6.3 Inverse and implicit function theorems for multifunctions

Inverse and implicit function theorems for multifunctions are powerful tools in set-valued analysis. They extend classical results to multifunctions, allowing us to study solution mappings of variational inequalities and generalized equations.

These theorems provide conditions for the existence of Lipschitz continuous single-valued localizations of inverse mappings. This enables local analysis of multifunction behavior, stability of solutions, and sensitivity to parameter changes in optimization and equilibrium problems.

Inverse Function Theorem for Multifunctions

Statement and Proof

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• The inverse function theorem for multifunctions states that if a multifunction $F$ is metrically regular at $(x̄, ȳ)$ with modulus $κ > 0$, then $F$ admits a Lipschitz continuous single-valued localization around $(x̄, ȳ)$ with a Lipschitz constant $κ$
• Metric regularity of a multifunction $F$ at $(x̄, ȳ)$ means that there exist neighborhoods $U$ of $x̄$ and $V$ of $ȳ$ and a constant $κ > 0$ such that $d(x, F⁻¹(y)) ≤ κd(y, F(x))$ for all $x ∈ U$ and $y ∈ V$
• The proof relies on the Ekeland variational principle and the construction of a sequence of approximate solutions to the inclusion $y ∈ F(x)$
  • The Ekeland variational principle is used to establish the existence of a nearby point that minimizes a certain function
  • The sequence of approximate solutions is constructed using the metric regularity property and the Ekeland point
• The theorem guarantees the existence of a single-valued localization of $F⁻¹$ around $(ȳ, x̄)$, denoted by $f$, which is Lipschitz continuous with a Lipschitz constant $κ$
  • The single-valued localization $f$ satisfies $f(ȳ) = x̄$ and $F(f(y)) ∋ y$ for all $y$ in a neighborhood of $ȳ$
  • This means that $f$ locally inverts the multifunction $F$ around the point $(ȳ, x̄)$

Metric Regularity and Its Implications

• Metric regularity is a key concept in the inverse function theorem for multifunctions
• It quantifies the stability of the inverse multifunction $F⁻¹$ near the point $(x̄, ȳ)$
• The modulus of metric regularity $κ$ measures how much the distance between a point $x$ and the inverse image $F⁻¹(y)$ can be controlled by the distance between $y$ and the image $F(x)$
  • A smaller value of $κ$ indicates a more stable inverse multifunction
  • A larger value of $κ$ suggests a less stable inverse multifunction
• Metric regularity ensures that the inverse multifunction $F⁻¹$ behaves well locally and can be approximated by a Lipschitz continuous single-valued function
• The Lipschitz constant of the single-valued localization $f$ is determined by the modulus of metric regularity $κ$
  • This connection highlights the importance of metric regularity in the local behavior of the inverse multifunction

Applying the Inverse Function Theorem

Local Analysis of Multifunctions

• The inverse function theorem for multifunctions enables the local analysis of the behavior of a multifunction $F$ around a point $(x̄, ȳ)$ where $F$ is metrically regular
• If $F$ is metrically regular at $(x̄, ȳ)$, the theorem guarantees the existence of a Lipschitz continuous single-valued localization $f$ of $F⁻¹$ around $(ȳ, x̄)$
  • This localization provides a local approximation of the inverse multifunction $F⁻¹$ near the point $(ȳ, x̄)$
  • It allows for the study of the local properties of $F⁻¹$, such as its stability, sensitivity, and behavior under perturbations
• The Lipschitz continuity of $f$ provides information about the local stability and sensitivity of the inverse multifunction $F⁻¹$ near $(ȳ, x̄)$
  • A smaller Lipschitz constant indicates a more stable and less sensitive inverse multifunction
  • A larger Lipschitz constant suggests a less stable and more sensitive inverse multifunction
• The modulus of metric regularity $κ$ serves as a measure of the local invertibility of $F$ and the Lipschitz constant of the single-valued localization $f$
  • It quantifies the relationship between the stability of $F⁻¹$ and the regularity of $F$ around the point $(x̄, ȳ)$

