# Solve the Global Minima Problem in Two Variable Functions

• Archimedess
In summary, determining the global minima of a two variable function can be difficult, especially if the function is not strictly convex. In order to find the global minima, one must first identify all the local minima and then analyze them to find the subset with the lowest values. This requires calculating partial derivatives and considering where they vanish, as well as looking at the Hesse matrix. The presence of periodic functions, such as ##\sin^2##, can also complicate the determination of global minima.
Archimedess
Homework Statement
Let ##f(x,y)=\arctan(4\sin^2(y)+3\ln(x^2+1))## show that it has ##\infty## global minima
Relevant Equations
No relevant equations
I'm always struggling understand how to determine if a two variable function has global minima, I know that if I find a local minima and the function is convex than the local minima is also a global minima, in this case is really difficult to determine if the function is convex.

Sorry if I don't post any attempt but I got no clue how to do this.

I'm afraid you will have to calculate the partial derivatives and consider where they vanish. Also look up the Hesse matrix.

Archimedess said:
no clue how to do this.
Not good enough per the PF guidelines !
Least you could do is find and discuss a few minima, remark that ##\sin^2## is periodical, etc ...

jbriggs444
No, it is not convex. A convex function has a UNIQUE global minimum.

Edit: OK, I'm going to back off on that statement. That's true for a strictly convex function. But you could imagine a convex function whose set of global minima was a finite flat region. For instance, a function with ##f(x,y) = 0## on the circle of radius 1, and positive elsewhere. That would qualify as a convex function with infinitely many global minima.

But that's not the reason for infinitely many minima here. @BvU has already identified the reason in comment #3.

To find the global minima, you're going to have to identify all the local minima. Then analyze them and find which subset of those have minimal values of ##f(x, y)##.

## 1. What is the Global Minima Problem in Two Variable Functions?

The Global Minima Problem in Two Variable Functions refers to finding the lowest point or the minimum value of a two variable function over a given domain. This is important in optimization problems where the goal is to find the most optimal solution.

## 2. Why is solving the Global Minima Problem important?

Solving the Global Minima Problem is important because it helps in finding the most efficient solution in various real-world problems. For example, it can be used in finance to find the best investment strategy or in engineering to optimize the design of a product.

## 3. What are the methods used to solve the Global Minima Problem in Two Variable Functions?

There are various methods that can be used to solve the Global Minima Problem in Two Variable Functions. Some of the commonly used methods include gradient descent, Newton's method, and simulated annealing.

## 4. How does gradient descent work in solving the Global Minima Problem?

Gradient descent is an iterative optimization algorithm that works by taking small steps towards the direction of the steepest descent. In the context of the Global Minima Problem, it starts at a random point and moves towards the direction of the minimum value until it reaches the global minimum.

## 5. What are some challenges in solving the Global Minima Problem in Two Variable Functions?

One of the main challenges in solving the Global Minima Problem is that the function may have multiple local minima, which can make it difficult to determine the global minimum. Another challenge is that the function may be complex and high-dimensional, making it computationally expensive to find the global minimum.

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