Set-valued mapping

Definition

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.

5 Must Know Facts For Your Next Test

1. Set-valued mappings can be represented graphically, showing the relationship between inputs and the corresponding sets of outputs.
2. They are essential in modeling situations where uncertainty or multiple possible results exists, such as in economic equilibrium and optimization problems.
3. Set-valued mappings can exhibit various properties like continuity, convexity, and upper or lower semi-continuity, which are crucial for analyzing their behavior.
4. They extend the concept of functions, allowing for a richer structure in mathematical analysis, especially in variational analysis and differential inclusions.
5. Applications of set-valued mappings include control theory, game theory, and economics, where decisions lead to a range of possible outcomes.

Review Questions

• How do set-valued mappings differ from single-valued mappings in terms of their application and representation?
  • Set-valued mappings differ from single-valued mappings primarily in that they assign a set of possible outputs for each input rather than just one output. This characteristic allows them to better model situations involving uncertainty or multiple choices, making them particularly useful in fields such as economics and optimization. The graphical representation of a set-valued mapping can illustrate how an input can correspond to various outputs, providing deeper insights into decision-making processes.
• Discuss the significance of continuity in set-valued mappings and how it influences their properties.
  • Continuity in set-valued mappings refers to the idea that small changes in the input should result in small changes in the output sets. This concept is important because it helps establish how stable or robust the outcomes are with respect to perturbations in inputs. Continuous set-valued mappings maintain their structure under small variations, allowing for predictable behavior in optimization scenarios and ensuring that solutions are reliable when conditions change slightly.
• Evaluate the role of closed-valued mappings in variational analysis and their implications for optimization problems.
  • Closed-valued mappings play a crucial role in variational analysis as they ensure that the image of compact sets remains closed, preserving essential topological properties. This characteristic is significant when analyzing optimization problems because it guarantees the existence of solutions within bounded regions. By maintaining closure, these mappings facilitate convergence results and ensure that optimal solutions can be found reliably, making them a fundamental tool for researchers working on complex decision-making models.

Related terms

single-valued mapping: A function that assigns exactly one value to each point in its domain.

graph of a set-valued mapping: The collection of points in the Cartesian product of the domain and the range, representing all pairs where each input maps to its corresponding set of outputs.

closed-valued mapping: A type of set-valued mapping where the image of every compact set is closed, ensuring that certain topological properties are preserved.