Glossary

6.1 Set-valued mappings and their properties

4 min read•august 14, 2024

Set-valued mappings are a key concept in variational analysis, assigning subsets of a codomain to elements in a domain. They generalize single-valued functions, allowing for more flexible modeling of complex relationships and uncertainties in various mathematical and real-world scenarios.

This topic explores the definition, properties, and examples of set-valued mappings. It covers important characteristics like , , and , which are crucial for understanding their behavior and applications in optimization, economics, and control theory.

Set-Valued Mappings: Definition and Properties

Definition and Notation

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  • , also known as multifunction or correspondence, assigns to each element of its domain a subset of its codomain
  • Represented using notation such as F:X⇒YF: X ⇒ Y, where XX is the domain and YY is the codomain
  • Domain of a set-valued mapping is the set of elements for which the mapping is defined
  • Range is the set of all elements that are assigned to at least one element in the domain
  • is the set of ordered pairs (x,y)(x, y) such that yy belongs to the image of xx under the mapping

Examples of Set-Valued Mappings

  • assigns to each set its power set
  • Inverse function of a non-injective single-valued function
  • Mapping that assigns to each real number xx the interval [x,x+1][x, x+1]
  • Mapping that assigns to each point in a metric space the set of its limit points

Properties of Set-Valued Mappings

  • Set-valued mapping is convex if its graph is a convex set, meaning that for any two points in the graph, the line segment connecting them is also contained in the graph
  • Convexity of set-valued mappings is preserved under various operations (intersection, addition, scalar multiplication)
  • Convex set-valued mappings have important applications in optimization and economics
  • Related properties include , , and

Closedness and Boundedness

  • Set-valued mapping is closed if its graph is a closed set, meaning that it contains all its limit points
  • have important properties (preservation of , existence of fixed points under certain conditions)
  • Set-valued mapping is bounded if its range is a bounded set, meaning that there exists a positive real number MM such that the distance between any two points in the range is less than MM
  • Boundedness of set-valued mappings is related to the concept of uniform boundedness and plays a role in the study of optimization problems
  • Examples of closed and include the mapping that assigns to each point in a compact set its closest point in another compact set

Continuity and Semicontinuity of Set-Valued Mappings

Definitions and Characterizations

  • Continuity of set-valued mappings is defined in terms of the behavior of the images of points close to a given point in the domain
  • Set-valued mapping is upper semicontinuous at a point if, for any open set containing the image of that point, there exists a neighborhood of the point such that the images of all points in the neighborhood are contained in the open set
  • is defined similarly, but with the requirement that the images of points in the neighborhood intersect the open set
  • Continuity of set-valued mappings is equivalent to the mapping being both upper and lower semicontinuous
  • Characterizations of continuity and semicontinuity in terms of sequences, nets, and filters

Implications and Applications

  • Continuity and semicontinuity of set-valued mappings have implications for the existence of fixed points, the stability of solutions to optimization problems, and the convergence of algorithms
  • Continuous set-valued mappings preserve compactness and connectedness
  • Upper semicontinuous set-valued mappings with closed convex values have fixed points under certain conditions ()
  • Lower semicontinuous set-valued mappings arise in the study of subdifferentials and normal cones in nonsmooth analysis
  • Applications in economics (equilibrium theory), control theory (reachable sets), and game theory (best response correspondences)

Set-Valued Mappings vs Single-Valued Functions

Relationships and Differences

  • Single-valued functions can be seen as a special case of set-valued mappings, where each element of the domain is assigned a singleton set (a set containing exactly one element)
  • Composition of set-valued mappings is defined differently from the composition of single-valued functions, as it involves taking the union of the images of the second mapping over the image of the first mapping
  • Inverse of a set-valued mapping is always defined, unlike the inverse of a single-valued function, which requires injectivity
  • Set-valued mappings can be used to study the properties of single-valued functions (existence of solutions to equations, behavior of optimization problems)

