Proper Convex Function

Definition

A proper convex function is a convex function defined on a convex set that is not identically infinite and has at least one point in its domain where it takes finite values. Proper convex functions are important in optimization and variational analysis, as they guarantee the existence of minimizers and help establish foundational concepts in convex analysis.

5 Must Know Facts For Your Next Test

1. Proper convex functions can be used to characterize optimality conditions and facilitate duality in optimization problems.
2. These functions often exhibit nice continuity properties, which are essential for convergence results in optimization algorithms.
3. If a proper convex function is lower semicontinuous, then its epigraph, which is the set of points lying on or above its graph, is a closed set.
4. The set of proper convex functions includes functions that are coercive, meaning they tend to infinity as the input goes to infinity, which ensures minimizers exist.
5. Examples of proper convex functions include quadratic functions and indicator functions of convex sets.

Review Questions

• How does the concept of proper convex functions relate to optimization problems?
  • Proper convex functions are crucial in optimization because they ensure that a minimizer exists under certain conditions. Since these functions are not identically infinite and have finite values at some points, they enable the formulation of meaningful optimization problems. This ensures that various optimization algorithms can converge to solutions effectively.
• Discuss the significance of lower semicontinuity for proper convex functions and their epigraphs.
  • Lower semicontinuity is significant for proper convex functions as it guarantees that the epigraph of the function is a closed set. This property helps in establishing important results in analysis and optimization, like ensuring that limits of sequences remain within the epigraph. It also plays a role in proving existence results for minimizers in optimization problems involving these functions.
• Evaluate how coerciveness of proper convex functions contributes to their properties in variational analysis.
  • Coerciveness is a key property of proper convex functions that ensures they do not approach infinity too quickly. This characteristic means that as you move far away from any compact set in their domain, the function values increase without bound. In variational analysis, this behavior guarantees that minimization problems have solutions since it prevents the existence of 'escaping sequences' that could lead to undefined behavior, thereby reinforcing stability and robustness in applied contexts.

Related terms

Convex Function: A function is convex if, for any two points in its domain, the line segment connecting the corresponding points on the graph lies above or on the graph.

Subdifferential:

The subdifferential of a convex function at a point is a set of slopes of supporting hyperplanes to the graph of the function at that point.

Lower Semicontinuous Function: A function is lower semicontinuous if, at every point in its domain, the limit infimum of the function values at nearby points is greater than or equal to the value of the function at that point.