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

• bebop_007
In summary, the property being described is a function that has a global minimum but no local minima. This means that it is possible to continuously follow the steepest descent to reach the global minimum, even if the function is non-convex.

#### bebop_007

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?

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.

## What is meant by "going downhill" in regards to function minimization?

Going downhill refers to the process of finding the minimum value of a function by following the steepest descent or gradient. This involves moving in the direction of the negative gradient, which leads to the minimum point of the function.

## What kind of functions can be minimized by going downhill?

Functions that can be minimized by going downhill are typically those with multiple variables and a smooth, continuous surface. Examples include cost functions, error functions, and energy functions.

## What is the role of gradient descent in minimizing a function?

Gradient descent is a popular optimization algorithm used to minimize functions by iteratively moving towards the direction of the negative gradient. This allows for a systematic approach to finding the minimum value of a function.

## Are there any limitations to using downhill minimization for functions?

While downhill minimization is effective for many functions, it may not always lead to the global minimum and can get trapped in local minima. Additionally, it may not be suitable for non-convex functions.

## How does "going downhill" relate to other optimization techniques?

"Going downhill" or gradient descent is one of many optimization techniques used to minimize functions. Other techniques include random search, simulated annealing, and genetic algorithms, each with their own advantages and limitations.