Revisiting augmented Lagrangian duals

• 成果类型: Article
• 署名作者: Cordova, M.; Oliveira, W. de; Sagastizabal, C.
• 署名单位: Universidade Federal de Santa Catarina (UFSC); Universite PSL; MINES ParisTech; Universidade Estadual de Campinas
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01703-5
• 发表日期: 2022
• 页码: 235-277
• 关键词: modified subgradient algorithm; proximal bundle methods

摘要

For nonconvex optimization problems, possibly having mixed-integer variables, a convergent primal-dual solution algorithm is proposed. The approach applies a proximal bundle method to certain augmented Lagrangian dual that arises in the context of the so-called generalized augmented Lagrangians. We recast these Lagrangians into the framework of a classical Lagrangian by means of a special reformulation of the original problem. Thanks to this insight, the methodology yields zero duality gap. Lagrangian subproblems can be solved inexactly without hindering the primal-dual convergence properties of the algorithm. Primal convergence is ensured even when the dual solution set is empty. The interest of the new method is assessed on several problems, including unit-commitment, that arise in energy optimization. These problems are solved to optimality by solving separable Lagrangian subproblems.