Analysis of the Alternating Direction Method of Multipliers for Nonconvex Problems

Introduction to IPOPT: a tutorial for downloading, installing, and using IPOPT. https://coin-or.github.io/Ipopt/. Accessed: 2017-7-21

Bai X, Scheinberg K (2015) Alternating direction methods for non convex optimization with applications to second-order least-squares and risk parity portfolio selection. http://www.optimization-online.org/DB_FILE/2015/02/4776.pdf. Accessed: 2019-1-22

Bertsekas DP (1979) Convexification procedures and decomposition methods for nonconvex optimization problems. J Optim Theory Appl 29(2):169–197

Bertsekas DP (1996) Constrained optimization and lagrange multiplier methods. Athena Scientific, Belmont

Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont

Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3(1):1–122

Chatzipanagiotis N, Dentcheva D, Zavlanos MM (2015) An augmented Lagrangian method for distributed optimization. Math Program 152 (1-2):405–434

Chatzipanagiotis N, Zavlanos MM (2017) On the convergence of a distributed augmented Lagrangian method for nonconvex optimization. IEEE Trans Autom Control 62(9):4405–4420

Chiang N, Petra CG, Zavala VM (2014) Structured nonconvex optimization of large-scale energy systems using PIPS-NLP. In: Power Systems Computation Conference (PSCC), 2014, pp 1–7

Curtis FE, Raghunathan AU (2017) Solving nearly-separable quadratic optimization problems as nonsmooth equations. Comput Optim Appl 67(2):317–360

Dinh QT, Necoara I, Diehl M (2013) A dual decomposition algorithm for separable nonconvex optimization using the penalty function framework. In: Decision and Control (CDC), 2013 IEEE 52nd annual conference on, pp 2372–2377

Eckstein J, Yao W (2015) Understanding the convergence of the alternating direction method of multipliers: theoretical and computational perspectives. http://www.optimization-online.org/DB_FILE/2015/06/4954.pdf. Accessed: 2019-1-22

Feng X, Mukai H, Brown RH (1990) New decomposition and convexification algorithm for nonconvex large-scale primal-dual optimization. J Optim Theory Appl 67(2):279–296

Fiacco AV (1976) Sensitivity analysis for nonlinear programming using penalty methods. Math Program 10(1):287–311

Fiacco AV, Ishizuka Y (1990) Sensitivity and stability analysis for nonlinear programming. Ann Oper Res 27(1):215–235

Hong M, Luo Z-Q, Razaviyayn M (2016) Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J Optim 26(1):337–364

Hours J-H, Jones CN (2014) An augmented Lagrangian coordination-decomposition algorithm for solving distributed non-convex programs. In: American control conference (ACC), vol 2014, pp 4312–4317

Houska B, Frasch J, Diehl M (2016) An augmented Lagrangian based algorithm for distributed nonconvex optimization. SIAM J Optim 26 (2):1101–1127

Kang J, Cao Y, Word DP, Laird C (2014) An interior-point method for efficient solution of block-structured NLP problems using an implicit Schur-complement decomposition. Comput Chem Eng 71:563–573

Li G, Pong TK (2016) Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Math Program 159 (1-2):371–401

Magnusson S, Chathuranga P, Rabbat M, Fischione C (2015) On the convergence of alternating direction Lagrangian methods for nonconvex structured optimization problems. IEEE Trans Control Netw Syst PP(99):1–1

Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York

Rodriguez JS, Nicholson B, Laird C, Zavala VM (2018) Benchmarking ADMM in nonconvex NLPs. Comput Chem Eng 119:315–325

Shapiro A, Sun J (2004) Some properties of the augmented Lagrangian in cone constrained optimization. Math Oper Res 29(3):479–491

Stephanopoulos G, Westerberg AW (1975) The use of Hestenes’ method of multipliers to resolve dual gaps in engineering system optimization. J Optim Theory Appl 15(3):285–309

Strang G (2006) Linear algebra and its applications, 4th edn. Thomson Brooks/Cole, Boston

Tanikawa A, Mukai H (1985) A new technique for nonconvex primal-dual decomposition of a large-scale separable optimization problem. IEEE Trans Autom Control 30(2):133–143

Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57

Wang Y, Yin W, Zeng J (2019) Global convergence of ADMM in nonconvex nonsmooth optimization. J Sci Comput 78(1):29–63

Zhang X, Byrd RH, Schnabel RB (1992) Parallel methods for solving nonlinear block bordered systems of equations. SIAM J Sci Stat Comput 13(4):841–859