|
|
| View previous topic :: View next topic |
| Author |
Message |
Bart Goddard
Guest
|
 Posted: Thu Jul 03, 2008 2:30 am Post subject: Examples of Hessians |
 |
|
I'm teaching vector calculus this Summer (yet again) and
finding (yet again) that it's darned hard to come up with
usable examples of finding extrema of a 3-variable function.
Does anyone have a small collection of f(x,y,z)'s
which have 1. by-hand-findable critical points
and 2. A decently interesting Hessian?
I'd take a 4-variable example too, and even give a good
cigar in exchange.
Bart
--
Cheerfully resisting change since 1959. |
|
| Back to top |
|
 |
| |
Ads |
Advertising
Sponsor
|
|
Ray Vickson
Guest
|
| Posted: Thu Jul 03, 2008 8:56 am Post subject: Re: Examples of Hessians |
|
|
On Jul 2, 2:30 pm, Bart Goddard <goddar...@netscape.net> wrote:
| Quote: |
I'm teaching vector calculus this Summer (yet again) and
finding (yet again) that it's darned hard to come up with
usable examples of finding extrema of a 3-variable function.
Does anyone have a small collection of f(x,y,z)'s
which have 1. by-hand-findable critical points
and 2. A decently interesting Hessian?
I'd take a 4-variable example too, and even give a good
cigar in exchange.
Bart
--
Cheerfully resisting change since 1959.
|
Here is an example from a management sciences optimization course I
used to teach.
There is an initial buyer population N0 = 10^5 for a durable good. In
any year the *fraction* of potential buyers who actually buy the good
is f(p) = (1/2) - p/200, (p = selling price in $). Since the good is
durable, buyers in year 1 are removed from the potential buyer
population, etc. The fixed production cost per item is c = 4 ($/unit).
Find the optimal selling prices p1, p2 and p3 in years 1, 2 and 3 to
maximize the three-year profit.
Formulation: number sold in year 1 is N1 = f(p1)*N0, number sold in
year 2 is N2 = f(p2)*(N0 - N1), and number sold in year 3 is N3 =
f(p3)*(N0 - N1 - N2). The three-year profit is F = (p1 - c)*N1 + (p2 -
c)*N2 + (p3 - c)*N3. Substituting and expanding, we get (using Maple,
for example, to do the algebra):
F = 50500*p1 - 500*p1^2 - 350000 + 25500*p2 + 255*p2*p1 - 250*p2^2
-5/2 * p2^2*p1
+ 13000*p3 + 130*p3*p1 + 130*p3*p2 + 13/10* p3*p2*p1 - 125*p3^2 - 5/4
* p3^2*p1
- 5/4 * p3^2*p2 - 1/80 * p3^2*p2*p1.
We have F1 = @F/@p1 = 50500 - 1000*p1 + 255*p2 - 5/2 * p2^2 + 130*p3
+ 13/10 * p3*p2 - 5/4 * p3^2 - 1/80 * p3^2*p2,
F2 = @F/@p2 = 25500 + 255*p1 - 500*p2 - 5*p2*p1 + 130*p3 + 13/10 *
p3*p1 - 5/4 * p3^2
- 1/80 * p3^2*p1,
and F3 = @F/@p3 = 13000 + 130*p1 + 130*p2 + 13/10 * p2*p1 - 250*p3
- 5/2 * p3*p1 - 5/2 * p3*p2 - 1/40 * p3*p2*p1.
It helps that F3 factors: F3 = (p1 + 100)*(130 + 13/10 * p2)*(1 -
p3/52). Setting F3 = 0 and requiring positive selling prices gives p3
= 52, then substituting that into F2 we have F2 = 28880 + 1444/5 *
p1-500*p2 - 5*p2*p1, which factors as F2 = (100+p1)*(1444/5 - 5p2), so
F2 = 0 gives p2 = 1444/25 = 57.76 (exactly), and substituting those
into F1 gives F1 = 7777568/125 - 1000*p1, so p1 = 972196/15625 =
62.22054400 (exactly). The Hessian of F is: H =
[-1000, 255-5*p2+13/10*p3-1/80*p3^2,
130+13/10*p2-5/2*p3-1/40*p3*p2], [255-5*p2+13/10*p3-1/80*p3^2,
-500-5*p1, 130+13/10*p1-5/2*p3-1/40*p3*p1],
[130+13/10*p2-5/2*p3-1/40*p3*p2, 130+13/10*p1-5/2*p3-1/40*p3*p1,
-250-5/2*p1-5/2*p2-1/40*p2*p1]
At the stationary point (p1,p2,p3) = (972196/15625, 1444/25, 52), this
becomes
H =[-1000, 0, 0],[0, -2534696/3125, 0], [0, 0, -1249605128/1953125]
= [-1000., 0., 0.], [0., -811.1027200, 0.], [0., 0., -639.7978255].