Extensions of Classical Results

  • Many concepts and results from the theory of single-valued functions have analogues in the context of set-valued mappings
  • Intermediate value theorem for set-valued mappings states that if a set-valued mapping is upper semicontinuous with compact convex values, then its range is connected
  • Implicit function theorem for set-valued mappings provides conditions under which a set-valued mapping can be locally represented as the graph of a single-valued function
  • for set-valued mappings, also known as the , gives conditions for the existence and uniqueness of fixed points of contractive set-valued mappings
  • Lyapunov stability theory for set-valued dynamical systems extends the classical stability results for single-valued systems

Key Terms to Review (24)

Baire Category Theorem: The Baire Category Theorem states that in a complete metric space, the intersection of countably many dense open sets is dense. This theorem highlights the importance of completeness in spaces and connects to various results in functional analysis, topology, and set-valued mappings, emphasizing how certain properties hold under specific conditions.
Banach Fixed Point Theorem: The Banach Fixed Point Theorem states that every contraction mapping on a complete metric space has a unique fixed point. This theorem is crucial in analysis and applied mathematics, particularly in proving the existence and uniqueness of solutions to various equations and optimization problems.
Bounded set-valued mappings: Bounded set-valued mappings are functions that associate each point in their domain with a bounded set of values in the codomain. These mappings ensure that the sets associated with each point do not extend beyond a certain limit, maintaining a form of control over the size of the outputs. This property is essential when studying continuity, compactness, and convergence in variational analysis, as it helps to establish well-defined behaviors of these mappings under various mathematical operations.
Closed Set-Valued Mappings: Closed set-valued mappings are functions that associate each point in a domain with a closed subset of a codomain. These mappings are essential in variational analysis as they help describe solutions to optimization problems where multiple outputs may correspond to a single input, especially when the output sets have closure properties that ensure limits of converging sequences stay within the mapped values.
Closedness: Closedness refers to a property of sets in topology, indicating that a set contains all its limit points. In optimization and analysis, it plays a crucial role in various contexts, such as characterizing the continuity and stability of solutions and providing necessary conditions for optimality. Understanding closedness helps connect concepts like subdifferentials, set-valued mappings, and operators, which are foundational to understanding the structure of convex analysis.
Compactness: Compactness is a property of a space that ensures every open cover has a finite subcover, meaning that from any collection of open sets that covers the space, one can extract a finite number of those sets that still cover the entire space. This property is crucial in various areas of analysis and optimization, ensuring that limits exist and solutions are bounded.
Connectedness: Connectedness refers to a property of a space in which any two points can be joined by a continuous path within that space. In the context of set-valued mappings, connectedness helps to understand the structure and relationships of sets, especially how they relate to continuity and path-connectedness in mathematical analysis.
Continuity: Continuity refers to the property of a function or mapping that ensures small changes in the input lead to small changes in the output. This concept is crucial for ensuring the stability of solutions and the behavior of functions in various mathematical contexts, such as optimization and analysis, influencing how problems are approached and solved.
Contractibility: Contractibility refers to a property of a topological space that indicates whether it can be continuously shrunk to a point. A space is contractible if there exists a homotopy that continuously deforms the entire space into a single point, which implies that the space is path-connected and simply connected. This concept is crucial in understanding the behavior of set-valued mappings, as contractible spaces can simplify the analysis of continuity and convergence in these mappings.
Convexity: Convexity refers to a property of sets and functions in which a line segment connecting any two points within the set or on the graph of the function lies entirely within the set or above the graph, respectively. This concept is crucial in optimization and variational analysis as it ensures that local minima are also global minima, simplifying the search for optimal solutions.
Graph of a set-valued mapping: The graph of a set-valued mapping is the collection of ordered pairs where each input is associated with a non-empty subset of outputs. This concept extends traditional functions, where instead of assigning a single output to each input, it assigns a set of possible outputs, allowing for more flexibility in modeling relationships. Understanding the graph of set-valued mappings is crucial for analyzing the properties and behaviors of these mappings in various mathematical contexts.
Kakutani's Fixed Point Theorem: Kakutani's Fixed Point Theorem states that if a compact convex set in a Euclidean space has a set-valued mapping that is upper hemicontinuous and has non-empty, convex values, then there exists at least one fixed point within that set. This theorem is significant in various fields such as economics and game theory, where it establishes the existence of equilibrium points in strategic situations.
