@book{livre1,
  title={Convex analysis and monotone operator theory in Hilbert spaces},
  author={Bauschke, Heinz H and Combettes, Patrick L},
  volume={408},
  year={2011},
  publisher={Springer}
}
@article{combettes2005signal,
	title={Signal recovery by proximal forward-backward splitting},
	author={Combettes, Patrick L and Wajs, Val{\'e}rie R},
	journal={Multiscale Modeling $\&$ Simulation},
	volume={4},
	number={4},
	pages={1168--1200},
	year={2005},
	publisher={SIAM}
}

@article{Nes05,
  title={Smooth minimization of non-smooth functions},
  author={Nesterov, Yu},
  journal={Math. Program.},
  volume={103},
  number={1},
  pages={127--152},
  year={2005},
  publisher={SIAM}
}

@Article{Chambolle2011,
author="Chambolle, Antonin
and Pock, Thomas",
title="A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging",
journal="Journal of Mathematical Imaging and Vision",
year="2011",
month="May",
day="01",
volume="40",
number="1",
pages="120--145"
}
@Article{Combettes2012,
author="Combettes, Patrick L.
and Pesquet, Jean-Christophe",
title="Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators",
journal="Set-Valued and Variational Analysis",
year="2012",
month="Jun",
day="01",
volume="20",
number="2",
pages="307--330"
}

@Article{Burer2005,
author="Burer, Samuel
and Monteiro, Renato D.C.",
title="Local Minima and Convergence in Low-Rank Semidefinite Programming",
journal="Mathematical Programming",
year="2005",
month="Jul",
day="01",
volume="103",
number="3",
pages="427--444"
}

@Article{Burer2003,
author="Burer, Samuel
and Monteiro, Renato D.C.",
title="A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization",
journal="Mathematical Programming",
year="2003",
month="Feb",
day="01",
volume="95",
number="2",
pages="329--357"
}
@article{park2016provable,
  title={Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems},
  author={Park, Dohyung and Kyrillidis, Anastasios and Bhojanapalli, Srinadh and Caramanis, Constantine and Sanghavi, Sujay},
  journal={arXiv preprint arXiv:1606.01316},
  year={2016}
}
@inproceedings{bhojanapalli2016dropping,
  title={Dropping convexity for faster semi-definite optimization},
  author={Bhojanapalli, Srinadh and Kyrillidis, Anastasios and Sanghavi, Sujay},
  booktitle={Conference on Learning Theory},
  pages={530--582},
  year={2016}
}
@article{chen2015fast,
  title={Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees},
  author={Chen, Yudong and Wainwright, Martin J},
  journal={arXiv preprint arXiv:1509.03025},
  year={2015}
}
@inproceedings{zhao2015nonconvex,
  title={A nonconvex optimization framework for low rank matrix estimation},
  author={Zhao, Tuo and Wang, Zhaoran and Liu, Han},
  booktitle={Advances in Neural Information Processing Systems},
  pages={559--567},
  year={2015}
}
@inproceedings{jain2010guaranteed,
  title={Guaranteed rank minimization via singular value projection},
  author={Jain, Prateek and Meka, Raghu and Dhillon, Inderjit S},
  booktitle={Advances in Neural Information Processing Systems},
  pages={937--945},
  year={2010}
}
@article{becker2013randomized,
  title={Randomized low-memory singular value projection},
  author={Becker, Stephen and Cevher, Volkan and Kyrillidis, Anastasios},
  journal={arXiv preprint arXiv:1303.0167},
  year={2013}
}
@inproceedings{yurtsever2015universal,
  title={A universal primal-dual convex optimization framework},
  author={Yurtsever, Alp and Dinh, Quoc Tran and Cevher, Volkan},
  booktitle={Advances in Neural Information Processing Systems},
  pages={3150--3158},
  year={2015}
}
@article{tran2018smooth,
  title={A smooth primal-dual optimization framework for nonsmooth composite convex minimization},
  author={Tran-Dinh, Quoc and Fercoq, Olivier and Cevher, Volkan},
  journal={SIAM Journal on Optimization},
  volume={28},
  number={1},
  pages={96--134},
  year={2018},
  publisher={SIAM}
}
@inproceedings{boumal2016non,
  title={The non-convex Burer-Monteiro approach works on smooth semidefinite programs},
  author={Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso},
  booktitle={Advances in Neural Information Processing Systems},
  pages={2757--2765},
  year={2016}
}
@article{boumal2016global,
  title={Global rates of convergence for nonconvex optimization on manifolds},
