4.1. Introduction in optimization.

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From the course by National Research University Higher School of Economics

Mathematics for economists

1 rating

National Research University Higher School of Economics

Mathematics for economists

1 rating

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

From the lesson

Unconstrained and constrained optimization.

Week 4 of the Course is devoted to the problems of constrained and unconstrained optimization. Of special attention are quadratic forms, critical points and their classification.

4.1. Introduction in optimization.10:13

4.2. Global max. Local max. Saddle point.9:34

4.3. Unconstrained optimization.9:17

4.4. Critical point. Taylor's formula.12:22

4.5. Quadratic forms. Positive definiteness. Negative definiteness.7:45

4.6. Sylvester's criterion (part 1).7:17

4.7. Sylvester's criterion (part 2).5:50

4.8. Examples of Hessians (part 1).7:47

4.9. Example of Hessians (part 2).4:01

4.10. Sufficient condition for a critical point to be a local maximum, a local minimum and neither of both.7:13

4.11. Examples of finding and classification of critical points (part 1).6:34

4.12. Examples of finding and classification of critical points (part 2).4:38

4.13. Constrained optimization.9:11

4.14. Lagrangian.5:56

4.15. Example of constrained optimization problem.4:06

Meet the Instructors

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

Optimization plays a major role in economic analysis.

That's why this part of mathematics is

absolutely important and we'll pay a lot of attention to it in this course.

In order to understand optimization in the case of many variables,

we have to recall what an optimization means for the functions of a single variable.

So, let's start refreshing

the definitions of a maximum and a minimum of such a function.

So, suppose we have a function of one variable only and this function is

defined on some interval ranging from a to b.

So, if we consider a graph of the function,

in order to refresh the concepts of maximum and minimum values.

So, let's consider a graph like this.

These are the end points but they're not included,

a and b, because this is an open interval.

At this point, which is X0,

a function takes the value which is the

greatest within the local neighborhood of this point,

but at point X1 it takes the global maximum.

So, let us formalize the concepts.

So, we say that X0 is a maximum point of a function if there is

some interval such that

for all X from this interval,

the values of the function are no greater

than the value of the function taken at X0 point.

So, this is a definition in the case of one variable of a local maximum.

At X1, the function takes the greatest value if we compare the values from I interval.

So then, in this definition we say that X1 is the global maximum because

the value of the function taken at any point from

the domain is no greater than the value taken at X1.

So, this inequality holds for all values of X from the domain.

Now, what tools do we use in order to find these points?

By the way, talking about minimum,

all we have to do, we need to reverse the signs over these inequalities.

That's how we get the definition of a local minimum and respectively, a global minimum.

So, the tools for finding maximum and minimum points using calculus.

So, if we additionally know that the function f

of x is continuously differentiable on I,

then the necessary condition for

a function to have

maximum or minimum value at some points will look as follows,

and it can be stated as a theorem.

So, suppose a function f of x is continuously differentiable on I, it's written here.

Let the point x*

from I is either a maximum or a minimum of this function on I.

Then, derivative at this point turn 0.

This theorem provides all the necessary condition for

a function to have a maximum or minimum value at a given point.

We can provide an example of a function like y equals x cube

whose derivative turn 0

at x equals 0.

But this point is not maximum nor its minimum because we can draw

the graph of a cubic parabola and

clearly at 0 point is the point where the function increases.

So, any solution of this equation

f' equals 0 gives us x,

which is a critical point of a function.

In order to check whether a critical point is really a point of extremum,

we need to perform a test,

which is another theorem which provides

a sufficiency condition for a maximum or a minimum.

It is based on the analysis or the second order derivative of this function.

So from now on,

let's suppose that the function f

of x is twice continuously differentiable on interval I.

If we have a critical point

and after the second differentiation we

substitute this critical point into the second derivative,

unless the value is 0,

we can analyze the nature

of this critical point based on the sign of the second order derivative.

So, let's state it as a theorem.

So, this is a theorem which provides sufficiency.

So once again, the statement starts with

the information regarding the differentiability of the function f of x.

Let x* from this interval be

a critical point of

the function f of x.

Then, given a negative value

of the second order derivative at this point,

that x* is a point of local maximum;

given second order derivative,

which takes a positive value,

then x* is a point

of local minimum.

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