Mathematical Economics - Part I -...

Optimization

FilomenaGarcia

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OptimizationProblems in Rn

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The objectivesof OptimizationTheory

Existence ofSolutions

UnconstrainedOptima

EqualityConstraints

InequalityConstraints

ConvexStructures inOptimizationTheory

QuasiconvexityinOptimization

ParametricContinuity:TheMaximumTheorem

SupermodularityandParametricMonotonicity

Mathematical Economics - Part I -Optimization

Filomena Garcia

Fall 2009

Filomena Garcia Optimization

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1 Optimization in Rn

Optimization Problems in Rn

Optimization Problems in Parametric FormOptimization Problems: Some ExamplesThe objectives of Optimization Theory

2 Existence of Solutions

3 Unconstrained Optima

4 Equality Constraints

5 Inequality Constraints

6 Convex Structures in Optimization Theory

7 Quasiconvexity in Optimization

8 Parametric Continuity: The Maximum Theorem

9 Supermodularity and Parametric Monotonicity


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Optimization Problems in Rn

An optimization problem in Rn is one where the values of agiven function f : Rn → R are to be maximized or minimizedover a given set D ⊂ Rn. The function f is called objectivefunction and the set D is called the constraint set.We denote the optimization problem as:

max {f (x) |x ∈ D}

A solution to the problem is a point x ∈ D such thatf (x) ≥ f (y) for all y ∈ D.We call f (D) the set of attainable values of f in D.


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It is worth noting the following:

1 A solution to an optimization problem may not exist

Example: Let D = R+ and f (x) = x , then f (D) = R+ andsupf (D) = +∞, so the problem max {f (x)|x ∈ D} has nosolution.

2 There may be multiple solutions to the optimizationproblem

Example: Let D = [−1, 1] and f (x) = x2, then themaximization problem max {f (x)|x ∈ D} has twosolutions: x = 1 and x = −1.


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We will be concerned about the set of solutions to theoptimization problem, knowing that this set could be empty.

argmax [f (x)|x ∈ D] = {x ∈ D |f (x) ≥ f (y) ,∀y ∈ D}

Two important results to bear in mind:

1 x is a maximum of f on D if and only if it is a minimumof −f on D.

2 Let ϕ : R→ R be a strictly increasing function. Then x isa maximum of f on D if and only if x is also a maximumof the composition ϕ ◦ f .


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Optimization Problems in Parametric Form

Optimization problems are often presented in parametric form,i.e. both the objective function and/or the feasible set dependon some parameter θ from a set of feasible parameter values Θ.In this case, we denote the optimization problem as:

max {f (x , θ)|x ∈ D(θ)}

In general, this problem has a solution which also depends onθ, i.e.

x(θ) = argmax {f (x , θ)|x ∈ D(θ)}


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Optimization Problems: Utility Maximization

1 Utility Maximization

In consumer theory the agent maximizes his utility fromconsuming xi units of commodity i .The utility is given by u(x1, ..., xn).The constraint set is the set of feasible bundles given thatthe agent has an income I and the commodities are pricedp = (p1, ...pn), i.e.

B(p, I ) ={

x ∈ Rn+|p · x ≤ I

}Notice that this is a parametrized optimization problem,the parameters being (p, I ).1.

1Other parameters may exist within the utility function, namely CobbDouglas weights


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Optimization Problems: Expenditure Minimization

2 Expenditure Minimization

It is the dual problem to the utility maximization.Consists on minimizing the expenditure constrained onattaining a certain level of utility u.The problem is given by:

min {p · x |u(x) ≥ u}


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Optimization Problems: Profit Maximization

3 Profit Maximization

It is the problem of a firm which produces a single outputusing n inputs through the production relationshipy = g(x1, ..., xn).Given the production of y units of good, the firm maycharge the price p(y) 2.w denotes the vector of input prices.The firm chooses the input mix which maximizes herprofits, i.e.,

max{

p(g(x))g(x)− w · x |x ∈ Rn+

}

2p(y) is the inverse demand functionFilomena Garcia Optimization

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Optimization Problems: Cost Minimization

