lecture 5 – review of optimization...

Lecture 5 – Review of Optimization Methods

Prof. Massimo Guidolin

Prep Course in Quant Methods for Finance

August-September 2017

Plan of the lecture

2Lecture 5 – Review of Optimization Methods

Statement of the problem: extrema, maxima, and minima

First-, second-, Nth-order derivative tests for y = f(x)

Differential conditions

First- and second-order derivative and differential tests for z = f(x, y)

Differential conditions expressed in matrix (quadratic) forms

Constrained optimization (case of equality constraints)

Motivation

3

In your MSc. studies you will encounter a few occasions in which rational behavior will be equivalent to the optimization, possibly subject to feasibility constraints, of a given objective function• E.g., in 20135 equilibrium stock prices will be the result of

investors optimizing their portfolios• In 20191 and 20192 you will discover that the most sensible (and

also best in a number of definitions) way to estimate parameters in statistical models, is by maximization of the likelihood function

While some of these problems will have an elementary nature and will consist of one choice variable only, other applications will be multivariate in nature

Sometimes it will be useful and natural—because they are used everywhere in econometrics and asset pricing—to express these conditions in matrix form

Often constraints will appear: for instance, that ptf. weights sum to 1


Optimization: Statement of the Problem Optimization == maximizing or minimizing some objective

function, y = f(x), by picking one or more appropriate values of the control (aka choice) variable x• Maxima and minima are also called extremums

o E.g., a risk-averse investor wants to maximize a mean-variance objective by picking an appropriate set of portfolio weights

• Optimization == maximizing /minimizing an objective function, y = f(x), picking one or more values of the control (aka choice) variable x

Key assumption: f(x) is n times (n2) continuously differentiable• This of course also implies continuity

4

5

• In the leftmost graph, optimization is trivial: the function is a constant and as such all points are at the same time maxima and minima, in a relative sense

• In the second plot, f(x) is monotonically increasing, there is no finite maximum if the set of nonnegative real numbers is the domaino Point D is at best the global minimum on the domain shown

• The points E and F on the right are examples of a relative (local) extrema, in the sense that each of these points represents an extremum in the immediate neighborhood of the point only

• A function can well have several relative extrema some of which may be maxima while others are minima


Optimization: Statement of the Problem

• We shall continue our discussion mainly with reference to the search for relative extrema such as points E and F

• This does not imply we do not care for global extrema because they must also be local extrema or end points of f(x) on its domaino E.g., if we know all the relative maxima, it is necessary only to select

the largest of these and compare it with the end points in order to determine the absolute maximum

Key Result 1 (First-Derivative Test): If a relative extremum of the function occurs at x = x0, then either f'(x0) = 0, or f'(x0) does not exist; this is a necessary condition

The First-Derivative Test

6

7

• Because we are assuming that y = f(x) is continuous and possesses a continuous derivative, we are ruling out sharp points

• For smooth functions, relative extreme values can occur only where f ’(x0) = 0

• However, zero slope while necessary, is not sufficient

Key Result 1 Qualified: If f ’(x0) = 0 thenthe value of f(x0) will be:(a) A relative maximum if the derivativef '(x) changes its sign from > 0 to <0 from the immediate left of the point x0 to its immediate right(b) A relative minimum if f '(x) changes its sign from negative to positive from the immediate left of x0 to its immediate right(c) Neither a relative maximum nor a relative minimum if f '(x) has the same sign on both the immediate left and right of point x0 (inflection point)


The First-Derivative Test

Inflection point


One Example

9

• A strictly concave (convex) function is such that if we pick any pair of points M and N on the function and join them by a straight line, the line segment MN must lie entirely below (above) the curve, except at points M and N

• Strict concavity f ’’(x) 0; strict concavity f ’’(x) 0

• The second derivative may take both positive and negative values, depending on the value of x; in this case, inflection points will have to appear


Concavity, Convexity, and Second-Order Derivatives

Concave Convex

Inflection point

10

Key Result 2 (Second-Derivative Test): If the first derivative of a function at x = x0 is f ’(x0) = 0 (first-order, necessary condi-tion), then f(x0 0 ), will be:(a) A relative maximum if f"(x0) < 0(b) A relative minimum if f"(x0) > 0• This test is in general more convenient to use than the first-

derivative test, because it does not require us to check the derivative sign to both the left and the right of x0

• Drawback: no unequivocal conclusion in the event that f"(x0) = 0 when the stationary value f(x0) can be either a relative maximum, or a relative minimum, or even an inflection point

