Good morning or good afternoon.
I am Esteve Codina now and I'm going to present this video
video 8T based on basic ideas on convex
functions and optimization corresponding to block three calculus.
The content of the video will follow
these points.
We'll go into the learning objectives.
We will go next through an introduction and an example.
Then we will go to the definition of what are global
and local maximum and minima of univariate functions.
We will have a notion of what is the concavity and convexity conditions
and next we will go into the first order conditions
for an interior global and or local minimum or maximum.
We will solve the the example presented in introduction and then we will
finally go into the learning concepts and give some references and extensions.
Okay.
So the learning objectives are to get a notion of what is a global local
maximum minimum of a real univariate function
and what are
real univariate functions, convex and concave,
on an interval
in the case that it's possible to calculate the derivatives
of these univariate functions, we will see a condition that maxima minima
must verify and how these conditions allow us to obtain
this points. Okay.
There are many problems in engineering that are solved.
Finding the values of variables of a given system
that maximize or minimize some given function.
In this video, we will characterize maximum minima for real univariate functions
and show how can they be obtained
analytically by applying the first order conditions.
Let's see an example.
Imagine a highway lane and the flow of vehicles per unit of time,
which depends on the speed of the cars on that lane.
Assume that engineers have experimentally found this relationship in the form
of a function that looks like in the graph that is presented in the slide.
So it may be interesting to advise the drivers on the speed they have to follow.
So the maximum
amount of cars per unit of time is achieved on that lane.
By inspecting the graph, we can see that this speed must be around
80 kilometers per hour and then the maximum flow
that will be reached will be approximately 2200 vehicles per hour.
Okay. So, this leads us to the definition of
what is the global maximum or minimum of a real univariate function.
Assume that 'f' is such a real univariate
function defined on an interval a, b of the real numbers
and assume that a given point 'x*' in the interval.
a, b it is verified that the value of the function at 'x*'
is larger equal than any other point in
in the interval.
Then we say that 'x*' is a global maximum of 'f' on the interval.
And if instead of being the largest possible, it's the smallest possible,
we will say that these are a global minimum
of the function on the interval.
Next, we have to give a definition of
what is the local maximum or minimum of a real univariate function.
Assume also that 'f' is a real univariate function defined on an interval
and assumed that at 'x*' it's possible to find a small distance
such that for any point 'x´' within the interval around 'x*'
and closer to it at least a distance 'delta',
it is verified that for this point 'x´',
it always falls from below
to the function evaluated at 'x*'.
Then we say that 'x*' is a local maximum of 'f'.
If instead we have a smaller equal than we would say that
'f' is a local minimum of the function at the point 'x*'.
We see here graphical examples.
We see here that the interval is 0, 120
and at point 20 we get that
this is the maximum possible value of the function on the interval
so that the global maximum. Instead, for instance, at the point 120.
We can see that this is the local maximum because around it, around the point 120,
any other value? Yes?.
any other value is smaller than that value of the function.
The same happens with global minimum. Yes?.
But instead of being greater equal, they are smaller equal.
Okay, so let's
go into the concavity and convexity ideas.
Let's consider two arbitrary points.
'x´' and 'x´´' contained in a given interval a, b
and the point 'x' between them,
so 'x' is greater equal than 'x´' in smaller equal than 'x´´'
then necessarily, there must exist some scalar alpha, between zero and one,
so that this third point must be
expressed as
'alpha' 'x´' + (1-'alpha') 'x´´'.
Then we say that 'f' is concave on the interval,
if there always holds that for two arbitrary points
'x´' and 'x´´', any third point lying
between 'x´' and 'x´´' verifies the following inequality.
So the function evaluated at the third point is always
greater equal than the value obtained on the secant
of the
values of 'f' evaluated at 'x´' and 'x´´'.
If instead of being larger equal, we get the condition as a
smaller equal, then we have the definition of a convex function.
We see here in this slide graphical representation of this concept.
