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320.313 Mathematics for EconomicsNonlinear Programming

Ronald Wendner

Department of EconomicsUniversity of Graz, Austria

The Plan

� Nonlinear programming

Problem statementInequality constraint(s)

Karush-Kuhn-Tucker ConditionsLagrange functionNecessary first-order conditions for choice variablesNecessary first-order conditions for Lagrange multiplier(s)

R. Wendner (U Graz, Austria) Mathematics 4 Econ 2 / 15

Problem Statement

Ingredients

� Setup

Choice variables x = (x1, ..., xN )

Objective function f(x) = f(x1, ..., xN )

Inequality constraint b− g(x1, ..., xN ) ≥ 0

constraint function g(x1, ..., xN )constraint constant b ∈ R

Nonnegativity constraints x1, x2, ..., xN ≥ 0

Opportunity set X ⊂ RN+

X ≡ {x ∈ RN+ | b− g(x1, ..., xN ) ≥ 0}


Problem Statement

Ingredients

� NLP can handle cases that CP cannot

Inequality constraints

Non-binding constraints

Corner solutions

x∗i = 0

x∗i is at boundary of constraint (and necessary FOC cannot be applied)

� CP problem is special case of NLP problem


Problem Statement

Ingredients

� Example

Choice variables x = (x1, x2)

Inequality constraint 10− (2x1 + x2) ≥ 0, x1 ≥ 0, x2 ≥ 0

Opportunity set X = {x ∈ R2+ | 10− (2x1 + x2) ≥ 0} ⊂ R2

+

ℝ+2

X ⊂ ℝ+2

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Problem Statement

Problem statement

� Optimization (maximization) problem

Let x = (x1, x2, ..., xN ) ≥ 0, b− g(x) ≥ 0,

X ≡ {x ∈ RN+ | b− g(x) ≥ 0}

Choose x so to max f(x) s.t. x ∈ X

maxx∈X

f(x)

� Global solution (maximum): All x∗ ∈ X, for which the following holds

f(x∗) ≥ f(x) for all x ∈ X

We call vector x∗ an optimizer (maximizer) of the NLP problem

� x∗ is maximizer ⇔ (∀x ∈ X)(f(x∗) ≥ f(x))


Karush-Kuhn-Tucker Conditions

Lagrange function

� To solve constrained optimization (maximization) problem

L(x1, ..., xN , λ) = f(x1, ..., xN ) + λ [b− g(x1, ..., xN )]︸︷︷︸≥0

L(x, λ) 6= f(x), generally

L(x∗, λ∗) = f(x∗), in optimum

Necessary first-order conditions: Karush-Kuhn-Tucker conditions

Constraints can be binding or non-binding (slack)



Binding constraint

� λ > 0

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X = {x ∈ R+ |x ≤ b}, x∗ = b

b ↑: b̂, X̂ = {x ∈ R+ |x ≤ b̂}, x∗ = b̂

As f(x∗)|b̂ = f(b̂) > f(b) = f(x∗)|b ⇒ λ > 0 (constraint is binding)



Slack constraint

� λ = 0

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X = {x ∈ R+ |x ≤ b}, x∗ 6= b

b ↑: b̂, X̂ = {x ∈ R+ |x ≤ b̂}, x∗ 6= b̂

As f(x∗)|b̂ = f(x∗) = f(x∗)|b ⇒ λ = 0 (constraint is slack)



Karush-Kuhn-Tucker conditions (maximum)

� Choice variables

∂ L(x, λ)∂ xi

≤ 0 , xi ≥ 0 , ∂ L(x, λ)∂ xi

xi = 0︸︷︷︸complementary slackness conditions

, i = 1, ..., N , (1)

� Lagrange multiplier(s)

∂ L(x, λ)∂ λ

≥ 0 , λ ≥ 0 , ∂ L(x, λ)∂ λ

λ = 0︸︷︷︸complementary slackness condition

. (2)

If (x∗, λ∗) is a solution (maximum) ⇒ (1) and (2)



Karush-Kuhn-Tucker conditions

� Examples

1. f(x) = x2, 0 ≤ x ≤ 1. Find (x∗, λ∗)!

Two solutions satisfy the KKT conditions: find the maximizers.

2. Consider the utility function u(x1, x2) = x1 + x2. Let the budget constraintbe given by I − p1x1 − p2x2, with p1 = 1, p2 = 2. Find (x∗

1, x∗2).




� Points to note

(i) KKT conditions generalize FOCs of classical programming

(ii) # constraints (M) need not be < # choice variables (N)

(iii) maxx f(x) = minx−f(x) : our KKT conditions can easily applied for minproblems

(iv) λ = 0 or λ > 0, but λ ≮ 0




� Examples

1. Minimize f(x) = (1− x), 0 ≤ x ≤ 1. Find x∗!

2. Minimize f(x) = x2, 0 ≤ x ≤ 1. Find x∗!




� Further points to note

(v) Optimum: L(λ∗, x∗1, ..., x

∗N ) = f(x∗

1, ..., x∗N )

→ complementary slackness

(vi) Existence: f(.) cont., X nonempty, clsd, bdd

(vii) Sufficient conditions

f(.) concaveg(.) convex

→ L(x, λ) is concave

(viii) V (b) = L(x∗, λ∗; b) then: λ∗ = ∂ V (b)/(∂ b)

(ix) Several inequality constraints




� Examples

1. Envious individuals. Government decides to give transfers to two types ofhouseholds: x1, x2. The total amount of transfers must not exceed x̄, i.e.,x1 + x2 ≤ x̄. Utility of the households is given by ui(xi) = xi − kx2

j ,i, j = 1, 2, i 6= j. We observe the parameter restriction k > 1/x̄. Thegovernment chooses x1, x2 so to maximize welfare: u1 + u2. Derive x∗

1, x∗2.

2. Quasilinear utility. Let the utility function of an individual be given byu(x1, x2) = x1 + a ln x2. Their budget constraint is I − p1x1 − p2x2, whereI, p1, p2, a > 0. We observe the parameter restriction I < ap1. Derive x∗

1, x∗2.


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