Unconstrained Optimization(Maximization)

• Ingredients

x choice variable (control v., instrument, decision v.)

X opportunity set: x ∈ X

unconstrained problems: X = Rconstrained problems: X ⊂ R

y objective (related to x)

objective function: y = f (x)

• choose x such as to max f (x)

& observe x ∈ X

• solution: x∗ ∈ X f (x∗) ≥ f (x) for all x ∈ X

x∗ “optimizer” or “maximizer”

• f ′(x) = 0 necessary condition for interior optimum

c© Ronald Wendner Summary M4M-II-1 v1.3

• points to note

– necessary vs. sufficient

– inner vs. boundary solution

– existence

• Weierstrass existence theorem

f (x) continuous

X nonempty, closed, bounded

– sufficient!

• local vs. global solutions

– global solution: x∗ ∈ X f (x∗) ≥ f (x) for all

x ∈ X

– local solution: x∗ ∈ X f (x∗) ≥ f (x) for all

x ∈ Nε(x)

– every global solution = local solution.

– sufficient condition(s) for x∗ to be global sol.?

c© Ronald Wendner Summary M4M-II-2 v1.3

• Local-global theorem

– f (x) concave function & X is convex set then:

(i) every local opt. is global opt.

(ii) set of global opt. is convex

– strict quasiconcavity is sufficient as well

– uniqueness

f (x) strictly (quasi)concave

• Second order conditions (suppose f ′(x) = 0)

– x is a solution ⇒ f ′′(x) ≤ 0 (necessary SOC)

– f ′(x) = 0 and f ′′(x) < 0 ⇒ x is a solution

(sufficient SOC)

c© Ronald Wendner Summary M4M-II-3 v1.3

Sufficient condition ⇒ Necessary condition

1. x∗ is maximizer ⇒ f ′(x∗) = 0, f ′′(x∗) ≤ 0

2. f ′(x∗) = 0, f ′′(x∗) < 0 ⇒ x∗ is maximizer

c© Ronald Wendner Summary M4M-II-4 v1.3

Several Choice Variables

• Ingredients

N choice variables: x ≡ (x1, x2, ..., xN)′

X opportunity set: x ∈ X

unconstrained problems: X = RN

constrained problems: X ⊂ RN

objective (related to x)

objective function: f (x)

• choose x ∈ X such as to max f (x)

• 2 complications

– necessary FOC: @ f ′(x)

– nec., suff. SOC: @ f ′′(x)

c© Ronald Wendner Summary M4M-II-5 v1.3

• FOC: derivative → partial derivative

– fi(x) ≡ ∂f(x)∂xi

= 0 for i = 1, ..., N

– N necessary FOC

• SOC: second order partial derivatives

– necessary: f (x) concave ⇔ H negative semidef-

inite

– sufficient: f (x) strictly concave ⇔ H negative

definite

→ linear algebra sessions

c© Ronald Wendner Summary M4M-II-6 v1.3