We see our function defined on the interval 0, 120
and we see, for instance, we focus on the point
x = 60 and two arbitrary points.
For instance, 'x´'= 40 and 'x´´' = 100.
Then 'x', which is equal
to 60, must be expressed, taking 'alpha'= 2/3 as
2/3 times 40 + 1/3 times 100.
So if we pick up the evolution of the function at the point 'x' = 60,
we see that it's always from below, from from above, sorry, than the value
obtained at the second expression in the red line of the graph.
This is for a concave function.
In the case of convex functions, we have precisely the opposite
and this condition is verified with a secant which always lies
from above of the real value of
the function on any two points
'x´' and 'x´´'.
However,
of course, there are functions which are neither convex nor concave,
which maybe verify the concavity
condition or the convexity condition for arbitrary points
'x´', 'x´´' but not for other choice of points 'x´' and 'x´´',
as is shown in the graph.
So the question is why are concave
and convex functions important? for a concave function
Any look on maximum is also a global maximum.
In other words, they have only global maximum on a given interval.
So for an arbitrary function, finding a global maximum,
not necessarily a concave function is a costly computational problem,
much higher than the finding than finding a local maximum.
So for concave functions, using procedures for finding a local maximum,
we will always obtain a global maximum. That's why are they important.
And the same apply for convex functions and global minima and local minima.
Okay, so
now we will go to the first order conditions.
In this example, we see that
points at 'x´' = 20
and 'x´´'= 100 which are local or global
maximum and local or global minimum of the function,
the slope, or equivalently, the derivative vanishes; is equal to zero.
Yeah, and notice that at 'x´'
and 'x´´' are points in the interior of the interval.
Yeah, they are larger than 'a' and smaller than 'b'.
So the first order condition is precisely to annihilate,
to make the derivative of the function, in case it's differentiable,
to make the derivative of the function to take the value zero.
Let's go now, using this condition,
to the example we have seen in the introduction and let's solve it,
we see that the example in introduction was the flow speed relationship,
and we see that it
can be expressed in terms of this function, for which
the coefficients 'b', 'c', and 'd' are given on the right hand side of the slide
and the interval, as we said, is between 0 and 120.
So the procedure tells us that we may have to consider
the derivative of the function 'j'.
Yes?, So we obtain the analytically the derivative of the function,
and then we equate it to zero and look for points 'v' which verify
this condition.
This leads us to the second degree a question.
We solve the second degree equation and we find two solutions,
80 and -182.85.
So the first one is the only value because we are considering points
in the interval 0, 120 and the second must be discarded.
So, actually this is the real maximum of the function on the interval.
0, 120.
A final word of caution should be said because actually
by employing this condition, for instance, for this function
and finding the solutions of the derivative of the function equal
to zero, instead of finding only minimums
or maximums, we can find another
type of points.
Yes. For instance, in this case we find that the solutions of the equation,
derivative of the function equal zero, are three points: 'x' = zero
and 'x' = +- square root of 3/2.
So we see that + square root of 3/2
is the right hand side point, which is a maximum.
- square root of 3/2 is on the left hand side.
We just a minimum, but zero is neither a maximum or minimum.
But that really is also a solution of the of the
of the condition derivative = 0.
So we have in this points
what all them, all these three points are called the stationary point.
Stationary points actually the solutions of the condition 'f´' equals zero.
Okay.
So the key learning concepts we have reached are: we have introduced
the idea of global and local maximum minimum of a real univariate function.
We have seen the notion of convex and concave
functions and why they are important.
And we have introduced using the example to the first order conditions
that verify maxima and minima and other points.
Such points are called stationary points.
Okay, as references.
There are these two basic references
in which we will be able to find examples and solve the exercises.
And if the extensions for a more analytical and in-depth
extension of the of these concepts, you will find this reference of calculus
volume one, which you can access to be
freely available on this, on this web page.
Okay.
So thank you very much for your attention that's all