Page MenuHomec4science

main.bbl
No OneTemporary

File Metadata

Created
Mon, Nov 11, 09:31

main.bbl

\begin{thebibliography}{10}
\bibitem{arora2018compressed}
S.~Arora, M.~Khodak, N.~Saunshi, and K.~Vodrahalli.
\newblock A compressed sensing view of unsupervised text embeddings,
bag-of-n-grams, and lstms.
\newblock 2018.
\bibitem{barvinok1995problems}
A.~I. Barvinok.
\newblock Problems of distance geometry and convex properties of quadratic
maps.
\newblock {\em Discrete \& Computational Geometry}, 13(2):189--202, 1995.
\bibitem{bertsekas1976penalty}
D.~P. Bertsekas.
\newblock On penalty and multiplier methods for constrained minimization.
\newblock {\em SIAM Journal on Control and Optimization}, 14(2):216--235, 1976.
\bibitem{bertsekas1982constrained}
D.~P. Bertsekas.
\newblock Constrained optimization and lagrange multiplier methods.
\newblock {\em Computer Science and Applied Mathematics, Boston: Academic
Press, 1982}, 1982.
\bibitem{bertsekas2014constrained}
D.~P. Bertsekas.
\newblock {\em Constrained optimization and Lagrange multiplier methods}.
\newblock Academic press, 2014.
\bibitem{bhojanapalli2018smoothed}
S.~Bhojanapalli, N.~Boumal, P.~Jain, and P.~Netrapalli.
\newblock Smoothed analysis for low-rank solutions to semidefinite programs in
quadratic penalty form.
\newblock {\em arXiv preprint arXiv:1803.00186}, 2018.
\bibitem{bhojanapalli2016dropping}
S.~Bhojanapalli, A.~Kyrillidis, and S.~Sanghavi.
\newblock Dropping convexity for faster semi-definite optimization.
\newblock In {\em Conference on Learning Theory}, pages 530--582, 2016.
\bibitem{birgin2016evaluation}
E.~G. Birgin, J.~Gardenghi, J.~M. Martinez, S.~Santos, and P.~L. Toint.
\newblock Evaluation complexity for nonlinear constrained optimization using
unscaled kkt conditions and high-order models.
\newblock {\em SIAM Journal on Optimization}, 26(2):951--967, 2016.
\bibitem{birgin2014practical}
E.~G. Birgin and J.~M. Mart\_nez.
\newblock {\em Practical augmented Lagrangian methods for constrained
optimization}, volume~10.
\newblock SIAM, 2014.
\bibitem{bolte2017error}
J.~Bolte, T.~P. Nguyen, J.~Peypouquet, and B.~W. Suter.
\newblock From error bounds to the complexity of first-order descent methods
for convex functions.
\newblock {\em Mathematical Programming}, 165(2):471--507, 2017.
\bibitem{bolte2018nonconvex}
J.~Bolte, S.~Sabach, and M.~Teboulle.
\newblock Nonconvex lagrangian-based optimization: monitoring schemes and
global convergence.
\newblock {\em Mathematics of Operations Research}, 2018.
\bibitem{boumal2016global}
N.~Boumal, P.-A. Absil, and C.~Cartis.
\newblock Global rates of convergence for nonconvex optimization on manifolds.
\newblock {\em arXiv preprint arXiv:1605.08101}, 2016.
\bibitem{boumal2014manopt}
N.~Boumal, B.~Mishra, P.-A. Absil, and R.~Sepulchre.
\newblock Manopt, a matlab toolbox for optimization on manifolds.
\newblock {\em The Journal of Machine Learning Research}, 15(1):1455--1459,
2014.
\bibitem{boumal2016non}
N.~Boumal, V.~Voroninski, and A.~Bandeira.
\newblock The non-convex burer-monteiro approach works on smooth semidefinite
programs.
\newblock In {\em Advances in Neural Information Processing Systems}, pages
2757--2765, 2016.
\bibitem{burer2003nonlinear}
S.~Burer and R.~D. Monteiro.
\newblock A nonlinear programming algorithm for solving semidefinite programs
via low-rank factorization.