4 Cost Minimization

It is the dual problem to the profit maximization.Consists on minimizing the cost of producing at least yunits of output.The problem is given by:

min {w · x |g(x) ≥ y}


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Optimization Problems: Consumption-LeisureChoice

5 Consumption-Leisure Choice

It is a model of the labor-supply decision process ofhouseholds in which the income becomes also an object ofchoice.Let H be the amount of hours that the agent has available.The wage rate of labor time is given by w . If working for Lunits of time, the agent earns wL.The agent obtains utility from consumption of x and fromleisure time, l .Then, the agent’s problem is:

max {u(x , l)|p · x ≤ wL, L + l ≤ H}


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The objectives of Optimization Theory

The objectives of Optimization Theory are:

1 Identify the set of conditions on f and D under which theexistence of solutios to optimization problems isguaranteed

2 Obtain a characterization of the set of optimal points,namely:

The identification of necessary conditions for an optimum,i.e. conditions that every solution must verify.The identification of sufficient conditions for an optimum,i.e. conditions such that any point that meets them is asolution.


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The objectives of Optimization Theory

The identification of conditions ensuring the uniqueness ofthe solution.

The identification of a general theory of parametricvariation in a parametrized family of optimizationproblems. For example:

The identification of conditions under which the solutionset varies continuously with the parameter θ.In problems where the parameters and actions have anatural ordering, the identification of conditions in whichparametric monotonicity is verified, i.e., increasing thevalue of the parameter, increases the value of the action.


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How we will proceed

Section 2: Studies the question of existence of solutions.The main result is the Weierstrass Theorem, whichprovides a general set of conditions under whichoptimization problems are guaranteed to have bothmaxima and minima.

From Section 3 to Section 5 we will examine the necessaryconditions for optima using differentiability assumptionson the undelying problem. We focus on local optima,given that differentiability is only a local property.

Section 3 focuses on the situation in which the optimumoccurs in the interior the feasible set D. The main resultare the necessary conditions that must be met by theobjective function at the optimum. Also sufficientconditions that identify optima are provided.


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How we will proceed

In Sections 4 and 5 some or all of the constraintsdetermine the optima. :

Section 4 focuses on equality constraints and the mainresult is the Theorem of Lagrange (both necessary andsufficient conditions for optima are presented.

Section 5 focuses on inequality constraints and the mainresult is the Theorem of Kuhn and Tucker which describesnecessary conditions for optima.

Sections 6 and 7 turn to the study of sufficient conditions,i.e., conditions that, when met, will identify points asbeing global optima. These are mainly related to theconvexity and quasiconvexity of the feasible sets andobjective functions.


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How we will proceed

Section 8 and 9 address respectivey the issue of parametriccontinuity and monotonicity of solutions.

The main result of Section 8 is the Maximum Theorem,which states that the continuity assumptions of theprimitives of the optimization problem are inherited,although not entirely, by the solutions.

Section 9 relies on the concept of supermodularity of theprimitives to conclude about the parametric monotonicityof solutions. The main result is the Theorem of Tarski.


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Existence of Solutions

Main Result:

The Weierstrass Theorem

Let D ∈ Rn be a compact and let f : D → R be a continuousfunction on D. Then f attains a max and a min on D, i.e.,∃z1, z2 ∈ D: ∀x ∈ D

f (z1) ≥ f (x) ≥ f (z2)


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Some Definitions:Compact set: closed and bounded

bounded set: is countained within a finite ball.

closed: contains its frontier. (valid even if the frontier isempty like in R)

Examples: closed interval in R.Continuous function: for any point a in the domain, we havethat:

limx→af (x) = f (a)


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Notice: The Weierstrass Theorem gives us sufficient conditionsfor maxima or minima to exist, however, we don’t know whathappens when some or even all of the conditions of theTheorem are violated.


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Failure of the Weierstrass Theorem:

1 Let D = R and f (x) = x3. f is continuous but D is notcompact (it is not bounded). Since f (D) = R, f does notattain a maximum or a minimum on D.