Applications to Variational Problems

• The inverse function theorem can be used to study the local properties of solution mappings to parameterized variational inequalities, generalized equations, and optimization problems
• In these contexts, the theorem provides conditions for the existence and stability of solutions near a given point
• For example, consider a parameterized variational inequality problem: Find $x ∈ C$ such that $⟨F(x, p), y - x⟩ ≥ 0$ for all $y ∈ C$, where $p$ is a parameter
  • The solution mapping $S(p) = \{x ∈ C | ⟨F(x, p), y - x⟩ ≥ 0, ∀y ∈ C\}$ associates each parameter value $p$ with the corresponding set of solutions
  • If the mapping $(x, p) ↦ F(x, p)$ is metrically regular at a point $(x̄, p̄)$ with $x̄ ∈ S(p̄)$, the inverse function theorem guarantees the existence of a Lipschitz continuous single-valued localization of $S$ around $p̄$
  • This localization provides information about the local behavior of the solution mapping and the stability of solutions with respect to parameter perturbations
• Similar applications can be found in the study of generalized equations, equilibrium problems, and optimization problems, where the inverse function theorem helps analyze the local properties of solution mappings and establish stability results

Implicit Function Theorem for Multifunctions

Statement and Assumptions

• The implicit function theorem for multifunctions generalizes the classical implicit function theorem to the setting of multifunctions
• Consider a multifunction $F(x, y)$ and a point $(x̄, ȳ)$ such that $ȳ ∈ F(x̄, ȳ)$
• The theorem states that if the partial multifunction $x ↦ F(x, ȳ)$ is metrically regular at $(x̄, ȳ)$ with modulus $κ > 0$, then there exists a Lipschitz continuous single-valued localization $y(x)$ of the solution mapping $S(x) := \{y | 0 ∈ F(x, y)\}$ around $x̄$ with $y(x̄) = ȳ$
  • The partial multifunction $x ↦ F(x, ȳ)$ fixes the second argument $y$ at $ȳ$ and varies only the first argument $x$
  • Metric regularity of this partial multifunction at $(x̄, ȳ)$ is a key assumption of the theorem
• The theorem provides conditions under which the solution mapping $S(x)$ admits a Lipschitz continuous single-valued localization around $x̄$
  • This localization, denoted by $y(x)$, satisfies $0 ∈ F(x, y(x))$ for all $x$ in a neighborhood of $x̄$ and $y(x̄) = ȳ$
  • It locally parameterizes the set of solutions to the inclusion $0 ∈ F(x, y)$ in terms of the variable $x$

Applications and Consequences

• The implicit function theorem for multifunctions has numerous applications in the study of parameterized variational inequalities, generalized equations, and equilibrium problems
• It can be used to analyze the local behavior of solution mappings and to establish the existence and stability of solutions to parameterized problems
• For instance, consider a parameterized generalized equation: Find $y$ such that $0 ∈ F(x, y)$, where $x$ is a parameter
  • The solution mapping $S(x) = \{y | 0 ∈ F(x, y)\}$ assigns to each parameter value $x$ the corresponding set of solutions
  • If the partial multifunction $x ↦ F(x, ȳ)$ is metrically regular at $(x̄, ȳ)$ with $0 ∈ F(x̄, ȳ)$, the implicit function theorem ensures the existence of a Lipschitz continuous single-valued localization $y(x)$ of $S$ around $x̄$
  • This localization describes the local behavior of the solution mapping and provides information about the stability of solutions with respect to parameter variations
• The theorem can also be applied to study the sensitivity of solutions to perturbations in the problem data
  • By examining the Lipschitz constant of the single-valued localization $y(x)$, one can quantify the local sensitivity of solutions to changes in the parameter $x$
  • A smaller Lipschitz constant indicates less sensitive solutions, while a larger Lipschitz constant suggests more sensitive solutions
• The implicit function theorem for multifunctions serves as a powerful tool for investigating the local properties of solution mappings and establishing stability and sensitivity results in various problem settings