• This is what makes the condition sufficient onlyo To see that a relative extremum may be consistent with f"(x0) = 0,

consider y = x4, that plots as a U-shaped curve and has a minimum at x = 0; the second derivative is f"(x0) = 12x2, so that f"(0) = 0


The Second-Order Derivative Test

Second-order, sufficient condition


Two Examples

12

Key Result 3 (Nth-Order Derivative Test): If the first derivative of a function at x = x0 is f ’(x0) = 0 and if the first nonzero derivative value at x0 encountered in successive derivation is that of the Nth derivative, f(N)(x0) 0,

then the stationary value f (x0) will be:(a) A relative maximum if N is an even number and f(N)(x0) < 0(b) A relative minimum if N is an even number but f(N) (x0) > 0(c) An inflection point if N is odd• Start from a Taylor expansion:

• All the derivative values are zero until we arrive at the Nth one:


Nth-Order Derivative Test

Reminder in Lagrange form

13

• f(N)(p) takes the same sign as f(N)(x0) because p is to be selected as close to x0 as necessary; the sign of (x - x0)N will vary if N is odd and will remain unchanged (positive) if N is even

• When N is odd, f(x) - f(x0) will change sign as we pass through the point x0 thereby violating the definition of a relative extremum (x0 must be an inflection point)

• When N is even, f(x) - f(x0) will not change sign from the left of x0to its right and this establishes the stationary value f(x0) as a relative maximum or minimum. depending on whether f(N)(x0) is negative or positive


Nth-Order Derivative Test

14

We now extend our earlier work to the case of many control variables: because it still allows graphical representation, we start with function of two choice variables, z = f(x, y)

We still assume that f(x, y) possesses continuous partial derivatives of any desired order

As before, extreme values may be both local (relative) or globalLecture 5 – Review of Optimization Methods

One Example

15

• Strategy remains looking for relative extrema to then try to isola-te global ones among them

Before proceeding, we need to re-interpret our earlier results• At the maximum point A as

well as the minimum point B, z must be stationary

• It is a necessary condition for an extremum of z that dz = 0 instantaneously as x varies

• This condition constitutes the differential version of the first-order condition for an extremum, but the two are equivalent because dz = f’(x)dx and f ’(x) = 0 dz = 0

• A maximum (minimum), such as A, has the property that as we slide along the curve infinitesimally to the left (dx < 0) and right (dx > 0) of A, we are descending (ascending) in both directions

• A sufficient condition is that dz < 0 (dz > 0) on both sides of ALecture 5 – Review of Optimization Methods

Optimization in Two Variables

16

• In other words, d(dz) < 0 (>0), or d2z < 0 (>0) for arbitrary nonzero values of dx, the second-order differential of z

• As usually, negativity (positivity) of d2z is sufficient, but not necessary, for a maximum (minimum) of z, as in certain cases, d2zmay be zero

• This derives from

The differential conditions, not the derivative conditions, are stated in forms that can be directly generalized from the one-variable case to cases with two or more choice variables


Differential Conditions

17

The first-order necessary condition for an extremum (either maximum or minimum) of a two-variable function, z = f(x, y), again involves dz = 0, but expressed as a total differential:

dz = fxdx + fzdz = 0for arbitrary values of dx and dy, not both zero

In order to satisfy this condition, it is necessary that the two partial derivatives fx and fy be simultaneously equal to zero:

• At point A, to have fx = 0 means that the tangent line Tx, drawnthrough A andparallel to the xz plane (holdingy constant), must have a zero slope


Necessary Differential Conditions

18

• fy = 0 at point A the tangent line Ty drawn through A and parallel to the yz plane (holding x constant), must have zero slope

• However, the first-order condition is necessary, but nor sufficient

• Point C in the left diagram have zero slopes, but this point does not qualify as an extremum: Whereas it is a minimum when viewed against the background of the yz plane, it turns out to be a maximum when looked at against the xz plane, a saddle point

• Point D on the right is an inflection point, whether viewed against the xz or the yz plane

Sufficient condition 2nd-order total differential, d2zLecture 5 – Review of Optimization Methods

Necessary Differential Conditions

Inflection pointSaddle point

19

The second-order sufficient conditionfor an extremum (either maximum or minimum) of a two-variable function, z = f(x, y), involves d2z: in the case of a maximum (minimum), d2z < 0 (> 0) for arbitrary values of dx and dy, not both zero


Sufficient Differential Conditions

Young’s theorem

20

• If an infinitesimal movement away from A in any direction along the surface invariably results in a decrease in z, A is peak of dome

