8.1 Monotone operators and their properties

Monotone operators are key players in variational analysis, linking convex functions and optimization. They're set-valued maps that satisfy a special condition, making them useful for solving complex problems.

Maximal monotone operators, like subdifferentials of convex functions, have unique properties. Their resolvents are single-valued and nonexpansive, which is super helpful in developing algorithms for optimization and variational inequalities.

Monotone Operators in Hilbert Spaces

Definition and Key Properties

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• A monotone operator is a set-valued mapping $T$ from a Hilbert space $H$ to its dual $H^*$ that satisfies the monotonicity condition: $\langle y - x, v - u \rangle \geq 0$ for all $(x, u)$, $(y, v)$ in the graph of $T$
• The graph of a monotone operator $T$, denoted by $G(T)$, is the set of all pairs $(x, u)$ in $H \times H^*$ such that $u \in T(x)$
• A monotone operator $T$ is said to be maximal if its graph is not properly contained in the graph of any other monotone operator
• The subdifferential of a proper, convex, and lower semicontinuous function is a maximal monotone operator
• The resolvent of a monotone operator $T$ with parameter $\lambda > 0$ is defined as $(I + \lambda T)^{-1}$, where $I$ is the identity operator
  • The resolvent is a single-valued, nonexpansive mapping

Relationship with Convex Functions

• The subdifferential of a proper, convex, and lower semicontinuous function $f : H \rightarrow (-\infty, +\infty]$ is a maximal monotone operator
  • The subdifferential of $f$ at $x \in H$ is the set $\partial f(x) = \{u \in H^* : f(y) \geq f(x) + \langle y - x, u \rangle$ for all $y \in H\}$
• If $T$ is a maximal monotone operator, then there exists a proper, convex, and lower semicontinuous function $f$ such that $T = \partial f$
  • This function $f$ is called the potential of $T$ and is unique up to an additive constant
• The Fenchel-Rockafellar duality theorem relates the convex conjugates of two proper, convex, and lower semicontinuous functions to their subdifferentials

Properties of Monotone Operators

Additivity and Scalar Multiplication

• Additivity: If $S$ and $T$ are monotone operators, then their sum $S + T$, defined by $(S + T)(x) = S(x) + T(x)$, is also a monotone operator
• Scalar multiplication: If $T$ is a monotone operator and $\alpha$ is a positive scalar, then $\alpha T$, defined by $(\alpha T)(x) = \alpha T(x)$, is also a monotone operator

Proofs of Additivity and Scalar Multiplication

• Proof of additivity: Let $(x, u)$ and $(y, v)$ be elements of the graphs of $S$ and $T$, respectively
  • Then, $\langle y - x, s - u \rangle \geq 0$ for all $s \in S(y)$ and $\langle y - x, t - v \rangle \geq 0$ for all $t \in T(y)$
  • Adding these inequalities yields $\langle y - x, (s + t) - (u + v) \rangle \geq 0$, which proves the monotonicity of $S + T$
• Proof of scalar multiplication: Let $(x, u)$ and $(y, v)$ be elements of the graph of $T$
  • Multiplying the monotonicity condition by $\alpha > 0$ gives $\alpha \langle y - x, v - u \rangle \geq 0$, which is equivalent to $\langle y - x, \alpha v - \alpha u \rangle \geq 0$, proving the monotonicity of $\alpha T$

Examples of Monotone Operators

Optimization and Variational Analysis

• The subdifferential of the $\ell_1$-norm, $\partial \|\cdot\|_1$, is a monotone operator used in sparse optimization problems (compressed sensing, feature selection)
• The normal cone operator to a nonempty, closed, and convex set $C \subseteq H$, defined as $N_C(x) = \{u \in H^* : \langle y - x, u \rangle \leq 0$ for all $y \in C\}$ if $x \in C$ and $N_C(x) = \emptyset$ if $x \notin C$, is a maximal monotone operator (projection onto convex sets, variational inequalities)
• The gradient of a differentiable, convex function is a monotone operator (gradient descent, convex optimization)
  • If the function is strictly convex, then its gradient is strictly monotone
• The proximal point algorithm for finding a zero of a maximal monotone operator $T$ involves iteratively applying the resolvent of $T$ to a starting point (optimization, variational inequalities)