2 Let D = (0, 1) and f (x) = x . f is continuous but D is notcompact (it is not closed). The set f (D) = (0, 1), so fdoes not attain a maximum or a minimum on D.

3 Let D = [−1, 1] and let f be given by

f (x) =

{0, x = −1, x = 1

1,−1 < x < 1

D is compact, but f fails to be continuous. f (D) is theopen interval (−1, 1)


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Intuition of the Proof of the Weierstrass Theorem:

1 First we show that under the stated conditions f (D) is acompact set, hence, closed and bounded.

2 Second, it is shown that if A is a compact set in R, thenmax A and min A are always well defined, never beeingthe case that A fails to contain the supremum and theinfimum. Hence max f (D) and min f (D) must exist on D.


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The Weierstrass Theorem in Applications

Consider the applications mentioned in section 1.The Utility Maximization Problem:

Maximize u(x) subject to x ∈ B(p, I ) ={

x ∈ Rn+|p · x ≤ I

}We assume u(x) to be continuous.

By the Weierstrass Theorem, if B is compact, we musthave a solution. ⇒ p >> 0


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The Weierstrass Theorem in Applications

The Cost Minimization Problem:

Minimize wx over x ∈ F (y) ={

x ∈ Rn+|g(x) ≥ y

}The objective function is continuous.

But the feasible action set is not compact because it is notbounded.

The Weierstrass Theorem does not guarantee or preclude asolution. What can we do to guarantee a solution? Bound theset.


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Unconstrained Optima

Unconstrained optima The constraints do not affect theoptimal point; in other words, the optimum is interior.

Local Maximum: A point x is a local maximum of f on ifthere is r > 0 such that f (x) ≥ f (y) for all y ∈ B(x , r).

Global Maximum: A point x is a global maximum of f iffor all r > 0, and all y ∈ D, f (x) ≥ f (y) for ally ∈ B (x , r).


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First Order Conditions

Necessary Conditions

If x∗ is a local maximum (or minimum) of the function f on Dand f is differentiable, then Df (x∗) = 0

Notice, the reverse is not necessarily true.Example:x = 0 for f (x) = x3

Any point that satisfies the first order conditions (FOC) is acritical point of f on D.


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Second Order Conditions

Sufficient Conditions

Suppose f is a C 2 function on D, and x∗ is a point in theinterior of D. Then:

If Df (x∗) = 0 and D2f (x∗) is negative definite at x∗, thenx∗ is a local strict local maximum of f on D.

If Df (x∗) = 0 and D2f (x∗) is positive definite at x∗, thenx∗ is a local strict local minimum of f on D.

Notice: This conditions only identify local optima (not global).


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Second Order Conditions

Example:Let f (x) = 2x3 − 3x2 and D = R. The first order conditionsare: f ′(x) = 0 and identify as critical points x = 0 and x = 1.At these points, the second order conditions are:f ′′(0) = −6 < 0 and f ′′(1) = 6 > 0. Hence, 0 is a localmaximum and 1 is a local minimum. However, these are notglobal max or min. In fact, this function has no global max normin (why?).

The existence of a global maximum (resp. minimum) in D isguaranteed only if the function is concave (resp. convex) in D


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Constrained Optimization Problems - EqualityConstraints

1 It is not often that optimization problems haveunconstrained solutions.

2 Typically some, or all of the constraints will matter.

3 In this chapter we will look into the constraints of theform: g(x) = 0 where g : Rn → Rk

The problem:

Maximize f (x) s.t. g(x)=0


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Lagrange Theorem - 2 variables, one restriction

We can solve graphically

The solution is:

∂f∂x1

∂f∂x2

=

∂g∂x1

∂g∂x2

g(x1, x2) = 0 (1)


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We can solve by substitutionSince g(x1, x2) = 0 in the solution, and ∂g

∂x26= 0, we have an

implicit funtion (what is this?) x2(x1).So, the problem becomes:

maxx1 f (x1, x2(x1))


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The FOC is:

∂f

∂x1+∂f

∂x2

∂x2

∂x1= 0

By the IMPLICIT FUNCTION THEOREM:

∂x2

∂x1= −

∂g∂x1

∂g∂x2

The solution is:

∂f∂x1

∂f∂x2

=

∂g∂x1

∂g∂x2

g(x1, x2) = 0 (2)


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We can rewrite the FOC of the previous problem as:

∂f

∂x1− ∂f

∂x2

∂g∂x1

∂g∂x2

= 0

or:∂f

∂x1+ λ

∂g

∂x1= 0

where

λ = −∂f∂x2

∂g∂x2

which is the FOC of the Lagrangean.