• The reason why d2z <0 (> 0) are only sufficient, but not necessary, conditions is that it is possible for d2z =0 and the point to be a maximum or a minimum

For this reason, second-order necessary conditions must be stated with weak inequalities:

These conditions may be written in terms second derivatives:

• Importantly, it is not just a matter of establishing the sign of fxxand fyy, but also of restricting the cross partial derivative fxy to ensure that the surface will yield (two-dimensional) cross sections in all possible directions


Sufficient Differential Conditions


One Example

22

The expression for d2z given byis a quadratic form: we need a criterion to sign them for arbitrary values of dx, dy, not both zero• A form is a polynomial expression in which each component term

has a uniform degree, e.g. 3x3y – 6x2y2 +x3y + 24y4


One Example

23

• A is a quadratic form in two variables, u and c

• In this form, dx = u and dy = v are cast in the role of variables, whereas the second partial derivatives are treated as constantso The reason is the changed nature of the problem: the 2nd order

condition for extremum stipulates d2z to be definitely positive (for a minimum) and definitely negative (for a maximum), regardless of the values that dx and dy may take (not both zero)

o The second partial derivatives, on the other hand, will assume specific values at the points we are examining as possible extrema

The following definition helps:

• If q changes signs when the variables assume different values, on the other hand, q is said to be indefinite


Quadratic Forms

24

• Clearly, the cases of positive and negative definiteness of q = d2zare related to the 2nd order sufficient conditions

• When q = d2z is indefinite we may face a saddle point• Because by completing the square, we can see

• Now that u and v appear only in squares, we can predicate the sign of q entirely on the values of a, b, and h as follows:

• The condition just derived may be stated more succinctly by use of determinants as

• Note that also |a| is a sub-determinant that consists of the first element on the principal diagonal, the first principal minor


Quadratic Forms and Second-Order Conditions

25

• The determinant of the full matrix is the other principal minor• One unified way to see this is to define

the matrix H as the Hessian matrix andto deal with her determinant:

• Are similar conditions possible for quadratic forms in 3 variables?



• Hence, for positive definiteness, the positivity conditions on the principal minors are 3:

• For positive definiteness, the 3 principal minors must alternate in sign in the specified manner:o The ith principal minor is a sub-determinant formed by deleting the

last (N - i) rows and columns of the matrix


> 0 > 0 > 0

26

27

For the quadratic form

The necessary-and-sufficient condition for positive definiteness is positivity of all the principal minors of |D|:

The condition for negative definiteness is that the principal minors alternate in sign:all the odd-numbered principal minors are negative and all even-numbered ones are positive


Generalizations to the N-Variable Case

• For z = f(x1, x2, x3, …, xN), the necessary FOCs are of course the standard ones: fx1 = fx2 = … = fxN = 0

• Sometimes we say that all internal (to the function domain) stationary points must set the gradient of the function to a zero vector, where gradient = row vector of first partial derivatives

Generalizations to the N-Variable Case

28

29

• Second-order conditions are concerned with the question of whether a stationary point is the peak of a hill (domed-shaped summit) or the bottom (bowl-shaped)of a valley

• I.e., they relate to how a curve, surface, or hypersurface (as the case may be) bends itself around a stationary point• With N variables, the hills and valleys are no longer graphable, but

we may nevertheless think of "hills" and "valleys" on hypersurfaces A function that gives rise to a hill (valley) over the entire

domain is said to be a concave (convex) function, either globally or over a portion of the domain

When FOCs Become SOCs: Concavity and Convexity


30

• In the non-strict (weak) case, the hill or valley is allowed to contain one or more flat portions, such as line segments (on a curve) or line segments and plane segments (on a surface)

• The word strict rules out such line or plane segments

• A strictly concave (strictly convex) function must be concave (convex), but the converse is not true

Key Result 4a (FOC=SOC): An extremum of a concave function must be a peak, a maximum; however, that absolute maximum may not be unique, because multiple maxima may occur if the hill contains a fiat horizontal top• Latter possibility can be dismissed if we specify strict concavity

Key Result 4b (FOC=SOC): An extremum of a convex function must be a bottom, a minimum, even though it may be not unique

When FOCs Become SOCs: Concavity and Convexity


Weakly convexfunction

31

Up to these points, all control variables have been independent of each other: the decision made regarding one variable does not impinge upon the choices of the remaining variables• E.g., a two-product firm can choose any value for Q1 and any Q2 it

wishes, without the two choices limiting each other• If the said firm is somehow required to observe a restriction

