Question

A popular technique arising in methods for minimizing functions in several variables involves a weak line search, where an approximate minimum \(x^{*}\) is found for a function in one variable, \(f(x)\), for which the values of \(f(0), f^{\prime}(0)\), and \(f(1)\) are given. The function \(f(x)\) is defined for all nonnegative \(x\), has a continuous second derivative, and satisfies \(f(0)

Short answer

Question: Based on the weak line search technique, create a short answer for finding the minimum point x* for given values of f(0), f'(0), and f(1). Answer: By interpolating the given values of f(0), f'(0), and f(1) using a quadratic polynomial, we find the minimum of the interpolating polynomial, which occurs at x* = 1/4.

Step-by-step solution

Write the quadratic interpolating polynomial for f(x)

Consider a quadratic interpolating polynomial given by \(P(x)=ax^2+bx+c\). Using the given information about f(x), we can write down a system of three equations as follows (remember \(f'(0)=-1\)): 1. \(P(0) = c = f(0) = 1\) 2. \(P'(0) = b = f'(0) = -1\) 3. \(P(1) = a+b+c = f(1) = 2\) Now we can solve this system of equations to obtain the coefficients a, b, and c.

Find the coefficients a, b, and c

From the system of equations in Step 1, we already know that \(c=1\) and \(b=-1\). We now just need to solve for a: \(a + b + c = f(1)\) \(a - 1 + 1 = 2\) \(a = 2\) Therefore, our interpolating polynomial is given by \(P(x) = 2x^2 - x + 1\).

Find the minimum of the interpolating polynomial P(x)

To find the minimum of P(x), we first find its derivative and then set it equal to zero: \(P'(x) = 4x - 1\) Setting \(P'(x)\) to zero and solving for x, we get: \(4x - 1 = 0\) \(x = \frac{1}{4}\) So, the minimum of the interpolating polynomial occurs at \(x^{*}=\frac{1}{4}\). (a) The value of \(x^{*}\) is \(\frac{1}{4}\).

Determine if the quadratic interpolant has a unique minimum satisfying 0 < x* < 1

To show that the minimum we found is unique, we can look at the second derivative of the interpolating polynomial: \(P''(x) = 4\) Since \(P''(x) > 0\), we know that the interpolating polynomial is convex, and there exists a unique minimum between 0 and 1. (b) Yes, the quadratic interpolant has a unique minimum satisfying \(0 < x^{*} < 1\). As for the function f, it is defined for all nonnegative x, has a continuous second derivative, and the given conditions \(f(0) < f(1)\) and \(f'(0) < 0\) ensure that there is a unique local minimum x* between 0 and 1. However, without further information about the function, we cannot be sure if the minimum at x* will be the only minimum of the function f.

Key concepts

Interpolating Polynomial

An interpolating polynomial is a polynomial that passes through a given set of points and is often utilized to estimate data points within the range of a discrete set of known data points. This polynomial provides a simpler function that can be used to approximate more complex functions. When dealing with numerical problems, it's essential to ensure that an interpolating polynomial not only passes through the data points but also reflects the function's behavior between and beyond those points.

For instance, if we know the value of a function and its derivatives at certain points, this information can be utilized to construct the polynomial. In the context of optimization, such as in a weak line search algorithm, an interpolating polynomial can help to find an approximate minimum of a function by using the known values of the function at certain intervals. In the provided exercise, the interpolating polynomial was used to estimate the minimum of the function based on given data rather than finding the exact minimum through more complicated calculations.

Quadratic Polynomial Interpolation

Quadratic polynomial interpolation is a specific case of interpolating polynomials where a second-degree polynomial is used. This polynomial takes the form of \(P(x) = ax^2 + bx + c\), which is determined by using three points or conditions. Since it is quadratic, the graph of the interpolating polynomial is a parabola, which can either open upward or downward, causing the shape to have either a single unique minimum or maximum.

In our exercise scenario, a quadratic polynomial was interpolated based on the known values of the function and its derivative. The determined coefficients reflect the rate of change and curvature of the original function. Since the interpolated polynomial has a positive leading coefficient (\(a = 2\) in the exercise), we know that the parabola opens upwards, indicating that any minimum point found is unique within the specified range.

Numerical Optimization Methods

Numerical optimization methods encompass a wide range of algorithms used to find the maximum or minimum of functions, often when analytical solutions are difficult to obtain or don't exist. These methods are at the core of many scientific and engineering challenges, including minimizing the cost, error, or energy, or maximizing efficiency, likelihood, or profit.

Optimization problems can often be visualized as the task of finding the lowest or highest point in a landscape of hills and valleys. A weak line search is one such numerical optimization technique, where we approximate the minimum of a function along a path or 'line' in the problem's domain. This method is particularly useful when dealing with high-dimensional spaces. The exercise we considered is an instance of this, as it involves using a quadratic polynomial to approximate the minimum based on a few given values, streamlining the process by reducing the complexity of the optimization task.