  author={Boumal, Nicolas and Absil, P-A and Cartis, Coralia},
  journal={arXiv preprint arXiv:1605.08101},
  year={2016}
}
@article{Lan16,
	title={Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization},
	author={S. Ghadimi,G. Lan, H. Xhang},
	journal={ Math. Program., Ser. A},
	volume = {155},
	pages = {267--305},
	year={2016}
}
@Article{Hare2009,
	title={Computing proximal points of nonconvex functions},
	author={Hare, Warren and Sagastiz{\'a}bal, Claudia},
	journal={Mathematical Programming},
	year={2009},
	volume={116},
	number={1},
	pages={221--258}
}
@book{arrow2012studies,
	title={Studies in Linear and Non-Linear Programming: Stanford Mathematical Studies in the Social Sciences},
	author={Arrow, K.J. and Hurwicz, L. and Uzawa, H.},
	number={no. 2},
	isbn={9781258450144},
	url={https://books.google.ch/books?id=cuJnMQEACAAJ},
	year={2012},
	publisher={Literary Licensing, LLC}
}
@article{chambolle_pock_2016, title={An introduction to continuous optimization for imaging}, volume={25}, DOI={10.1017/S096249291600009X}, journal={Acta Numerica}, publisher={Cambridge University Press}, author={Chambolle, Antonin and Pock, Thomas}, year={2016}, pages={161--319}
}
@article{huang2018mini,
	title={Mini-Batch Stochastic ADMMs for Nonconvex Nonsmooth Optimization},
	author={Huang, Feihu and Chen, Songcan},
	journal={arXiv preprint arXiv:1802.03284},
	year={2018}
}
	
	@article{Valkonen14,
		author={Tuomo Valkonen},
		title={A primal-dual hybrid gradient method for nonlinear operators with applications to MRI},
		journal={Inverse Problems},
		volume={30},
		number={5},
		pages={055012},
		url={http://stacks.iop.org/0266-5611/30/i=5/a=055012},
		year={2014}
	}
@inproceedings{zhou2010stable,
	title={Stable principal component pursuit},
	author={Zhou, Zihan and Li, Xiaodong and Wright, John and Candes, Emmanuel and Ma, Yi},
	booktitle={Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on},
	pages={1518--1522},
	year={2010},
	organization={IEEE}
}
@article{jain2017non,
	title={Non-convex optimization for machine learning},
	author={Jain, Prateek and Kar, Purushottam},
	journal={Foundations and Trends{\textregistered} in Machine Learning},
	volume={10},
	number={3-4},
	pages={142--336},
	year={2017},
	publisher={Now Publishers, Inc.}
}
@article{komodakis2015playing,
	title={Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems},
	author={Komodakis, Nikos and Pesquet, Jean-Christophe},
	journal={IEEE Signal Processing Magazine},
	volume={32},
	number={6},
	pages={31--54},
	year={2015},
	publisher={IEEE}
}


@article{mollenhoff2015primal,
	title={The primal-dual hybrid gradient method for semiconvex splittings},
	author={Mollenhoff, Thomas and Strekalovskiy, Evgeny and Moeller, Michael and Cremers, Daniel},
	journal={SIAM Journal on Imaging Sciences},
	volume={8},
	number={2},
	pages={827--857},
	year={2015},
	publisher={SIAM}
}
@article{artina2013linearly,
	title={Linearly constrained nonsmooth and nonconvex minimization},
	author={Artina, Marco and Fornasier, Massimo and Solombrino, Francesco},
	journal={SIAM Journal on Optimization},
	volume={23},
	number={3},
	pages={1904--1937},
	year={2013},
	publisher={SIAM}
}
@InProceedings{Matin17,
	author="Benning, Martin
	and Knoll, Florian
	and Sch{\"o}nlieb, Carola-Bibiane
	and Valkonen, Tuomo",
	editor="Bociu, Lorena
	and D{\'e}sid{\'e}ri, Jean-Antoine
	and Habbal, Abderrahmane",
	title="Preconditioned ADMM with Nonlinear Operator Constraint",
	booktitle="System Modeling and Optimization",
	year="2016",
	publisher="Springer International Publishing",
	address="Cham",
	pages="117--126",
	abstract="We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).",
	isbn="978-3-319-55795-3"
}
@article{beck2012smoothing,
	title={Smoothing and first order methods: A unified framework},
	author={Beck, Amir and Teboulle, Marc},
	journal={SIAM Journal on Optimization},
	volume={22},
	number={2},
	pages={557--580},
	year={2012},
	publisher={SIAM}
}
@article{combettes2018monotone,
	title={Monotone Operator Theory in Convex Optimization},
	author={Combettes, Patrick L},
	journal={arXiv preprint arXiv:1802.02694},
	year={2018}
}
@inproceedings{lin2011linearized,
	title={Linearized alternating direction method with adaptive penalty for low-rank representation},
