaranteed by ensuring that there are no strict inequalities in any of the constrains and that the constraints are continuous.