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Lagrange Theorem

Lagrange Theorem

Let f , g : R2 → R be C 1 functions. Suppose that x∗ = (x∗1 , x∗2 )

is a local maximizer or minimizer of f subject to g(x1; x2) = 0.Suppose also that Dg(x∗) 6= 0. Then, there exists a scalarλ∗ ∈ R such that (x∗1 , x

∗2 , λ

∗) is a critical point of thelagrangean function:

L(x1; x2;λ) = f (x1, x2) + λg(x1, x2)

In other words, at (x∗1 , x∗2 , λ

∗): ∂L∂x1

= 0, ∂L∂x2

= 0 and ∂L∂λ = 0


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Lagrange Theorem

The constraint qualification

We must assume Dg(x∗) 6= 0. (If there are many variablesand restrictions, this amounts to say that the Jacobianmatrix of g is invertible or that the rank is equal to thenumber of constraints).

This condition allows us to identify g as an implicitfunction.

The inverse of Dg is used in computing λ.


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Lagrange Theorem - Necessary conditions

Notice: The Lagrange Theorem provides necessary but notsufficient conditions for local optima.It is not said that if we find x and λ such that g(x) = 0 andthe FOC of the Lagrangean are equal to zero, that x , λ areindeed an optimum, even if Dg(x) 6= 0.


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Lagrange Theorem - Necessary conditions

Then why can we use the Lagrange method so often?

Proposition

Suppose that the following two conditions hold:

1 A global optimum x∗ exists in the given problem.

2 The constraint qualification is met at x∗.

Then, there exists a λ∗ such that (x∗, λ∗) is a critical point ofL.

So, under the conditions of Proposition 27, the Lagrangeanmethod does identify the optimum.


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Lagrange Theorem - Sufficient Conditions

Sufficient Conditions

Suppose that there is x∗ and λ∗ which satisfy the constraintqualification and solve the FOC of the Lagrangean. Then:

If the Bordered Hessean has a positive determinant, wehave a local maximum

If the Bordered Hessean has a negative determinant, wehave a local minimum


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Equality Constraints

An Example:

Maxx x2 − y2 s.t. x2 + y2 = 1

First we verify that the conditions of Proposition work:

1 The function f is continuous and the constraint set iscompact.

2 The Constraint qualification, namelyDg(x , y) = (−2x ,−2y) 6= 0, is met.

Then, we can rely on the Theorem of Lagrange to pinpoint theoptima:

L(x ; y ;λ) = f (x , y) + λg(x , y) = x2 − y2 + λ(1− x2 − y2)


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Equality Constraints

An Example (cont.): The first order condtitions are:

∂L∂x = 0⇔ 2x − λ2x = 0∂L∂y = 0⇔ −2y − λ2y = 0

∂L∂λ = 0⇔ x2 + y2 = 1

The critical points are: (1,0,1);(-1,0,1);(0,1,-1);(0,-1,-1). Thefirst two are global maximizers, and the second are globalminimizers.


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Inequality constraints and Kuhn Tucker

Most economic problems include non-negativity restrictions andinequalities in the optimization. For instance, it is natural toimpose that consumption be non negative, or work hours, orcapital.In this chapter we will study problems of the form:

maxxs.t.g(x) = 0andh(x) ≥ 0

Our tool will be the Kuhn-Tucker Theorem.


