On Optimality Conditions for Nonlinear Conic Programming

• 成果类型: Article; Early Access
• 署名作者: Andreani, Roberto; Gomez, Walter; Haeser, Gabriel; Mito, Leonardo M.; Ramos, Alberto
• 署名单位: Universidade Estadual de Campinas; Universidad de La Frontera; Universidade de Sao Paulo; Universidade Federal do Parana
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1203
• 发表日期: 2021
• 页码: 1-26
• 关键词: weak constraint qualifications; Augmented Lagrangian method; optimization problems; GLOBAL CONVERGENCE; differentiability; INEQUALITY; STABILITY; projection

摘要

Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush-Kuhn-Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.