Derives first- and second-order necessary and sufficient optimality conditions for directional local minimality in unconstrained nonsmooth optimization and adapts them to nondirectional local minimality using critical directions.
Second-order optimality conditions for optimization problems with generalized equation constraints
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abstract
This paper provides second-order optimality conditions for optimization problems with generalized equation constraints (GEPs), a framework that encompasses several important and challenging models in mathematical programming, including mathematical programs with variational inequality constraints (MPVIs) and bilevel programs. The obtained optimality conditions are novel even for these particular problem classes. As an application, second-order optimality conditions for MPVIs are detailed. The technical key lies in developing first- and second-order variational analysis of the highly intricate constraint system, which is needed to capture the local curvature of the feasible set entering these optimality conditions. Part of this task was already carried out in our companion paper \cite{BeGfrYeZhangZhou}, and here we complete the study. Comprehensive variational analysis results are derived, which are of independent interest.
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math.OC 1years
2026 1verdicts
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On directional local minimality and directional optimality conditions in nonsmooth optimization
Derives first- and second-order necessary and sufficient optimality conditions for directional local minimality in unconstrained nonsmooth optimization and adapts them to nondirectional local minimality using critical directions.