Therefore, we have a strict local maximum.
R.G. Vickson |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Ray Vickson
Guest
|
| Posted: Thu Jul 03, 2008 9:28 am Post subject: Re: Examples of Hessians |
|
|
On Jul 2, 2:30 pm, Bart Goddard <goddar...@netscape.net> wrote:
| Quote: |
I'm teaching vector calculus this Summer (yet again) and
finding (yet again) that it's darned hard to come up with
usable examples of finding extrema of a 3-variable function.
Does anyone have a small collection of f(x,y,z)'s
which have 1. by-hand-findable critical points
and 2. A decently interesting Hessian?
I'd take a 4-variable example too, and even give a good
cigar in exchange.
Bart
--
Cheerfully resisting change since 1959.
|
An even better example is obtained if we set c = 40 ($) instead of c =
4 in my previous example---better because it gives a non-diagonal
Hessian at the optimal solution. The new F is similar to the previous.
Now the derivative F3 = @F/@p3 = (p1 + 100)*(p2/40 + 5/2)*(p3 - 70),
so p3 = 70. Then @F/@p2 = F2(p1,p2,70) = (p1 + 100)*(5p2 - 1444/5), so
p2 = 1444/25 = 57.76 (exact), and then @F/@p1 = F1(p1,57.76,70) =
9206282/125-1000*p1, so p1 = 4603141/62500 = 73.65025600 (exact).
Now the Hessian H at (p1,p2,p3) = (73.650256, 57.76, 70) is not
diagonal:
H = [[-1000., 72.45000000, 0.], [72.45000000, -868.2512800, 0.], [0.,
0., -684.8766097]]
We can test that this is negative definite, so again we have a strict
local max.
R.G. Vickson |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Michael Press
Guest
|
| Posted: Thu Jul 03, 2008 9:43 am Post subject: Re: Examples of Hessians |
|
|
In article <Xns9ACFA7E8DA5C9goddardbenetscapenet@64.209.0.81>,
Bart Goddard <goddardbe@netscape.net> wrote:
| Quote: |
I'm teaching vector calculus this Summer (yet again) and
finding (yet again) that it's darned hard to come up with
usable examples of finding extrema of a 3-variable function.
Does anyone have a small collection of f(x,y,z)'s
which have 1. by-hand-findable critical points
and 2. A decently interesting Hessian?
I'd take a 4-variable example too, and even give a good
cigar in exchange.
|
This one seems trivial, yet for me had intrinsic interest.
In a Cartesian plane find the distance of a line from the origin.
Distance of a plane from (0,0,0)
Extemal values of xx + yy + zz on x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Maximum area of a triangle of perimeter 2.s
Extremal values of xyz subject to 1/x + 1/y + 1/z = c, c constant and > 0.
Characterize the nature of the extremal values.
Derive Snell's Law.
Maximum of (sum a_i x_i)^2 subject to sum (x_i)^2 = 1
Minimum distance between 256x^2 + 81y^2 = 2304 and 4x + 3y = 24.
Are these the kinds of exercises you are looking for?
--
Michael Press |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Ray Vickson
Guest
|
| Posted: Fri Jul 04, 2008 5:01 pm Post subject: Re: Examples of Hessians |
|
|
On Jul 4, 7:33 am, Bart Goddard <goddar...@netscape.net> wrote:
| Quote: |
Ray Vickson <RGVick...@shaw.ca> wrote in news:81aa3335-3bf3-46db-bc2b-
b24d5ea32...@q27g2000prf.googlegroups.com:
Now the Hessian H at (p1,p2,p3) = (73.650256, 57.76, 70) is not
diagonal:
H = [[-1000., 72.45000000, 0.], [72.45000000, -868.2512800, 0.], [0.,
0., -684.8766097]]
We can test that this is negative definite, so again we have a strict
local max.
Thanks.
B.
--
Cheerfully resisting change since 1959.
|
I removed that message (at least in Google Groups). The above so-
called optimal solution is erroneous; I think the error arose by
changing a parameter in a Maple worksheet but then not changing all
subsequent results. However, the matrix given above is, indeed, the
Hessian at the indicated point; it just unfortunate that the point
does not happen to be the optimal solution. (The Hessian is diagonal
at the optimal solution.) Note: the example in my original posting is
OK.
R.G. Vickson |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
|
|
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum
|
|
FAQ
Search
Memberlist
Usergroups
Register
Profile
Log in to check your private messages
Log in