Inverse and Implicit Multifunctions in Context

Optimization and Equilibrium Analysis

• Inverse and implicit function theorems for multifunctions find significant applications in optimization and equilibrium analysis
• In optimization, these theorems can be used to study the local behavior of solution mappings to parameterized optimization problems and to establish the existence and stability of optimal solutions
  • Consider a parameterized optimization problem: Minimize $f(x, p)$ subject to $x ∈ C(p)$, where $p$ is a parameter
  • The solution mapping $S(p) = \{x ∈ C(p) | f(x, p) = \min_{y ∈ C(p)} f(y, p)\}$ associates each parameter value $p$ with the corresponding set of optimal solutions
  • If the objective function $f$ and the constraint mapping $C$ satisfy suitable regularity conditions, the inverse or implicit function theorem can be applied to analyze the local behavior of $S$ around a given point
  • This analysis provides insights into the stability and sensitivity of optimal solutions with respect to parameter perturbations
• In equilibrium analysis, the implicit function theorem for multifunctions is useful for investigating the local properties of equilibrium mappings and proving the existence and stability of equilibria in parameterized equilibrium problems
  • Consider a parameterized equilibrium problem: Find $x ∈ C$ such that $F(x, x, p) ∈ K$, where $p$ is a parameter, $C$ is a constraint set, $F$ is an equilibrium mapping, and $K$ is a target set
  • The equilibrium mapping $E(p) = \{x ∈ C | F(x, x, p) ∈ K\}$ assigns to each parameter value $p$ the corresponding set of equilibria
  • If the equilibrium mapping $F$ satisfies appropriate regularity conditions, the implicit function theorem can be used to study the local behavior of $E$ around a given equilibrium point
  • This analysis helps understand the stability and sensitivity of equilibria with respect to parameter variations and provides conditions for the existence of locally unique and stable equilibria

Variational Inequalities and Generalized Equations

• The inverse and implicit function theorems for multifunctions are powerful tools for studying variational inequalities and generalized equations
• In the context of variational inequalities, these theorems can be employed to analyze the local behavior of solution mappings and to establish the existence and stability of solutions to parameterized variational inequalities
  • Consider a parameterized variational inequality problem: Find $x ∈ C$ such that $⟨F(x, p), y - x⟩ ≥ 0$ for all $y ∈ C$, where $p$ is a parameter, $C$ is a constraint set, and $F$ is a mapping
  • The solution mapping $S(p) = \{x ∈ C | ⟨F(x, p), y - x⟩ ≥ 0, ∀y ∈ C\}$ assigns to each parameter value $p$ the corresponding set of solutions
  • If the mapping $F$ satisfies suitable regularity conditions, the inverse or implicit function theorem can be applied to study the local behavior of $S$ around a given solution point
  • This analysis provides insights into the stability and sensitivity of solutions with respect to parameter perturbations and helps establish conditions for the existence and uniqueness of solutions
• Generalized equations, which encompass variational inequalities and complementarity problems, can also be studied using the inverse and implicit function theorems for multifunctions
  • Consider a parameterized generalized equation: Find $x$ such that $0 ∈ F(x, p)$, where $p$ is a parameter and $F$ is a multifunction
  • The solution mapping $S(p) = \{x | 0 ∈ F(x, p)\}$ assigns to each parameter value $p$ the corresponding set of solutions
  • If the multifunction $F$ satisfies appropriate regularity conditions, the inverse or implicit function theorem can be employed to investigate the local behavior of $S$ around a given solution point
  • This analysis helps understand the stability and sensitivity of solutions with respect to parameter variations and provides conditions for the existence and local uniqueness of solutions
• The inverse and implicit function theorems for multifunctions serve as essential tools for studying the local properties of solution mappings in variational inequalities and generalized equations, enabling a deeper understanding of the stability, sensitivity, and parametric dependence of solutions