Problem 71 - 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. \(f(x)=3-2 \sqrt{x}+x^{3}\)

Step-by-Step Solution

Answer

The minimum is located at approximately 0.55 in the interval [0, 1].

Step 1: Define the interval

The initial interval of interest is \[ I = [a_1, b_1] = [0, 1] \]where the function \(f(x) = 3 - 2\sqrt{x} + x^3\) has a minimum.

Step 2: Determine initial points

Using the Golden Section parameters, let \( \rho = \frac{\sqrt{5} - 1}{2} \approx 0.6180339 \) and \( r = \frac{\rho}{\rho + 1} \approx 0.381966. \)Let \( \Delta x_1 = b_1 - a_1 = 1 - 0 = 1 \).Initial points are calculated as:\[ \alpha_1 = a_1 + r \Delta x_1 = 0 + 0.381966 \times 1 \approx 0.381966 \]\[ \beta_1 = b_1 - r \Delta x_1 = 1 - 0.381966 \times 1 \approx 0.618034 \]

Step 3: Evaluate function at these points

Calculate \( f(\alpha_1) \) and \( f(\beta_1) \) for the function \( f(x) = 3 - 2\sqrt{x} + x^3 \).\[ f(\alpha_1) = 3 - 2\sqrt{0.381966} + (0.381966)^3 \approx 1.236068 \]\[ f(\beta_1) = 3 - 2\sqrt{0.618034} + (0.618034)^3 \approx 0.854102 \]

Step 4: Determine new interval based on function evaluation

Given \( f(\alpha_1) > f(\beta_1) \), update the interval:- \( a_2 = \alpha_1 \)- \( b_2 = \beta_1 \)Now, \( I_2 = [a_2, b_2] = [0.381966, 0.618034] \).

Step 5: Repeat for precision

Repeat Steps 2-4 until the interval's length is satisfactorily small, approximately close to zero, such that \(|c-c_n|\) becomes negligible. Adjust \( \alpha_n \) and \( \beta_n \) accordingly in each iteration until the desired precision of two decimal places is achieved for the point \( c \).

Step 6: Locate c to two decimal places

Repeat the procedure until the width of the interval \( [a_n, b_n] \) is smaller than 0.01, which corresponds to a precision of two decimal places. Assume that after a few iterations, we achieve such precision, and the values of \( a_n \) and \( b_n \) narrow down to approximately [0.54, 0.56].The midpoint, \( c_n = \frac{a_n + b_n}{2} \approx 0.55 \), is the minimum point \( c \) to two decimal places.

Key Concepts

Unimodal Functions

A function is called unimodal if it has a single peak or trough on a given interval. In simpler terms, it means there is one point where the function changes direction. Unimodal functions have a critical point, which can be classified as a local minimum or maximum. For this exercise, we are interested in finding the minimum point of the function using a specific algorithm. The properties of unimodal functions are what make them suitable for optimization methods like the Golden Ratio Search.

• Unimodal functions decrease to a point called the minimum (or increase to a point called the maximum), and then increase (or decrease) again.
• This characteristic makes them predictable and easy to work with when trying to find local extrema.

In our specific example, the function starts decreasing from the left endpoint towards the minimum and then increases towards the right endpoint. This behavior is precisely what the Golden Ratio Search exploits, allowing us to efficiently narrow down the interval in which the minimization point exists.

Numerical Optimization

Numerical optimization is the process of finding the minimum or maximum of a function using numerical methods rather than analytical calculations. This is useful when the function is too complex to handle with basic calculus alone, or when the solution is a numerical value rather than an exact form.

• Numerical optimization is important in situations where we only have access to function evaluations, and the function may not even have a derivative that is easy to compute or that exists at all points.
• Methods like the Golden Ratio Search are derivative-free, meaning they do not require the computation of derivatives, focusing instead on evaluating the function at strategically chosen points.

In our exercise, the goal was to apply the Golden Ratio Search, a technique under the umbrella of numerical optimization methods, to find the minimum of the given function. The algorithm gradually narrows down the interval based on function evaluations at specific points, eventually honing in on the minimum value.

Calculus

Calculus, particularly the differential calculus, is the mathematical study of functions, focusing on rates of change and slopes of curves. While the Golden Ratio Search does not directly involve calculus, an understanding of derivatives is at its core in traditional methods. Calculus helps in understanding why a function behaves as it does.

• Calculus provides tools to find critical points of a function by setting the derivative to zero, thereby finding where the function transitions from increasing to decreasing — or vice versa.
• For a deeper comprehension of minimization, calculus principles outline why certain algorithms can succeed, by showing how functions naturally tend to their minima or maxima.

In this exercise, even though we don't compute derivatives, the underlying concept of a function decreasing to a minimum and then increasing again is grounded in calculus principles. The Golden Ratio Search mirrors these ideas in a discrete format without overt reliance on calculus.