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Kuhn-Tucker Theorem

Kuhn-Tucker Theorem - Maximization

Let f : Rn → R and hj : Rn → R be C 1 functions, i = 1, ..., l .Suppose x∗ is a local maximum of f on

D = {x ∈ Rn|hi (x) ≥ i = 1, ..., l}

Let E denote the set of effective (binding) constraints at x∗,and let hE = (hi )i∈E . Suppose that the rank of DhE = |E |.Then, there exists a vector λ∗ = (λ∗1, λ

∗2..., λ

∗l ) such that the

following constraints are met:

1 λ∗i ≥ 0 and λ∗i hi (x∗) = 0, for i = 1, ..., l .

2 Df (x∗) +∑l

i=1 λ∗i Dhi (x∗) = 0


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Kuhn-Tucker Theorem

Kuhn-Tucker Theorem - Minimization

Let f : Rn → R and hj : Rn → R be C 1 functions, i = 1, ..., l .Suppose x∗ is a local minimum of f on D. Let E denote theset of effective (binding) constraints at x∗, and lethE = (hi )i∈E . Suppose that the rank of DhE = |E |. Then,there exists a vector λ∗ = (λ∗1, λ

∗2..., λ

∗z l) such that the

following constraints are met:

1 λ∗i ≥ 0 and λ∗i hi (x∗) = 0, for i = 1, ..., l .

2 Df (x∗)−∑l

i=1 λ∗i Dhi (x∗) = 0


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Kuhn-Tucker Theorem - Some Observations

1 A pair x∗, λ∗ meets the FOC condition for a maximum inan inequality constrained maximization problem if itsatisfies h(x∗) ≥ 0 and conditions 1 and 2 of the theorem.

2 Condition 1 is called the Complementary slacknesscondition.

3 K-T theorem provides only necessary conditions for localoptima which meet the constraint qualification.

4 Df (x∗)−∑l

i=1 λ∗i Dhi (x∗) = 0


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Kuhn-Tucker Theorem - Interpreting the conditions

Complementary Slackness conditionThe Lagrange multipliers give us the effect in the maximum ofrelaxing the restriction. Hence we can have one of threesituations:

1 the restriction is verified with strict inequality, and in thatcase, relaxing the restricting will not change the maximumvalue attained: λ = 0 and h > 0.

2 the restriction is verified with equality, and in that case,relaxing the restricting will at least increase the valueattained: λ > 0 and h = 0.

3 the restriction is verified with equality, and in that case,relaxing the restricting will maintain the value attained:λ = 0 and h = 0.


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Kuhn-Tucker Theorem - Constraint Qualification

Constraint Qualification

r(DhE (x∗)) = |E |

An example where the constraint qualification fails:Let: f (x , y) = −(x2 + y2) and h(x , y) = (x − 1)3 − y2.Consider the problem of maximizing f subject to h ≥ 0.By inspection we see that f attains its max when (x2 + y2)attains its min. We also must have x ≥ 1 and y ≥ 0. So themaximum is obtained at (x∗, y∗) = (1, 0), a point where theconstraint is binding.Dh(x∗, y∗) = (3(x∗ − 1)2, 2y∗) = (0, 0). The K-T theoremfails. In fact:

Df (x∗, y∗) + λ(x∗, y∗) = (2, 0)⇔ .


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Why the procedure usually works

In general we can rely on the K-T theorem due to the followingproposition:

Proposition

Suppose the following conditions hold:

1 A global maximum x∗ exists to the giveninequality-constrained problem.

2 The constraint qualification is met at x∗.

Then, there exists λ∗ such that (x∗, λ∗) is a critical point of L.


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An Example

Let g(x , y) = 1− x2 − y2. Consider the problem of maximizingf (x , y) = x2 − y over the set D = (x , y)|g(x , y) ≥ 0.

1 The conditions of proposition 2 are met: by weierstrassand the fact that if the constraint is met, x and y cannotbe simmultaneously zero, and hence r(Dg) = 1.

2 We can use the K-T theorem.

L(x , y , λ) = x2 − y + λ(1− x2 − y2)

∂L∂x = 0⇔ 2x − 2λx = 0∂L∂y = 0⇔ −1− 2λy = 0

λ ≥ 0,1− x2 − y2 ≥ 0, λ(1− x2 − y2) = 0


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An Example (cont.)

