[Paper Review] Metric Subregularity of Subdifferential and KL Property of Exponent 1/2

Shaohua Pan, Dongdong Zhang+1 more|arXiv (Cornell University)|Dec 3, 2018

Optimization and Variational Analysis4 citations

TL;DR

This paper establishes a deep connection between metric subregularity of the limiting subdifferential relative to the critical set and the Kurdyka-Łojasiewicz (KL) property of exponent 1/2 for proper lower semicontinuous functions. For convex functions, these two properties are equivalent; in the nonconvex case, the KL property of exponent 1/2 combined with quadratic growth on the critical set implies metric subregularity, and under primal-lower-niceness and a stationary value assumption, the converse holds.

ABSTRACT

For a proper lower semicontinuous function, we study the relations between the metric subregularity of its limiting subdifferential relative to the critical set and the KL property of exponent 1/2. When the function is convex, we establish the equivalence between them. When the function is nonconvex, we show that the KL property of exponent 1/2 along with the quadratic growth on the critical set implies the metric subregularity of the subdifferential relative to the critical set; and if the function is primal-lower-nice, under an assumption on stationary values, the latter implies the former. These results provide a bridge for the two kinds of regularity and contribute to enriching each other.

Motivation & Objective

• To clarify the relationship between metric subregularity of the limiting subdifferential relative to the critical set and the KL property of exponent 1/2.
• To investigate whether these two regularity concepts are equivalent in the convex case.
• To determine sufficient conditions under which the KL property of exponent 1/2 implies metric subregularity in the nonconvex setting.
• To identify conditions under which metric subregularity implies the KL property of exponent 1/2 in nonconvex functions, particularly under primal-lower-niceness and stationary value assumptions.

Proposed method

• Analyzing the limiting subdifferential of a proper lower semicontinuous function and its behavior relative to the critical set.
• Employing the Kurdyka-Łojasiewicz (KL) inequality with exponent 1/2 to characterize the local geometry of the function near critical points.
• Using quadratic growth conditions on the critical set as a key assumption to link the KL property to metric subregularity in nonconvex cases.
• Applying the concept of primal-lower-nice functions to establish reverse implications from metric subregularity to the KL property.
• Utilizing variational analytic tools, including subdifferential calculus and metric regularity theory, to derive the main results.
• Establishing equivalence in the convex case through structural properties of subdifferentials and the behavior of the function near minimizers.

Experimental results

Research questions

• RQ1 Are metric subregularity of the limiting subdifferential relative to the critical set and the KL property of exponent 1/2 equivalent for convex functions?
• RQ2 Does the KL property of exponent 1/2 combined with quadratic growth on the critical set imply metric subregularity of the subdifferential in the nonconvex case?
• RQ3 Under what conditions does metric subregularity of the subdifferential relative to the critical set imply the KL property of exponent 1/2 in nonconvex functions?
• RQ4 How does the primal-lower-nice property influence the relationship between metric subregularity and the KL property of exponent 1/2?
• RQ5 What role do stationary values play in establishing the implication from metric subregularity to the KL property in nonconvex settings?

Key findings

• For convex functions, the metric subregularity of the limiting subdifferential relative to the critical set is equivalent to the KL property of exponent 1/2.
• In the nonconvex case, the KL property of exponent 1/2 together with quadratic growth on the critical set implies metric subregularity of the subdifferential relative to the critical set.
• If the function is primal-lower-nice and the stationary values satisfy a certain assumption, then metric subregularity of the subdifferential relative to the critical set implies the KL property of exponent 1/2.
• The results establish a bidirectional bridge between two key regularity concepts in variational analysis, enriching both the theory of metric subregularity and the KL property.
• The findings extend the applicability of the KL property and metric subregularity to broader classes of functions, particularly through the introduction of quadratic growth and primal-lower-niceness as auxiliary conditions.
• The study provides a unified framework for understanding convergence behavior in nonsmooth optimization via the interplay of subdifferential regularity and KL-type inequalities.