Mathematical Optimization | Chapter 5.2 : Golden Section | Indonesian

Summary

The video explains the Golden Section method for optimizing single-variable functions, focusing on iterative evaluations and boundary adjustments. This method uses the Golden Ratio to determine midpoints within a given range, updating the interval based on comparisons of function values at the calculated points. The process continues iteratively until the optimal value is found, and convergence is achieved. Through a worked example involving a quadratic function, the script demonstrates how to refine the interval and calculate the optimal value, emphasizing how the method progressively narrows the search range to find the peak of the function.

Takeaways

• 😀 Golden Section Method is an iterative optimization technique for single-variable functions, used to find the optimal value by evaluating the function at specific points.
• 😀 The method relies on determining a range (interval) where the optimal value lies, defined by an upper bound (b) and a lower bound (a).
• 😀 Two points, x1 and x2, are chosen within this interval, and they are based on the golden ratio (approximately 0.618). The points are symmetrically placed within the interval.
• 😀 The golden ratio is used to calculate the distance (D) between the points x1 and x2. This distance helps to determine their locations relative to the interval's bounds.
• 😀 In the case of a quadratic function, such as f(x) = 18x - 2x² + 10, the method seeks the maximum value of the function by narrowing the search interval.
• 😀 The method compares the function values at x1 and x2. If the function value at x1 is greater, the optimal point lies between a and x2. Otherwise, it lies between x1 and b.
• 😀 Iterative updates are performed by adjusting the bounds (a or b) based on the function evaluations, progressively narrowing the search range towards the optimal point.
• 😀 The iterative process continues until a convergence condition is met or the maximum number of iterations is reached. Convergence occurs when the distance between the two points becomes negligible.
• 😀 After convergence, the optimal value of x can be found as the midpoint between the final bounds, and the corresponding optimal function value is calculated.
• 😀 An example case with the function f(x) = 2x * sin(x) - x² / 10 was demonstrated, where the method was used to find the maximum function value over the interval [0, 4].
• 😀 The method's efficiency and the decreasing distance between x1 and x2 illustrate how the algorithm zeroes in on the optimal point over successive iterations.

Q & A

What is the Golden Section method used for in optimization problems?

The Golden Section method is used to iteratively find the optimal value of a function by evaluating its values and adjusting the search interval. It is particularly useful for single-variable optimization problems.

How does the Golden Section method differ from the Newton method?

The Golden Section method uses the concept of dividing the search interval based on the Golden Ratio and evaluates function values at these points. In contrast, the Newton method uses derivatives to approximate the root of a function, adjusting the interval based on the slope.

What is the Golden Ratio used in the Golden Section method?

The Golden Ratio used in the Golden Section method is approximately 0.618, and it is used to divide the search interval into two sub-intervals of specific proportion to optimize the function evaluation process.

How do you calculate the positions of X1 and X2 in the Golden Section method?

The positions of X1 and X2 are calculated based on the initial interval [a, b]. X1 is found by subtracting the Golden Ratio multiplied by the difference between b and a from b, while X2 is calculated by adding the same value to a.

What happens if the function value at X1 is greater than the function value at X2?

If the function value at X1 is greater than at X2, the optimal point lies between a and X2. In this case, the upper bound of the interval is updated to X2, and the process is repeated with the new interval.

What does it mean when the function value at X2 is greater than the value at X1?

When the function value at X2 is greater than that at X1, the optimal point lies between X1 and b. In this case, the lower bound of the interval is updated to X1, and the process continues with the new interval.

How do you know when to stop the Golden Section method?

You stop the Golden Section method when either the solution converges to a specific value or when the maximum number of iterations is reached. Convergence is typically indicated when the difference between the bounds of the interval becomes negligible.

What is the significance of the initial range [a, b] in the Golden Section method?

The initial range [a, b] defines the boundaries within which the optimal value of the function is sought. The method iteratively narrows this interval based on function evaluations until the optimal point is found.

In the provided example, what function is being maximized using the Golden Section method?

The function being maximized in the example is f(x) = 18x - 2x^2 + 10, and the goal is to find the x value that maximizes this function.

How does the interval update during each iteration in the Golden Section method?

During each iteration, based on the comparison of function values at X1 and X2, the interval is updated by either adjusting the lower or upper bound to move closer to the optimal value. This iterative process continues until the solution converges.