[Paper Review] Second order optimality conditions for strong local minimizers via subgradient graphical derivative

Nguyễn Huy Chiêu, Le Van Hien+2 more|arXiv (Cornell University)|Mar 13, 2019

Optimization and Variational Analysis31 references4 citations

TL;DR

This paper establishes second-order optimality conditions for strong local minimizers in nonsmooth optimization using the subgradient graphical derivative, showing that positive definiteness of this derivative at proximal stationary points is sufficient and necessary under certain regularity conditions. It extends classical results by replacing Robinson’s constraint qualification with the weaker metric subregularity constraint qualification in $C^2$-cone reducible programs, linking quadratic growth to strong metric subregularity of the subdifferential.

ABSTRACT

This paper is devoted to the study of second order optimality conditions for strong local minimizers in the frameworks of unconstrained and constrained optimization problems in finite dimensions via subgradient graphical derivative. We prove that the positive definiteness of the subgradient graphical derivative of an extended-real-valued lower semicontinuous proper function at a proximal stationary point is sufficient for the quadratic growth condition. It is also a necessary condition for the latter property when the function is either subdifferentially continuous, prox-regular, twice epi-differentiable or variationally convex. By applying our results to the $\mathcal{C}^2$-cone reducible constrained programs, we establish no-gap second order optimality conditions for (strong) local minimizers under the metric subregularity constraint qualification. These results extend the classical second order optimality conditions by surpassing the well-known Robinson's constraint qualification. Our approach also highlights the interconnection between the strong metric subregularity of subdifferential and quadratic growth condition in optimization problems.

Motivation & Objective

• To characterize strong local minimizers in nonsmooth, unconstrained optimization via the subgradient graphical derivative.
• To establish no-gap second-order optimality conditions for constrained $C^2$-cone reducible programs under weaker constraint qualifications than Robinson’s.
• To connect the quadratic growth condition with the strong metric subregularity of the subdifferential using primal-dual structures.
• To extend classical second-order conditions beyond convex and semi-algebraic settings using graphical derivative techniques.

Proposed method

• The paper uses the subgradient graphical derivative as a primal-dual second-order structure to analyze strong local minimizers.
• It proves that positive definiteness of the subgradient graphical derivative at a proximal stationary point implies the quadratic growth condition.
• The method relies on the equivalence between quadratic growth and strong metric subregularity of the subdifferential in specific function classes.
• For constrained problems, the approach applies to $C^2$-cone reducible programs under the metric subregularity constraint qualification (MSCQ).
• The analysis leverages tools from variational analysis, including prox-regularity, subdifferential continuity, and twice epi-differentiability.
• Key results are derived by connecting the graphical derivative to the limiting subdifferential and using stability properties in optimization.

Experimental results

Research questions

• RQ1 Can the subgradient graphical derivative serve as a second-order structure to characterize strong local minimizers in nonconvex, nonsmooth optimization?
• RQ2 Is the positive definiteness of the subgradient graphical derivative both necessary and sufficient for quadratic growth under weaker regularity assumptions than convexity?
• RQ3 Does the metric subregularity constraint qualification (MSCQ) suffice to establish no-gap second-order optimality conditions in $C^2$-cone reducible programs?
• RQ4 How does the strong metric subregularity of the subdifferential relate to the quadratic growth condition in nonconvex settings?
• RQ5 Can the classical second-order sufficient condition be fully characterized under MSCQ, replacing Robinson’s constraint qualification?

Key findings

• Positive definiteness of the subgradient graphical derivative at a proximal stationary point is sufficient for the quadratic growth condition in unconstrained, nonsmooth optimization.
• This condition becomes necessary for quadratic growth when the function is subdifferentially continuous, prox-regular, twice epi-differentiable, or variationally convex.
• For $C^2$-cone reducible programs, the classical second-order sufficient condition fully characterizes strong local minimizers under the metric subregularity constraint qualification (MSCQ).
• MSCQ is strictly weaker than Robinson’s constraint qualification (RCQ), allowing broader applicability of second-order optimality conditions.
• The strong metric subregularity of the subdifferential is equivalent to the quadratic growth condition for the two broad classes of functions under consideration.
• The results provide a no-gap characterization of strong local minimizers in constrained programs under MSCQ, extending prior results that required stronger constraint qualifications.