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

  • Context: Graduate 
  • Thread starter Thread starter bebop_007
  • Start date Start date
  • Tags Tags
    Function
Click For Summary
SUMMARY

The discussion centers on identifying a specific property of functions that allows for continuous descent to a global minimum, even in non-convex scenarios. The example provided is the 2-D Rosenbrock function, which, despite being non-convex, possesses a unique characteristic: it has a global minimum with no local minima. This property enables numerical algorithms to effectively reach the valley of the function, although they may struggle to navigate towards the global minimum. The term sought to describe such functions is "unimodal."

PREREQUISITES
  • Understanding of convex and non-convex functions
  • Familiarity with optimization algorithms
  • Knowledge of gradient descent techniques
  • Basic concepts of mathematical analysis
NEXT STEPS
  • Research the properties of unimodal functions in optimization
  • Explore the characteristics of the Rosenbrock function in detail
  • Learn about gradient descent and its applications in non-convex optimization
  • Investigate numerical algorithms for finding global minima in complex functions
USEFUL FOR

Mathematicians, data scientists, and optimization engineers interested in advanced function properties and optimization techniques.

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.
 

Similar threads

Undergrad Gradient descent algorithm and optimizers
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Discussion about least squares method
  • · Replies 2 ·
Replies
2
Views
2K
MHB Q(x) is a strictly convex function, show that G is positive definite
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad What Exactly is Step Size in Gradient Descent Method?
  • · Replies 3 ·
Replies
3
Views
3K
Weber-Fermat Problem, degenerate cases
  • · Replies 7 ·
Replies
7
Views
3K
High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
How to use functions defined in an external header-only library?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Ballentine on the "multicomponent state function"
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Optimization Solver - BFGS method with bound constraints
  • · Replies 12 ·
Replies
12
Views
13K
Cryptographic attacks as minimization of degree 4 polynomial
  • · Replies 7 ·
Replies
7
Views
2K