What kind of function can be minimized by going "downhill"

  • Thread starter Thread starter bebop_007
  • Start date Start date
  • Tags Tags
    Function
AI Thread Summary
The discussion focuses on identifying a specific property of functions that allows for continuous steepest descent to reach a global minimum. While smoothness is necessary, it is not sufficient for this property. The example of the 2-D Rosenbrock function illustrates that it is non-convex yet possesses the desired characteristic, where any numerical algorithm can lead to the global minimum despite potential challenges in navigating the valley. The property being sought is that of a function with a global minimum but no local minima. This type of function is often referred to as a "global minimum function."
bebop_007
Messages
1
Reaction score
0
Hi all:
I'm looking for the name of a property that a function (of arbitrary dimension) has when you can continuously follow the steepest descent to get to the global minimum. Being "smooth" is necessary but not sufficient.

For example, for just 1-D, this property is equivalent to being "convex", however the 2-D Rosenbrock function is a non-convex function, but still does have the property I'm looking for: any numerical algorithm will bring you quickly to the valley --it will then struggle to follow the valley to get to the global minimum, but at least in principle there is always a small-but-finite gradient that, if followed, will lead you to the global minimum which is the only stationary point. Therefore, the Rosenbrock function is a ... function.

Can anyone tell me what word I should use to finish that sentence?
 
Mathematics news on Phys.org
I don't know a single word, but you can say that it is a function that has a global minimum but no local minima.
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

I Gradient descent algorithm and optimizers
Replies
5
Views
2K
I Discussion about least squares method
Replies
2
Views
2K
MHB Q(x) is a strictly convex function, show that G is positive definite
Replies
2
Views
3K
Python Scipy minimizer function and function executions
Replies
16
Views
2K
I What Exactly is Step Size in Gradient Descent Method?
Replies
3
Views
3K
I Understand the delta rule increment in gradient descent
Replies
18
Views
2K
Minimizing systems of boolean functions as per specific gate costs
Replies
6
Views
1K
Weber-Fermat Problem, degenerate cases
Replies
7
Views
2K
I Are there any two pairs of integers with the same result in a specific function?
Replies
7
Views
2K
B How do I invert this exponential function?
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top