Lower Hemicontinuous

Definition

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.

5 Must Know Facts For Your Next Test

1. Lower hemicontinuity is important for proving the existence of solutions to variational problems since it helps to establish compactness and convergence properties.
2. If a set-valued mapping is lower hemicontinuous at a point, it implies that the image of sequences converging to that point has certain limit points related to the mapping.
3. This property can be visually interpreted as a 'lower boundary' where outputs do not decrease suddenly, thus ensuring more reliable mappings.
4. Lower hemicontinuity is often discussed alongside upper hemicontinuity to describe different behaviors of set-valued mappings.
5. In many practical applications, such as optimization and game theory, lower hemicontinuity ensures that small changes in parameters lead to predictable changes in outcomes.

Review Questions

• How does lower hemicontinuity relate to the behavior of solutions in optimization problems?
  • Lower hemicontinuity ensures that small perturbations in input values do not lead to sudden drops in output values for set-valued mappings. This stability is vital when analyzing solutions to optimization problems because it implies that as one approaches an optimal solution, the feasible region does not unexpectedly lose potential solutions. This behavior allows us to work with continuity arguments when establishing the existence of solutions.
• Compare and contrast lower hemicontinuity and upper hemicontinuity with examples illustrating their differences.
  • Lower hemicontinuity involves outputs that remain stable or do not drop suddenly when inputs change, while upper hemicontinuity concerns outputs that do not rise unexpectedly. For instance, if we consider a set-valued mapping where inputs represent resource allocation and outputs represent feasible allocations, lower hemicontinuity would ensure that slight changes in resource allocation do not lead to loss of feasible outcomes. Conversely, upper hemicontinuity would ensure that increasing resource allocation does not create new constraints abruptly.
• Evaluate how lower hemicontinuity impacts the overall stability and robustness of economic models involving set-valued mappings.
  • Lower hemicontinuity plays a crucial role in ensuring stability within economic models that utilize set-valued mappings. By guaranteeing that small changes in input lead to predictable outputs, models become more robust against fluctuations, enhancing reliability in predictions. This stability allows economists to draw meaningful conclusions about equilibrium and policy impacts while minimizing adverse effects caused by unexpected variations in inputs. Thus, understanding lower hemicontinuity is essential for building sound economic theories and practices.

Related terms

Upper Hemicontinuous: A property of set-valued mappings where the inverse image of every closed set is closed in the domain, often describing a kind of stability in the upper direction.

Continuity: In the context of functions, continuity means that small changes in input lead to small changes in output, ensuring a predictable relationship.

Set-valued Mapping: A function that assigns a set of values rather than a single value to each point in its domain, commonly used in optimization and variational problems.