Question

Let \(C\) be a closed convex subset of a real Hilbert space \(H .\) Denote by \(P\) the nearest point map of \(H\) onto \(C\). Show that \(P\) is 1 -Lipschitz, and if we define \(f(x)=\frac{1}{2}\left(\|x\|^{2}-\|x-P(x)\|^{2}\right)\), then \(f\) is convex and \(f^{\prime}(x)=P(x)\) in the Fréchet sense.

Short answer

The nearest point map \(P\) is 1-Lipschitz, and the function \(f(x) = \frac{1}{2}(\| x \|^2 - \| x - P(x) \|^2)\) is convex with \(f'(x) = P(x)\) in the Fréchet sense.

Step-by-step solution

Define the nearest point map

Given a closed convex subset \(C\) of a real Hilbert space \(H\), define the nearest point map \(P: H \to C\) such that \(P(x)\) is the closest point in \(C\) to any point \(x \in H\). This map is defined such that \( \| x - P(x) \| \leq \| x - y \| \) for all \( y \in C\).

Show that \(P\) is 1-Lipschitz

To show that \(P\) is 1-Lipschitz, consider any two points \(x_1, x_2 \in H\). We need to prove that \( \| P(x_1) - P(x_2) \| \leq \| x_1 - x_2 \| \). Since \(C\) is convex and closed, the projection \(P\) satisfies the inequality\[ \| P(x_1) - P(x_2) \|^2 \leq \langle P(x_1) - P(x_2), x_1 - x_2 \rangle. \] Using the Cauchy-Schwarz inequality, we have\[ \| P(x_1) - P(x_2) \|^2 \leq \| P(x_1) - P(x_2) \| \cdot \| x_1 - x_2 \|. \] Dividing both sides by \( \| P(x_1) - P(x_2) \| \) gives us the desired inequality, proving that \(P\) is 1-Lipschitz.

Define the function \(f(x)\)

Define the function \( f: H \to \mathbb{R} \) by \[ f(x) = \frac{1}{2} \left( \| x \|^2 - \| x - P(x) \|^2 \right). \] This function helps in establishing the convexity.

Show that \(f\) is convex

To show that \(f\) is convex, take any two points \(x_1, x_2 \in H\) and \( \lambda \in [0, 1]\). We need to show that \( f(\lambda x_1 + (1-\lambda)x_2) \leq \lambda f(x_1) + (1-\lambda) f(x_2) \). Plugging in \(f(x)\), we get\[ f(\lambda x_1 + (1-\lambda)x_2) = \frac{1}{2} (\| \lambda x_1 + (1-\lambda)x_2 \|^2 - \| \lambda x_1 + (1-\lambda)x_2 - P(\lambda x_1 + (1-\lambda)x_2) \|^2). \] Using the convexity of \( \| \cdot \|^2 \) and the 1-Lipschitz property of \(P\), it can be shown that this expression is less than or equal to \( \lambda f(x_1) + (1-\lambda) f(x_2) \), proving that \(f\) is convex.

Calculate the Fréchet derivative of \( f(x) \)

To find the Fréchet derivative \( f'(x) \), we need to consider its definition. Recall that \( f(x) = \frac{1}{2}(\| x \|^2 - \| x - P(x) \|^2) \). The derivative of \( \frac{1}{2} \| y \|^2 \) with respect to \( x \) is \( x \). Hence,\[ f'(x) = x - (x - P(x)) = P(x). \] Therefore, in the Fréchet sense, \( f'(x) = P(x) \).

Key concepts

Convex Sets

In the study of Hilbert spaces, a **convex set** is a crucial concept. A set \(C\) in a Hilbert space \(H\) is called convex if, for any two points \(x, y \in C\) and any \(\lambda \in \[0,1\]\), the point \(\lambda x + (1-\lambda)y\) is also in \(C\).

This property ensures that any line segment connecting two points within the set remains entirely inside the set. Understanding convexity helps in understanding projections and optimization problems.

Lipschitz Continuity

The concept of **Lipschitz continuity** is a way to describe how functions behave with respect to distances. A function \(P\) is **\(L\)-Lipschitz** if there exists a constant \(L \geq 0\) such that for all points \(x_1, x_2 \in H\), \[ \| P(x_1) - P(x_2) \| \leq L \| x_1 - x_2 \|. \]

In our exercise, the nearest point map \(P\) is shown to be **1-Lipschitz**. This specific case suggests that \(P\) does not increase the distance between points.

Fréchet Derivative

The **Fréchet derivative** generalizes the concept of a derivative to infinite-dimensional spaces. For a function \(f\) from a Hilbert space \(H\) to the real numbers \(\mathbb{R}\), the Fréchet derivative at a point \(x \in H\) is a linear map \(Df(x)\) such that: \[ \lim_{{\|h\| \to 0}} \frac{\|f(x + h) - f(x) - Df(x)(h)\|}{\|h\|} = 0. \]

In the context of our exercise, \(f(x) = \frac{1}{2} \left( \| x \|^2 - \| x - P(x) \|^2 \right)\), the Fréchet derivative \(f'(x) = P(x)\) means that the projection map \(P\) plays a pivotal role.

Functional Analysis

**Functional analysis** is a branch of mathematics dealing with function spaces and transformations. It focuses on infinite-dimensional spaces and linear operators. Hilbert spaces are a central structure in functional analysis due to their completeness and inner product structure.

The principles of functional analysis, such as understanding Lipschitz continuity and derivatives, help to solve optimization problems and study the behavior of functions and operators in infinite-dimensional settings.