From (1) either x = 0 or λ = 1. If λ = 1, (2) implies that

y = −1/2 and x2 + y2 = 1⇔ x = ±√

32 .

If x = 0, we may have λ = 0 or λ > 0. Take the first case:

λ > 0, then from the CSC we must have y = ±1. y = 1 isinconsistent from (2). so y = −1 and λ = 1

2 .

λ = 0 is inconsistent with (2).

So, the critical points are: (x , y , λ) = (±√

32 ,−1

2 , 1) and(x , y , λ) = (0,−1, 1

2) the value of the function is respectively5/4 and 1, hence we obtain the maxima. (why not theminimum)?


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Interpreting the Lagrange Multipliers

The Lagrange multipliers of the Lagrange Theorem and of theKuhn-Tucker Theorem can be interpreted as the sensitivity ofthe value of the function in the optimum to relaxing therestriction.Case 1: Equality restrictionsSuppose that we have the following optimization problem:

maxx f (x)s.t.g(x) + c = 0

We would solve the problem by writing the FOC of theLagrangean w.r.t. x and λ, namely:

∂L

∂x=∂f (x)

∂x+ λ

∂g(x)

∂x; g(x) + c = 0

The optimum is x∗(c) and the value of the function in theoptimum is f (x∗(c)).


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We can compute the effect of c on the optimum of thefunction.

∂f

∂c=∂f (x∗(c))

∂x

∂x∗(c)

∂c

∂f (x∗(c))

∂x= −λ∂g(x∗(c))

∂x

Also: ∂g(x∗(c))∂x = −1 (because the constraint must be observed

in the optimum), hence:

∂f

∂c= λ


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Interpreting the Lagrange Multipliers

Case 2: Inequality restrictionsSuppose that we have the following optimization problem:

maxx f (x)s.t.h(x) + c ≥ 0

We would solve the problem by writing the FOC of theLagrangean w.r.t. x and λ, namely:

∂L

∂x=∂f (x)

∂x+ λ

∂g(x)

∂x

λ ≥ 0, h(x) + c ≥ 0, λ(h(x) + c) = 0

The optimum is x∗(c) and the value of the function in theoptimum is f (x∗(c)).


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We can compute the effect of c on the optimum of thefunction.

∂f

∂c=∂f (x∗(c))

∂x

∂x∗(c)

∂c

∂f (x∗(c))

∂x= −λ∂h(x∗(c))

∂x

If the restriction is not binding, λ = 0 and, hence, ∂f (x∗(c))∂x = 0

If the restriction is binding,h(x) + c = 0⇒ ∂h(x∗(c))

∂x∂x∗(c)∂c = −1

So: ∂f∂c = λ.


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Convexity and Optimization

The notion of convexity occupies a central position in the studyof optimization theory. It encompasses not only the idea ofconvex sets, but also of concave and convex functions. Theattractiveness of convexity for optimization theory arises fromthe fact that when an optimization problem meets suitableconvexity conditions, the same first-order conditions that wehave shown in previous to be necessary for local optima, alsobecome sufficient for global optima. Moreover, if the convexityconditions are tightened to what are called strict convexityconditions, we get the additional bonus of uniqueness of thesolution.


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Convexity Defined

We say that an optimization problem is convex if the objectivefunction is concave (max) or convex (min) and the feasible setis convex.Convex setA set D is said to be convex if given x1, x2 ∈ D we haveλx1 + (1− λ)x2 ∈ D.concave functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) ≥ λf (x1) + (1− λ)f (x2)

convex functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) ≤ λf (x1) + (1− λ)f (x2)


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Convexity Defined (cont.)

Strictly concave functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) > λf (x1) + (1− λ)f (x2)

Strictly convex functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) < λf (x1) + (1− λ)f (x2)


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Implications of convexity

Every concave or convex function must also be continuouson the interior of its domain.

Every concave or convex function must possess minimaldifferentiability properties.

