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@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}
}
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