In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficie...In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.展开更多
基金Chao Ding’s research was supported by the National Natural Science Foundation of China(Nos.11671387,11531014,and 11688101)Beijing Natural Science Foundation(No.Z190002)+6 种基金Xu-Dong Li’s research was supported by the National Key R&D Program of China(No.2020YFA0711900)the National Natural Science Foundation of China(No.11901107)the Young Elite Scientists Sponsorship Program by CAST(No.2019QNRC001)the Shanghai Sailing Program(No.19YF1402600)the Science and Technology Commission of Shanghai Municipality Project(No.19511120700)Xin-Yuan Zhao’s research was supported by the National Natural Science Foundation of China(No.11871002)the General Program of Science and Technology of Beijing Municipal Education Commission(No.KM201810005004).
文摘In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.