The concavity or convexity of an everywhere differentiablefunction f can be completely characterized in terms of thebehavior of its second derivative D2f


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Some General Observations

Theorem 1

Suppose D ⊂ Rn is convex and f : D → R is concave. Then

1 Any local maximum of f is a global maximum of f .

2 The set argmax {f (x)|x ∈ D} of maximizers of f on D iseither empty or convex.


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Some General Observations

Theorem 2

Suppose D ⊂ Rn is convex and f : D → R is strictly concave.Then

1 Any local maximum of f is a global maximum of f .

2 The set argmax {f (x)|x ∈ D} of maximizers of f on D iseither empty or or contains a single point (uniqueness).


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Convexity and Unconstrained Optimization

Theorem 3

Suppose D ⊂ Rn is convex and f : D → R is concave anddifferentiable function on D. Then, x is an unconstrainedmaximum of f on D if and only if Df (x) = 0.


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Convexity and the Theorem of Kuhn-Tucker

Theorem 4

Suppose D ⊂ Rn is open and convex and f : D → R is concaveand differentiable function on D. For i = 1, 2...l , lethi : D → R also concave. Suppose that there is some x ∈ Dsuch that hi (x) > 0, i = 1, ...l . Then, x∗ maximizes f over theconstraint set, if and only if there is λ∗ ∈ Rk such that theKuhn Tucker first order conditions hold.


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The Slater Condition The condition that there is some x ∈ Dsuch that hi (x) > 0 is called the Slater Condition

1 The SC is used only in the proof that the K-T conditionsare necessary at an optimum. It is not related tosufficiency.

2 The rank condition established in the K-T theorem can bereplaced by the SC and concavity of the constraints.


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Quasiconvexity in Optimization

Convexity carries important implications for optimization.However it is a quite restrictive assumption. For instance, sucha commonly used utility function, such as the Cobb Douglasfunction is not concave unless

∑ni=1 αi ≤ 1.

u(x1, ..., xn = xα11 ...xαn

n .

In this chapter we examine optimization under a weakening ofthe condition of convexity, which is called quasi-convexity.


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Quasiconvexity and Quasiconcavity defined

quasi-concave functionA function f defined on a convex set D is quasi-concave if andonly if: for all x1, x2 ∈ D and for all λ ∈ (0, 1) it is the case that

f (λx1 + (1− λ)x2) ≥ minf (x1), f (x2)

quasi-convex functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) ≤ maxf (x1), f (x2)


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Quasiconvexity and Quasiconcavity defined

strict quasi-concave functionA function f defined on a convex set D is quasi-concave if andonly if: for all x1, x2 ∈ D and for all λ ∈ (0, 1) it is the case that

f (λx1 + (1− λ)x2) > minf (x1), f (x2)

strict quasi-convex functionA function defined on a convex set D is concave if: givenx1, x2 ∈ D

f (λx1 + (1− λ)x2) < maxf (x1), f (x2)


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Quasiconvexity as a generalization of convexity

If a function f is concave on a convex set, it is alsoquasi-concave on the convex set. Same for concavity. Thereverse does not hold.Let f : R→ R.If f is a nondecreasing function on R, then, f isboth quasi concave and quasi-convex.


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Implications of quasi-convexity

Quasi-concave and quasi-convex functions are notnecessarily continuous on the interior of their domains.

Quasi-concave and quasi-convex functions may have localoptima that are not global optima.

First order conditions are not necessary to identify everylocal optima under quasi-convexity.


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Quasiconvexity and Optimization

Theorem

Suppose that f : D → R and hi are strictly quasiconcave andD is convex. Suppose that there are x∗ and λ∗ such that theK-T conditions are satisfied. Then, x∗ maximizes f over Dprovided at least one of the following conditions holds:

1 Df (x∗) 6= 0

2 f is concave


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Quasiconvexity and Optimization

Theorem

Suppose that f : D → R is strictly quasiconcave and D isconvex. Then, any local maximum of f on D is also a globalmaximum of f on D. Moreover, the set or maximizers of f iseither empty or a singleton. Same holds for a minimum andstrictly quasiconvex functions.


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