\newblock {\em Mathematical Programming}, 95(2):329--357, 2003.
\bibitem{burer2005local}
S.~Burer and R.~D. Monteiro.
\newblock Local minima and convergence in low-rank semidefinite programming.
\newblock {\em Mathematical Programming}, 103(3):427--444, 2005.
\bibitem{candes2008introduction}
E.~J. Cand{\`e}s and M.~B. Wakin.
\newblock An introduction to compressive sampling.
\newblock {\em IEEE signal processing magazine}, 25(2):21--30, 2008.
\bibitem{cartis2011evaluation}
C.~Cartis, N.~I. Gould, and P.~L. Toint.
\newblock On the evaluation complexity of composite function minimization with
applications to nonconvex nonlinear programming.
\newblock {\em SIAM Journal on Optimization}, 21(4):1721--1739, 2011.
\bibitem{cartis2012complexity}
C.~Cartis, N.~I. Gould, and P.~L. Toint.
\newblock Complexity bounds for second-order optimality in unconstrained
optimization.
\newblock {\em Journal of Complexity}, 28(1):93--108, 2012.
\bibitem{cartis2018optimality}
C.~Cartis, N.~I. Gould, and P.~L. Toint.
\newblock Optimality of orders one to three and beyond: characterization and
evaluation complexity in constrained nonconvex optimization.
\newblock {\em Journal of Complexity}, 2018.
\bibitem{chambolle2011first}
A.~Chambolle and T.~Pock.
\newblock A first-order primal-dual algorithm for convex problems with
applications to imaging.
\newblock {\em Journal of mathematical imaging and vision}, 40(1):120--145,
2011.
\bibitem{chen2001atomic}
S.~S. Chen, D.~L. Donoho, and M.~A. Saunders.
\newblock Atomic decomposition by basis pursuit.
\newblock {\em SIAM review}, 43(1):129--159, 2001.
\bibitem{clason2018acceleration}
C.~Clason, S.~Mazurenko, and T.~Valkonen.
\newblock Acceleration and global convergence of a first-order primal--dual
method for nonconvex problems.
\newblock {\em arXiv preprint arXiv:1802.03347}, 2018.
\bibitem{dai2002convergence}
Y.-H. Dai.
\newblock Convergence properties of the bfgs algoritm.
\newblock {\em SIAM Journal on Optimization}, 13(3):693--701, 2002.
\bibitem{fernandez2012local}
D.~Fernandez and M.~V. Solodov.
\newblock Local convergence of exact and inexact augmented lagrangian methods
under the second-order sufficient optimality condition.
\newblock {\em SIAM Journal on Optimization}, 22(2):384--407, 2012.
\bibitem{fletcher2013practical}
R.~Fletcher.
\newblock {\em Practical methods of optimization}.
\newblock John Wiley \& Sons, 2013.
\bibitem{flores2012complete}
F.~Flores-Baz{\'a}n, F.~Flores-Baz{\'a}n, and C.~Vera.
\newblock A complete characterization of strong duality in nonconvex
optimization with a single constraint.
\newblock {\em Journal of Global Optimization}, 53(2):185--201, 2012.
\bibitem{ge2016efficient}
R.~Ge, C.~Jin, P.~Netrapalli, A.~Sidford, et~al.
\newblock Efficient algorithms for large-scale generalized eigenvector
computation and canonical correlation analysis.
\newblock In {\em International Conference on Machine Learning}, pages
2741--2750, 2016.
\bibitem{ghadimi2016accelerated}
S.~Ghadimi and G.~Lan.
\newblock Accelerated gradient methods for nonconvex nonlinear and stochastic
programming.
\newblock {\em Mathematical Programming}, 156(1-2):59--99, 2016.
\bibitem{Goodfellow2014}
I.~J. {Goodfellow}, J.~{Pouget-Abadie}, M.~{Mirza}, B.~{Xu}, D.~{Warde-Farley},
S.~{Ozair}, A.~{Courville}, and Y.~{Bengio}.
\newblock {Generative Adversarial Networks}.
