Optimization第四讲

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xi
x max i
Week 2
Optimization with equality constraints on variables.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
This week: Concave Programming
Optimization with inequality constraints on variables.
L(x1, x2, ) f (x1, x2 ) + g(x1, x2 )
Thus, we want this as large as possible
And this as small as possible
9
Now: maximization with an inequality constraint.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
Some rough intuition for the method we are going to use: Again write down the Lagrangean:
L(x1, x2, ) f (x1, x2 ) + g(x1, x2 )
10
How to minimize a function with a restriction?
The global maximum x* of a function f(x) with x ≥ 0 has either one of the following properties: • f ’(x*) = 0 • x* = 0 This implies that the following must hold: • f ’(x*) ≤ 0 and x*f ’(x*) = 0
Some rough intuition for the method we are going to use: Again write down the Lagrangean:
L(x1, x2, ) f (x1, x2 ) + g(x1, x2 )
Ultimately, we want to maximize this. Thus we want to have it as close to the Lagrangean as possible.
Faculty of Economics
Optimization Lecture 4
Marco Haan March 7, 2005
Week 1
Optimization with direct restrictions on variables.
max f (x1, x2 )
s.t.
xi
x min i
11
Thus
The global minimum x* of a function f(x) with x ≥ 0 has either one of the following properties: • f ’(x*) = 0 • x* = 0
Hence, we want to maximize the Lagrangean with respect to the x’s, and we want to minimize it with respect to lambda. We are looking for a saddle point of the Lagrangean.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
2
From week 1: a global optimum may also involve a corner solution...
xmax
x
3
xmin
xmax
x
4
Thus (again from lecture 1)
The global maximum x* of a function f(x) on some interval [xmin,xmax] has either one of the following properties: • f ’(x*) = 0 • x* = xmin • x* = xmax This implies that one or both of the following must hold: • f ’(x*) ≤ 0 and (x* – xmin)f ’(x*) = 0 • f ’(x*) ≥ 0 and (xmax – x*)f ’(x*) = 0
This implies that the following must hold: • f ’(x*) ≤ 0 and x*f ’(x*) = 0
Such constraints have to be satisfied in almost all economic problems.
6
Recall: maximization with an equality constraint.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
L(x1, x2, ) f (x1, x2 ) + g(x1, x2 )
7
Now: maximization with an inequality constraint.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
8
Now: maximization with an inequality constraint.
max f (x1, x2 ) s.t. g(x1, x2 ) 0
Some rough intuition for the method we are going to use: Again write down the Lagrangean:
5
Now suppose we only have the
constraint x ≥ 0.
The global maximum x* of a function f(x) with x ≥ 0 has either one of the following properties: • f ’(x*) = 0 • x* = 0