Augmented Lagrangian Methods for Solving Composite Conic Programming

The paper by [Xinyuan Zhao, Defeng Sun, and Kim Chuan Toh, titled “A Newton-CG augmented Lagrangian method for semidefinite programming”, published in SIAM Journal on Optimization 20 (2010), pp. 1737–1765], initiated the research on using the semismooth Newton-CG augmented Lagrangian method (ALM) for solving semidefinite programming (SDP). SDPNAL+ is a MATLAB software for solving large scale SDP with bound constraints (click here for an introduction on how to use the package). This software was awarded [the triennial triennial Beale–Orchard-Hays Prize for Excellence in Computational Mathematical Programming by the Mathematical Optimization Society at Bordeaux, France, July 2-6, 2018. See Picture 1, Picture 2, and Picture 3. For detailed information about the software, please refer to the papers by [Defeng Sun, Kim Chuan Toh, Yancheng Yuan, and Xinyuan Zhao, titled “SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0)”, published in Optimization Methods and Software 35 (2020) 87–115] and by [Liuqin Yang, Defeng Sun, and Kim Chuan Toh, titled “SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints”, published in Mathematical Programming Computation 7 (2015), pp. 331-366.] For extensions to convex quadratic SDP, see the work by [Xudong Li, Defeng Sun, and Kim Chuan Toh, titled “QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming”, published in Mathematical Programming Computation 10 (2018) 703–743.] In the paper by [Ying Cui, Defeng Sun, and Kim Chuan Toh, titled “On the R-superlinear convergence of the KKT residuals generated by the augmented Lagrangian method for convex composite conic programming”, published in Mathematical Programming 178 (2019) 381—415], we provide a fairly comprehensive treatment of the theoretical convergence rates as well as practical implementations of the ALM for solving linear SDP and convex quadratic SDP.