One optimization problem

[ValeryOchkov]

Do you know one symbolic solution of this problem?

See please one numerical solution in attach (M15):

PS At A=B x=A/6

[ValeryOchkov]

First guess - x1=x2=x3=x4

[ValeryOchkov]

Sorry, I cannot correct solve this problem numerical

In M15 and P3 - see please the first picture, this picture and attach.

[Werner_E]

Set up four different variables to solve for: x ... the smallest of the four and you may say its at the side A - that way changing A and B could (not in this case) yield a different result. The other three variables are the differences Delta1 .. Delta3. So wie have V(A,B,x,delta1..delta3):=(A-2x-delta1)*(B-2x-delta2-delta3)*x

That way you get rid of the min() function which is the problem.

[ValeryOchkov]

Thanks, Werner, but see the picture and attach:

[Werner_E]

Yes, the system is very sensible with respect to the guess values. Probably a very "flat" function.

[AlanStevens]

For the life of me I can't see why you'd want to optimise on four variables here when symmetry dictates that they must all be the same (or, if you prefer, the desire not to waste material means they must all be the same). However, if you insist, here is a way to get the right answer (numerically):

Alan

[ValeryOchkov · Accepted solution]

Thanks, Alan.

To create one program for optimization (Minimize, Maximize etc) - it is one (a little) haft of problem. The second (main) haft is to find functions for the testing of this programs

I think/hope that my simple function is one good test for it!

[Werner_E]

Maybe I just got a knot in the brain, but shouldn't the four partial derivatives of V() wrt the variables x1..x4 be zero at the solution point? After all its obviously not a boundary value so it wil be a local max.

Afterthought: Set up that way its only a problem in three variables, as the only thing the result depends on is the sum of x3 and x4.

[ValeryOchkov]

Nice a test problem

[ValeryOchkov]

Valery Ochkov wrote:

Nice a test problem

[Werner_E]

This shows exactly ... nothing , or?

[ValeryOchkov]

Werner Exinger wrote:

This shows exactly ... nothing , or?

Or!

We have a simple function of four arguments with a maximum and can not find direct this maximum with the maximize function.

[Werner_E]

Valery Ochkov wrote:

Werner Exinger wrote:

This shows exactly ... nothing , or?

Or!

We have a simple function of four arguments with a maximum and can not find direct this maximum with the maximize function.

Yes, because you didn't constrain.

[ValeryOchkov]

Werner Exinger wrote:

Valery Ochkov wrote:

Werner Exinger wrote:

This shows exactly ... nothing , or?

Or!

We have a simple function of four arguments with a maximum and can not find direct this maximum with the maximize function.

Yes, because you didn't constrain.

Can you constrain this problem?

[Werner_E]

Alan did

[Werner_E]

????????????

Am I missing the point or is it just playing around w/o deeper insight?

[Werner_E]

The main question remains http://communities.ptc.com/message/223269#223269

[Werner_E]

OK, sometimes I need a little longer to figure it out what is behind a problem.

In case of Valery's 4 variable extreme value problem (which in reality is only a one variable problem) the various objective function would represent a 4dimensional surface in 5-domensional space. This is a little bit difficult for the imagination, so I reduced the problem to just two parameters. I assume that the bent-over lateral surfaces of the box on opposite sides are equal. That way we can visualize the objective functions as 2D-surfaces in 3D space - a bit more comfortable.

The problems that have so far emerged in this thread are:

1) Why does Maximize fail with Valery's objective function using the min() function?

2) Why do the partial derivatives don't evaluate to zero at the solution point

3) Why do we not arrive at the solution by setting the partial derivatives to zero, neither with Valery's, nor with Alan's objective function.

I played around and the attached sheet emerged. As far as I am concerned the answers to the questions are:

1) because min() introduces a discontinuity in the derivation of the objective function, there would not be a unique derivative in the area of the solution and so the CG algorithm, which uses the gradients to approach the solutions, fails

2) & 3) because the solution is not a local maximum of a continous differentiable function.

Any suggestions?

Here the 2D/3D objective functions of Valery and Alan

[ValeryOchkov]

2D variant of this problem:

[ValeryOchkov]

Fractal variant (A=B=1) of this problem - one, five and n boxes from one quadrate sheet:

[AlanStevens]

Attached is a simplified version of Valery's box fractal optimisation, but with an infinite number of boxes. It uses only the symbolic maths of Mathcad, not the numerical maths (apart from the graph plots).

Alan

[ValeryOchkov]

Thanks Alan!

You solution uses the same ratio of corner length to square side - this is the simplification

But I have a solution for infinite number of a - a.infinite=1/6, a.infinite-1=see above my picture etc.

It is a problem with infinite number of optimization parameters.

See http://communities.ptc.com/message/198664

[ValeryOchkov]

AlanStevens wrote:

with an infinite number of boxes.

My old solution:

[ValeryOchkov]

Werner Exinger wrote:

OK, sometimes I need a little longer to figure it out what is behind a problem.

Thanks, Werner!

See one part of my article "Work on forums - an important aspect of engineering

or

The first step to cloud Universities"

http://twt.mpei.ac.ru/ochkov/CloudUniOchkov.pdf

"Another form of academic work, which is practiced by the author - is hosted lecture material on the forum PlanetPTC for his extensive discussions with fellow teachers from all over the World. One example of such a discussion (one solution to the optimization problem, the comparison of analytical and numerical methods of solution) can be seen here http://communities.ptc.com/message/223224

[ValeryOchkov]

Werner Exinger wrote:

1) Why does Maximize fail with Valery's objective function using the min() function?

Try please this sheet at n>2:

[Werner_E]

Nice, I like it. I am no specialist but the algorithm demonstrated here seems to be neither Conjugate Gradients nor Quasi Newton.

It would be better and informative to trace and graph the behaviour of Mathcad's built-in algorithms - maximize() in our case here. Should not be that difficult to do. We would see that those algorithms are more complex and behave somewhat chaotic and b yno means so "good" and straightforeward as the one we see in your pic.

[ValeryOchkov]

Werner Exinger wrote:

Nice, I like it.

I am glad.

See please one animation here http://communities.ptc.com/docs/DOC-1979

[Werner_E]

How about tracing the algorithm actually used in the problems - Mathcad's built in Maximize()!?

[ValeryOchkov]

Werner Exinger wrote:

How about tracing the algorithm actually used in the problems - Mathcad's built in Maximize()!?

See bellow please