Curve Enclosing Max Area

[ozlegolas]

Homework Statement

Hi all,
I have a project for optimization class and the problem is that:

A curve $$\alpha$$(t) = (x(t),y(t)) of length L = 6 units is to be found so that the area enclosed by the curve and the x-axis is maximum. Curve will be start at (1, 0) and end at (3, 0).

I am supposed to solve this problem by using various kind of optimization algorithms which are Steepest descent, Newton’s method, Fletcher-Reeves and Davidon-Fletcher-Powell. I'll apply these methods in Matlab.

The Attempt at a Solution

Since these methods are for unconstrained optimization problems, I think I should treat this constrained problem as an unconstrained problem using some methods like penalty and barrier methods. However, I could't find out which method I should apply.

Thanks in advance..

 

[mighty2000]

Hello,

Thank you for sharing your project with us. Finding the maximum enclosed area for a given curve is indeed an interesting optimization problem. As you have mentioned, this problem can be treated as an unconstrained optimization problem by using penalty or barrier methods.

In this case, I would suggest using the barrier method. This method introduces a barrier function that penalizes points outside the feasible region, thus enforcing the constraints. The barrier function is usually chosen such that it approaches infinity as the point approaches the boundary of the feasible region.

You can then use one of the optimization algorithms mentioned (Steepest descent, Newton’s method, Fletcher-Reeves and Davidon-Fletcher-Powell) to solve the unconstrained problem with the barrier function. It is important to note that the choice of the barrier function can affect the convergence of the algorithm, so it is crucial to choose it carefully.

I hope this helps and wish you all the best with your project.