	author={Lin, Zhouchen and Liu, Risheng and Su, Zhixun},
	booktitle={Advances in neural information processing systems},
	pages={612--620},
	year={2011}
}
@article{gabay1976dual,
	title={A dual algorithm for the solution of nonlinear variational problems via finite element approximation},
	author={Gabay, Daniel and Mercier, Bertrand},
	journal={Computers and Mathematics with Applications},
	volume={2},
	number={1},
	pages={17--40},
	year={1976},
	publisher={Elsevier}
}
@article{Gabay83,
	title={Applications of the method of multipliers to variational inequalities},
	author ={D. Gabay},
	journal={ in: M. Fortin and R. Glowinski (eds.), Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, North-Holland, Amsterdam},
    year ={1983}
}
@article{banert2016fixing,
	title={Fixing and extending some recent results on the ADMM algorithm},
	author={Banert, Sebastian and Bot, Radu Ioan and Csetnek, Ern{\"o} Robert},
	journal={arXiv preprint arXiv:1612.05057},
	year={2016}
}
@article{shefi2014rate,
	title={Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization},
	author={Shefi, Ron and Teboulle, Marc},
	journal={SIAM Journal on Optimization},
	volume={24},
	number={1},
	pages={269--297},
	year={2014},
	publisher={SIAM}
}
@incollection{burachik2010deflected,
	title={A deflected subgradient method using a general augmented Lagrangian duality with implications on penalty methods},
	author={Burachik, Regina S and Kaya, C Yal{\c{c}}{\i}n},
	booktitle={Variational Analysis and Generalized Differentiation in Optimization and Control},
	pages={109--132},
	year={2010},
	publisher={Springer}
}
@article{gasimov2002augmented,
	title={Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming},
	author={Gasimov, Rafail N},
	journal={Journal of Global Optimization},
	volume={24},
	number={2},
	pages={187--203},
	year={2002},
	publisher={Springer}
}
@article{burachik2010primal,
	title={A primal dual modified subgradient algorithm with sharp Lagrangian},
	author={Burachik, Regina S and Iusem, Alfredo N and Melo, Jefferson G},
	journal={Journal of Global Optimization},
	volume={46},
	number={3},
	pages={347--361},
	year={2010},
	publisher={Springer}
}
@article{bauschke2017projecting,
	title={Projecting onto the intersection of a cone and a sphere},
	author={Bauschke, Heinz H and Bui, Minh N and Wang, Xianfu},
	journal={arXiv preprint arXiv:1708.00585},
	year={2017}
}
@article{shechtman2015phase,
	title={Phase retrieval with application to optical imaging: a contemporary overview},
	author={Shechtman, Yoav and Eldar, Yonina C and Cohen, Oren and Chapman, Henry Nicholas and Miao, Jianwei and Segev, Mordechai},
	journal={IEEE signal processing magazine},
	volume={32},
	number={3},
	pages={87--109},
	year={2015},
	publisher={IEEE}
}
@article{jiang2017robust,
	title={Robust Phase Retrieval via ADMM with Outliers},
	author={Jiang, Xue and So, Hing-Cheung and Liu, Xingzhao},
	journal={arXiv preprint arXiv:1702.06157},
	year={2017}
}
@article{bolte2014proximal,
	title={Proximal alternating linearized minimization or nonconvex and nonsmooth problems},
	author={Bolte, J{\'e}r{\^o}me and Sabach, Shoham and Teboulle, Marc},
	journal={Mathematical Programming},
	volume={146},
	number={1-2},
	pages={459--494},
	year={2014},
	publisher={Springer}
}
@article{birgin2016evaluation,
	title={Evaluation complexity for nonlinear constrained optimization using unscaled KKT conditions and high-order models},
	author={Birgin, Ernesto G and Gardenghi, JL and Mart{\'i}nez, Jos{\'e} Mario and Santos, SA and Toint, Ph L},
	journal={SIAM Journal on Optimization},
	volume={26},
	number={2},
	pages={951--967},
	year={2016},
	publisher={SIAM}
}
@article{rosasco2014convergence,
	title={Convergence of stochastic proximal gradient algorithm},
	author={Rosasco, Lorenzo and Villa, Silvia and V{\~u}, Bang C{\^o}ng},
	journal={arXiv preprint arXiv:1403.5074},
	year={2014}
}
@inproceedings{hu2009accelerated,
	title={Accelerated gradient methods for stochastic optimization and online learning},
	author={Hu, Chonghai and Pan, Weike and Kwok, James T},
	booktitle={Advances in Neural Information Processing Systems},
	pages={781--789},
	year={2009}
}
@article{rosasco2017first,
