The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions

• 成果类型: Article
• 署名作者: Izmailov, AF; Solodov, MV
• 署名单位: Russian Academy of Sciences
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.27.3.614.308
• 发表日期: 2002
• 页码: 614-635
• 关键词: nonlinear complementarity; mathematical programs; extremum conditions; local-structure; singular point; equality-type; neighborhood; stationarity; set

摘要

We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to right-hand side perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2-regular) equality constraints and optimality conditions for related optimization problems. The class mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation-based reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.