Extending a 2-dimensional nonlinear objective problem to 50-dimensions

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In summary, the speakers are discussing a two-dimensional nonlinear objective function with a well-defined minimum and maximum. They are also mentioning a MATLAB function they are trying to create and test, and are looking for a larger objective function with a well-defined global minimum to test it on. One speaker suggests using a nonlinear function with a global minimum at the origin, while the other suggests using quartic polynomials with positive leading coefficients. They also mention the difficulty in constructing nonlinear functions without constraints that are bounded and suggest using sine and cosine functions to ensure a unique global maximum and minimum.
  • #1
FOIWATER
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Hi,

I have a small two dimensional nonlinear objective that has a very well defined minimum and maximum. Here is the function:
$$f(x,y)=2(1-x)^{2}e^{-x^{2}-(y+2)^{2}}-9(\frac{x}{5}-x^{3}-y^{5})e^{-x^{2}-y^{2}}-\frac{1}{5}e^{-(x-1)^{2}-y^{2}}$$

Attached is it's plot and contour.

Notice this nonlinear function is only 2 dimensional in its arguements.

I am trying to code my own MATLAB function that will solve nonlinear optimizations (no constraints, at the moment) faster than the standard solver.

I want to test it against a large objective, I am thinking 50 or 60 arguements. I would like a function which has a well defined global minimum. Does anyone know of one which is?

I could extend the one above, but I want to be sure before I test with it..

Any help appreciated!

Plot1.png
Contour1.png
 
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  • #2
What about the following, for ##\vec{x}## a ##n##-vector?

$$f(\vec{x})\equiv \exp\left(\sum_{i=1}^n {x_i}^2\right)$$

It's nonlinear and has a global minimum at ##\vec{x}=\vec{0}##.

Or did you want something that's not monotonic and hence has local extrema distinct from the global ones?
 
  • #3
That's a good function to keep in mind I will definitely do it thanks. I would love one that has various well defined local minima as well as a well defined global one, if I can find one
 
  • #4
I think this might give you what you want, for ##n## of the order of 50 or 60:

$$
f(\vec{x})\equiv \prod_{i=1}^n\left( a_i\prod_{j=1}^4 (x_i-b_{ij})\right)
$$

where ##\forall i:\,a_{i}>0##. Just choose any bunch of coefficients ##a_i## and ##b_{ij}##. You could get the computer to do this with random number generation, or you could choose a pattern for them. All that matters is that they are mostly different from one another.

##f## is a product of ##n## quartic polynomials with positive leading coefficient. Each of the quartics will have a global minimum as well as possibly a local non-global minimum. Let the two local minima of the ##i##th quartic be ##m_i^1## and ##m_i^2##, where those two numbers are equal if the quartic has only one local minimum (eg ##x^4##).

Then the set of local minima of ##f## is

$$S=\left\{(m_1^{k_1},m_2^{k_2},...,m_n^{k_n})\ |\ \forall i\ k_i\in\{1,2\}\right\}$$

##S## has cardinality of at most ##2^n##, so you can easily identify all the local minima and work out which one gives the global minimum.

You can then run it through your optimiser to see if it finds the correct minimum.
 
  • #5
That's awesome, is it a function that has any actual real world application or just something you've constructed
 
  • #6
I just made it up - especially for you. :smile:
 
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  • #7
I am going to test it against this. Thankyou.

I am also trying to find a nonlinear function of about 50 variables, could be more could even be 100, that has a real world application with no constraints. Could even be a constrained problem where the constraints have been added to the objective as penalties. Do you know of any? I think most real world problems are going to be constrained at such a large number of variables, though.
 
  • #8
You see, I am trying to construct some of my own, but everytime I test them, they are unbounded. for example, I have been trying variations on:
$$\sum_{i=1}^{25}(0.25+0.5x_{i}+x_{i}^2)+constant\times \sum_{i=26}^{50}\sum_{j=26}^{50}x_{i}(x_{i}-x_{j})*constant$$
Or something like that. But when I test them in 'fminunc' it's always unbounded. I am noticing I need to make them with exp(-ve arguement) so that as I move away from the origin in any direction, the function goes to zero. So the optimum is close to the origin.
 
  • #9
I am starting to realize the difficulty in constructing nonlinear functions without constraints that are bounded? Everything I am creating...is unbounded. Everything that's mostly nonlinear and not quadratic
 
  • #10
Try using sin and cos functions, which are bounded, non-linear and non-quadratic.

eg ##\sum_{i=1}^n a_i\sin(b_i x_i+c_i)## will have an infinite number of local extrema, but will be bounded by ##\pm\sum_{i=1}^n |a_i|##.

If you want to ensure a unique global maximum and minimum, multiply it by
$$e^{-\sum_{i=1}^n {x_i}^2}$$.
 

1. What is the purpose of extending a 2-dimensional nonlinear objective problem to 50-dimensions?

The purpose of extending a 2-dimensional nonlinear objective problem to 50-dimensions is to increase the complexity and dimensions of the problem, allowing for a more accurate and realistic representation of real-world scenarios. This can lead to better solutions and insights for optimization problems.

2. How is this extension achieved?

This extension is achieved by adding additional variables and constraints to the original 2-dimensional problem, effectively increasing the number of dimensions. This can be done manually or through mathematical techniques such as variable transformation or decomposition.

3. Are there any benefits to extending a 2-dimensional problem to 50-dimensions?

Yes, there are several benefits to extending a 2-dimensional problem to 50-dimensions. It allows for a more accurate and comprehensive representation of real-world scenarios, which can lead to better solutions and insights. It also provides a more challenging and interesting problem for researchers and scientists to work on.

4. What are some challenges that may arise from extending a 2-dimensional problem to 50-dimensions?

One of the main challenges is the increased computational complexity. As the number of dimensions increases, the problem becomes more difficult to solve and may require more advanced techniques and algorithms. There may also be a higher risk of overfitting the model to the data, leading to less accurate solutions.

5. Can this extension be applied to any type of objective problem?

Yes, this extension can be applied to any type of objective problem, including linear, nonlinear, and multi-objective problems. However, the complexity and feasibility of the extension may vary depending on the specific problem and its constraints. It is important to carefully consider the benefits and challenges before extending a problem to a higher dimension.

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