On the directional asymptotic approach in optimization theory

Area of Science:

• Optimization Theory
• Nonsmooth Analysis
• Variational Analysis

Background:

• Local minimizers in nonsmooth optimization can exhibit complex stationarity properties.
• Existing optimality conditions may not fully capture the behavior of minimizers in generalized settings.

Purpose of the Study:

• To derive new necessary optimality conditions for nonsmooth optimization problems.
• To introduce and analyze higher-order stationarity and regularity conditions.
• To extend existing concepts to a broader class of constraints and mappings.

Main Methods:

• Utilizing coderivative constructions of varying orders ($$).
• Applying constraint qualifications, including directional metric subregularity.
• Extending directional pseudo- and quasi-normality concepts.
• Developing novel coderivative-like variational tools.

Main Results:

• Established new necessary optimality conditions combining limiting variational tools of orders 1 and $$.
• Introduced directional asymptotic regularity conditions as constraint qualifications.
• Demonstrated that pseudo- and quasi-normality properties imply directional asymptotic regularity.
• Showcased applicability to complementarity-constrained and nonlinear semidefinite optimization.

Conclusions:

• The derived conditions offer a refined characterization of local minimizers in nonsmooth optimization.
• New regularity conditions provide valuable tools for analyzing and solving complex optimization problems.
• The findings contribute to the advancement of variational analysis and optimization theory.