Problem 72 - Calculus Single Variable

Question

Exercises 71 and 72 utilize a minimization algorithm called the Golden Ratio Search. Let \(\rho\) denote the Golden Ratio, \((\sqrt{5}-1) / 2 .\) Let \(r=\rho /(\rho+1) \approx 0.381966 .\) Suppose that \(f\) is continuous on \(I=\left[a_{1}, b_{1}\right],\) has a minimum at \(c\) in \(\left(a_{1}, b_{1}\right),\) is decreasing on \(\left(a_{1}, c\right),\) and is increasing on \(\left(c, b_{1}\right) .\) (Such a function is said to be unimodal.) Let \(\Delta x_{1}=b_{1}-a_{1}\), let \(a_{2}=\alpha_{1}\) and \(b_{2}=b_{1} .\) Repeat the procedure, letting \(\Delta x_{2}=b_{2}=a_{2} \equiv \rho \Delta x_{1} \quad \alpha_{2}=a_{2} \neq r, \Delta x_{2},\) and \(\beta_{2}=b_{2}=r, \Delta x_{2}\). Calculate \(f\left(\alpha_{2}\right)\) and \(f\left(\beta_{2}\right) .\) (From the point of view of efficiency, it is useful to observe that one of these two values has been computed in the last step: If \(a_{2}=a_{1},\) then \(\beta_{2}=\alpha_{1} ;\) if \(a_{2}=\alpha_{1},\) then \(\left.\alpha_{2}=\beta_{1} .\right)\) If \(f\left(\alpha_{2}\right)>f\left(\beta_{2}\right),\) then let \(a_{3}=a_{2}\) and \(b_{3}=\beta_{2} .\) Otherwise, let \(a_{3}=\alpha_{2}\) and \(b_{3}=b_{2} .\) Repeat the procedure. After every step, the point \(c\) is in the interval \(I_{n}=\left[a_{n}, b_{n}\right]\). Because \(b_{n}=a_{n}=\rho^{n=1} \cdot(b-a),\) the midpoint) \(c_{n}=\left(a_{n}+b_{n}\right) / 2\) of \(I_{n}\) satisfies \(\left|c-c_{n}\right| \leq \rho^{n-1} \cdot(b-a) / 2 .\) Thus \(n\) can be chosen large enough so that \(c_{n}\) approximates \(c\) with any specified accuracy. For the given function \(f\) on [0,1] ), locate \(c\) to two decimal places. \(f(x)=2+x^{2}-\sin (x)\)

Step-by-Step Solution

Answer

The minimum of the function is approximately at \( x = 0.62 \).

Step 1: Define the Initial Interval

We start with the interval \( I=[a_1, b_1] = [0, 1] \). Let \(\Delta x_1 = b_1 - a_1 = 1 - 0 = 1 \).

Step 2: Calculate the Initial Points

Using the golden ratio, set \( r = \frac{\sqrt{5}-1}{2} \approx 0.618033 \), therefore \( \alpha_1 = a_1 + r \times \Delta x_1 = 0.618033 \) and \( \beta_1 = b_1 - r \times \Delta x_1 = 0.381966 \).

Step 3: Evaluate Function Values

Compute the function \( f(x) = 2 + x^2 - \sin(x) \) at the points: \( f(\alpha_1) = f(0.618033) \) and \( f(\beta_1) = f(0.381966) \).

Step 4: Compare Function Values

Evaluate \( f(\alpha_1) \) and \( f(\beta_1) \). If \( f(\alpha_1) > f(\beta_1) \), then the new interval is \([a_2, b_2] = [a_1, \alpha_1]\). Otherwise, it is \([a_2, b_2] = [\beta_1, b_1]\).

Step 5: Repeat the Process

Continue repeating the evaluation and interval reduction steps with \( \Delta x_{n+1} = \rho \Delta x_n \) to iteratively update \( \alpha_n \) and \( \beta_n \) until the interval length is smaller than the desired precision of 0.01.

Step 6: Determine the Approximate Minimum

The point \( c_n \approx c \) at the middle of the final interval \([a_n, b_n]\) after iterative refinement gives the approximate minimum.

Key Concepts

Unimodal Function

A unimodal function is a special type of function that displays a single peak or trough within a specified interval. This very characteristic makes it particularly suitable for optimization algorithms like the Golden Ratio Search. The defining feature of a unimodal function is that it has only one local minimum or maximum.
For a function to be unimodal, the interval must be divided into two sub-intervals: one where the function decreases and another where it increases. This creates a definitive low (or high) point within the interval, making the task of locating the minimum or maximum much easier.
In our Golden Ratio Search exercise, the function \( f(x) = 2 + x^2 - \sin(x) \) is defined over the interval [0, 1]. It's decreasing on one side of the minimum point and increasing on the other. Thus, the definition of unimodal functions plays a crucial role in simplifying the search for the function's minimum value within the interval.

Minimization Algorithm

The Golden Ratio Search is a specific type of minimization algorithm aimed at efficiently finding the minimum value of a unimodal function. The algorithm takes advantage of the properties of the Golden Ratio to reduce the number of evaluations needed to arrive at an approximation to the minimum value.
Here's how it works:

• First, choose two points within the initial interval and evaluate the function at these points.
• Based on the evaluations, you reduce the interval, focusing on the side that still contains the minimum.
• This step is repeated, each time reducing the interval length according to the Golden Ratio, leading to an increasingly precise estimate of the minimum.

The Golden Ratio, approximately 0.618, provides the most efficient reduction of the search interval, ensuring that less time and fewer calculations are necessary compared to other methods. Continued iteration with this method results in a small interval where the true minimum lies, making this method both efficient and powerful for optimization tasks.

Continuous Function

A continuous function is a foundational concept in calculus and optimization, important for ensuring that algorithms like the Golden Ratio Search work properly. Essentially, a function is continuous if it doesn't have any jumps, breaks, or holes in its domain. The value of the function changes smoothly as you move along the interval of interest.
The function \( f(x) = 2 + x^2 - \sin(x) \) given in the exercise is continuous on the interval [0,1], meaning there aren't any abrupt changes in its value. This continuous nature guarantees that the optimization algorithm can function correctly and find the minimum within [0, 1].
In practical terms, the continuity ensures a steady transition through each computed point, thus enabling the algorithm to progressively pin down an approximate value of the minimum efficiently. Without continuity, these strategic choices based on function evaluations could miss the true minimum, leading to incorrect or inefficient results.