	title={A first-order stochastic primal-dual algorithm with correction step},
	author={Rosasco, Lorenzo and Villa, Silvia and V{\~u}, B?ng C{\^o}ng},
	journal={Numerical Functional Analysis and Optimization},
	volume={38},
	number={5},
	pages={602--626},
	year={2017},
	publisher={Taylor \& Francis}
}
@article{combettes2015stochastic,
	title={Stochastic quasi-Fej{\'e}r block-coordinate fixed point iterations with random sweeping},
	author={Combettes, Patrick L and Pesquet, Jean-Christophe},
	journal={SIAM Journal on Optimization},
	volume={25},
	number={2},
	pages={1221--1248},
	year={2015},
	publisher={SIAM}
}
@article{vu2016almost,
	title={Almost sure convergence of the forward--backward--forward splitting algorithm},
	author={V{\~u}, B?ng C{\^o}ng},
	journal={Optimization Letters},
	volume={10},
	number={4},
	pages={781--803},
	year={2016},
	publisher={Springer}
}
@inproceedings{ouyang2013stochastic,
	title={Stochastic alternating direction method of multipliers},
	author={Ouyang, Hua and He, Niao and Tran, Long and Gray, Alexander},
	booktitle={International Conference on Machine Learning},
	pages={80--88},
	year={2013}
}
@article{huang2018mini,
	title={Mini-Batch Stochastic ADMMs for Nonconvex Nonsmooth Optimization},
	author={Huang, Feihu and Chen, Songcan},
	journal={arXiv preprint arXiv:1802.03284},
	year={2018}
}
@inproceedings{palaniappan2016stochastic,
	title={Stochastic variance reduction methods for saddle-point problems},
	author={Palaniappan, Balamurugan and Bach, Francis},
	booktitle={Advances in Neural Information Processing Systems},
	pages={1416--1424},
	year={2016}
}
@article{liang2004gradient,
	title={A gradient search method to round the semidefinite programming relaxation solution for ad hoc wireless sensor network localization},
	author={Liang, Tzu-Chen and Wang, Ta-Chung and Ye, Yinyu},
	journal={Sanford University, formal report},
	volume={5},
	year={2004}
}
@article{tu2014practical,
	title={Practical first order methods for large scale semidefinite programming},
	author={Tu, Stephen and Wang, Jingyan},
	year={2014}
}
@article{yang1998second,
	title={Second-order global optimality conditions for convex composite optimization},
	author={Yang, XQ},
	journal={Mathematical Programming},
	volume={81},
	number={3},
	pages={327--347},
	year={1998},
	publisher={Springer}
}
@article{burke1987second,
	title={Second order necessary and sufficient conditions for convex composite NDO},
	author={Burke, James V},
	journal={Mathematical Programming},
	volume={38},
	number={3},
	pages={287--302},
	year={1987},
	publisher={Springer}
}
@article{bolte2018nonconvex,
	title={Nonconvex Lagrangian-based optimization: monitoring schemes and global convergence},
	author={Bolte, J{\'e}r{\^o}me and Sabach, Shoham and Teboulle, Marc},
	journal={Mathematics of Operations Research},
	year={2018},
	publisher={INFORMS}
}
@article{poliquin1998tilt,
	title={Tilt stability of a local minimum},
	author={Poliquin, RA and Rockafellar, R Tyrrell},
	journal={SIAM Journal on Optimization},
	volume={8},
	number={2},
	pages={287--299},
	year={1998},
	publisher={SIAM}
}
@article{gfrerer2015complete,
	title={Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions},
	author={Gfrerer, Helmut and Mordukhovich, Boris S},
	journal={SIAM Journal on Optimization},
	volume={25},
	number={4},
	pages={2081--2119},
	year={2015},
	publisher={SIAM}
}
@article{poliquin1996prox,
	title={Prox-regular functions in variational analysis},
	author={Poliquin, Ren{\'e} and Rockafellar, R},
	journal={Transactions of the American Mathematical Society},
	volume={348},
	number={5},
	pages={1805--1838},
	year={1996}
}
@book{mordukhovich2006variational,
	title={Variational analysis and generalized differentiation I: Basic theory},
	author={Mordukhovich, Boris S},
	volume={330},
	year={2006},
	publisher={Springer Science \& Business Media}
}
@book{rockafellar2009variational,
	title={Variational analysis},
	author={Rockafellar, R Tyrrell and Wets, Roger J-B},
	volume={317},
	year={2009},
	publisher={Springer Science \& Business Media}
}
@book{luenberger1984linear,
	title={Linear and nonlinear programming},
	author={Luenberger, David G and Ye, Yinyu and others},
	volume={2},
	year={2007}, Third edition, 
	publisher={Springer}
}