Lower Hemicontinuous: Lower hemicontinuous refers to a property of set-valued mappings where the inverse image of every open set is open in the domain. In simpler terms, it means that for any point in the target space, if you slightly change the input, the output does not suddenly drop down, ensuring a kind of stability in the mapping. This property is crucial in variational analysis as it helps in understanding how solutions to optimization problems behave under small perturbations.
Lower Semicontinuity: Lower semicontinuity refers to a property of a function where, intuitively, the value of the function does not jump upwards too abruptly. Formally, a function is lower semicontinuous at a point if, for any sequence approaching that point, the limit of the function values at those points is greater than or equal to the function value at the limit point. This concept connects with various ideas like subgradients and generalized gradients, as well as with set-valued mappings and their continuity, making it essential in optimization and variational analysis.
Nadler Fixed Point Theorem: The Nadler Fixed Point Theorem states that if a set-valued mapping is continuous and maps a compact convex set into itself, then there exists a fixed point in that set. This theorem extends classical fixed point theorems by considering set-valued mappings, which assign multiple values to each point in their domain. It highlights the importance of continuity and compactness in establishing the existence of fixed points, serving as a bridge between topology and analysis.
Optimal Control Problems: Optimal control problems involve finding a control policy that minimizes or maximizes a certain objective, typically related to a dynamic system's performance over time. These problems often utilize variational analysis to establish the existence of optimal solutions and to analyze the properties of control strategies within various applications such as economics, engineering, and biology.
Pointwise Convergence: Pointwise convergence refers to a type of convergence of a sequence of functions where each function in the sequence converges to a limiting function at each point in the domain. This means that for every point in the domain, the values of the sequence of functions approach the corresponding value of the limit function as the index goes to infinity. Understanding pointwise convergence is crucial when discussing properties such as continuity and differentiability, and it is also foundational when analyzing set-valued mappings and their behaviors under various convergence criteria.
Power Set Function: The power set function is a mathematical operation that takes a set as an input and outputs the set of all possible subsets of that input set, including the empty set and the set itself. This concept is crucial in understanding set-valued mappings because it provides a comprehensive way to represent all potential selections or combinations from a given set, which can be essential in optimization and decision-making processes.
Set convergence: Set convergence refers to the process in which a sequence or a net of sets approaches a limit set, meaning that for any point in the limit set, there exists a point in the sequence or net that gets arbitrarily close to it. This concept is essential for understanding how collections of subsets behave in various mathematical contexts, particularly in analyzing the properties and continuity of set-valued mappings and their implications in optimization problems.
Set-valued mapping: A set-valued mapping is a function that assigns to each point in its domain a set of values in its range, rather than a single value. This concept allows for the modeling of situations where multiple outcomes or decisions can be associated with a given input, which is particularly useful in optimization and equilibrium problems.
Star-shapedness: Star-shapedness refers to a geometric property of a set where there exists at least one point in the set such that a line segment connecting this point to any other point in the set lies entirely within the set. This concept is particularly important in understanding the structure and properties of set-valued mappings, as it helps define certain characteristics of sets that are crucial for analysis.
Upper Hemicontinuous: Upper hemicontinuity is a property of set-valued mappings where, roughly speaking, the values of the mapping change in a controlled manner as the input varies. Specifically, a set-valued mapping is upper hemicontinuous at a point if, for every open neighborhood around the image of that point, there exists a corresponding neighborhood around the point such that all images of points in that neighborhood are contained within the original open neighborhood. This concept is important in understanding the stability and continuity of solutions in optimization and variational analysis.
Upper Semicontinuity: Upper semicontinuity refers to a property of set-valued mappings where, intuitively, the image of a point under the mapping does not jump upward as the input approaches that point. This means that for any point in the domain, if you look at nearby points, the values in the image either stay the same or decrease, making it easier to analyze convergence and stability in various mathematical contexts. This concept plays a critical role in understanding the behavior of multifunctions and their differentiability, as well as establishing existence results for equilibrium problems.
Variational Inequalities: Variational inequalities are mathematical expressions that describe the relationship between a function and its constraints, typically involving an inequality condition. They often arise in optimization problems where one seeks to find a solution that minimizes a given functional while satisfying certain constraints, thus connecting to broader concepts in variational analysis.
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