(such as a production quota) in the form of Q1 + Q2 = k, however, the independence between the choice variables will be lost

• The new optimum satisfying the production quota constitu-tes a constrained optimum, which, in general, may be expected to differ from the free optimum

Key Result 5: A constrained maxi-mum can never exceed the free maximum

Constrained Optimization


32

• In general, a constrained maximum can be expected to have a lower value than the free maximum, although, by coincidence, the two maxima may happen to have the same valueo We had added another constraint intersecting the first constraint at

a single point in the xy plane, the two constraints together would have restricted the domain to that single point

o Then the locating of the extremum would become a trivial matter• In a meaningful problem, the number and the nature of the

constraints should be such as to restrict, but not eliminate, the possibility of choiceo Generally, the number of constraints should be less than the

number of choice variables Under C < N equality constraints, when we can write a sub-set

of the choice variables as an explicit function of all others, the former can be substituted out:

max𝑥𝑥1,𝑥𝑥2,…,𝑥𝑥𝑁𝑁

𝑓𝑓 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑁𝑁𝑠𝑠. 𝑡𝑡. 𝑥𝑥1 = 𝑔𝑔 𝑥𝑥2, … , 𝑥𝑥𝑁𝑁 , … , 𝑥𝑥𝐶𝐶 = 𝑔𝑔 𝑥𝑥𝐶𝐶+1, … , 𝑥𝑥𝑁𝑁

Constrained Optimization


33

becomes: max𝑥𝑥𝐶𝐶+1,𝑥𝑥𝐶𝐶+2,…,𝑥𝑥𝑁𝑁

𝑓𝑓 𝑥𝑥𝐶𝐶+1, 𝑥𝑥𝐶𝐶+2, … , 𝑥𝑥𝑁𝑁an unconstrained problem

However, the direct substitution method cannot be applied when the C constraints do not allow us to re-write the objective functions in N – C free control variables• Even if some of the variables become implicit functions of others,

it would be complex to proceed because the objective would become “highly composite”

In such cases, we often resort to the method of Lagrange (undetermined) multipliers• The goal is to convert a constrained extremum problem into a

form such that the first-order condition of the free extremum problem can still be applied

• For instance, consider an objective function z = f(x.y) subject to the constraint g(x,y)=c where c is a constant

Lagrange Multiplier Method


the Lagrangian problem is: max𝑥𝑥,𝑦𝑦,𝜆𝜆

𝑓𝑓 𝑥𝑥,𝑦𝑦 − 𝜆𝜆[𝑐𝑐 − 𝑔𝑔 𝑥𝑥, 𝑦𝑦 ]𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑓𝑓𝑓𝑓𝐿𝐿𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝐿𝐿

The necessary FOC is then:

Lagrange Multiplier Method

The stationary values of the Lagrangian function Z will

automatically satisfy the constraint

34

The optimal value * provides a measure of the sensitivity of the Lagrangian function to a shift of the constraint• To the standard differential statement of the free FOC, we now

add a differential in the constraint g(x,y), to capture the fact that we can no longer take dx and dy both as "arbitrary" changes

• To satisfy this necessary condition:

• This condition together with the constraint g(x,y)=c, will provide two equations from which to find the critical values of x and y.

• Yet, the differential approach just gives stationary points for the control variables, but does not yield a value for the multipliers

In the general case:

Lagrange Multiplier Method in Differential Form

35

36

The Lagrangian is: with necessary FOCs

• When there is more than one constraint, the Lagrange-multiplier method is equally applicable, provided we introduce as many multipliers as there are constraints in the Lagrangian function

• It is tempting to go a step further and borrow the second-order necessary and sufficient conditions from a free problem

• This should not be done: even though Z is indeed a standard type of extremum with respect to the choice variables, it is not so with respect to the Lagrange multiplier

• Unlike x and y, if is replaced by any other value, no effect will be produced on Z because [c - g(x,y)] is identically zero

General Case


System of N+1 simultaneous equations

37

• To find an appropriate new expression for d2z, we must treat dyas a variable dependent on x and y during differentiation (if dx is to be considered a constant)

• Algebra reveals that

• The coefficients are simply the second partial derivatives of Z with respect to the choice variables x and y

Key Result 6:

• As in the case of free extremum, it is possible to express the second-order sufficient condition in determinant-driven form

• In place of the Hessian determinant, in the case we shall encounter what is known as the Bordered Hessian

Second-Order Necessary and Sufficient Conditions


Sufficient conditions

Second-Order Necessary and Sufficient Conditions

38

lecture 5 – review of optimization...

Documents