\newblock {\em ArXiv e-prints}, June 2014.
\bibitem{hestenes1969multiplier}
M.~R. Hestenes.
\newblock Multiplier and gradient methods.
\newblock {\em Journal of optimization theory and applications}, 4(5):303--320,
1969.
\bibitem{Ilyas2017}
A.~{Ilyas}, A.~{Jalal}, E.~{Asteri}, C.~{Daskalakis}, and A.~G. {Dimakis}.
\newblock {The Robust Manifold Defense: Adversarial Training using Generative
Models}.
\newblock {\em arXiv e-prints}, page arXiv:1712.09196, Dec. 2017.
\bibitem{jaggi2013revisiting}
M.~Jaggi.
\newblock Revisiting frank-wolfe: Projection-free sparse convex optimization.
\newblock In {\em ICML (1)}, pages 427--435, 2013.
\bibitem{karimi2016linear}
H.~Karimi, J.~Nutini, and M.~Schmidt.
\newblock Linear convergence of gradient and proximal-gradient methods under
the polyak-{\l}ojasiewicz condition.
\newblock In {\em Joint European Conference on Machine Learning and Knowledge
Discovery in Databases}, pages 795--811. Springer, 2016.
\bibitem{khot2011grothendieck}
S.~Khot and A.~Naor.
\newblock Grothendieck-type inequalities in combinatorial optimization.
\newblock {\em arXiv preprint arXiv:1108.2464}, 2011.
\bibitem{kulis2007fast}
B.~Kulis, A.~C. Surendran, and J.~C. Platt.
\newblock Fast low-rank semidefinite programming for embedding and clustering.
\newblock In {\em Artificial Intelligence and Statistics}, pages 235--242,
2007.
\bibitem{lan2016iteration}
G.~Lan and R.~D. Monteiro.
\newblock Iteration-complexity of first-order augmented lagrangian methods for
convex programming.
\newblock {\em Mathematical Programming}, 155(1-2):511--547, 2016.
\bibitem{lovasz2003semidefinite}
L.~Lov{\'a}sz.
\newblock Semidefinite programs and combinatorial optimization.
\newblock In {\em Recent advances in algorithms and combinatorics}, pages
137--194. Springer, 2003.
\bibitem{mascarenhas2004bfgs}
W.~F. Mascarenhas.
\newblock The bfgs method with exact line searches fails for non-convex
objective functions.
\newblock {\em Mathematical Programming}, 99(1):49--61, 2004.
\bibitem{mixon2016clustering}
D.~G. Mixon, S.~Villar, and R.~Ward.
\newblock Clustering subgaussian mixtures by semidefinite programming.
\newblock {\em arXiv preprint arXiv:1602.06612}, 2016.
\bibitem{mossel2015consistency}
E.~Mossel, J.~Neeman, and A.~Sly.
\newblock Consistency thresholds for the planted bisection model.
\newblock In {\em Proceedings of the forty-seventh annual ACM symposium on
Theory of computing}, pages 69--75. ACM, 2015.
\bibitem{nedelcu2014computational}
V.~Nedelcu, I.~Necoara, and Q.~Tran-Dinh.
\newblock Computational complexity of inexact gradient augmented lagrangian
methods: application to constrained mpc.
\newblock {\em SIAM Journal on Control and Optimization}, 52(5):3109--3134,
2014.
\bibitem{nesterov2009primal}
Y.~Nesterov.
\newblock Primal-dual subgradient methods for convex problems.
\newblock {\em Mathematical programming}, 120(1):221--259, 2009.
\bibitem{nesterov1983method}
Y.~E. Nesterov.
\newblock A method for solving the convex programming problem with convergence
rate o (1/k\^{} 2).
\newblock In {\em Dokl. Akad. Nauk SSSR}, volume 269, pages 543--547, 1983.
\bibitem{nocedal2006numerical}
J.~Nocedal and S.~Wright.
\newblock {\em Numerical Optimization}.
\newblock Springer Series in Operations Research and Financial Engineering.
Springer New York, 2006.
\bibitem{nouiehed2018convergence}
M.~Nouiehed, J.~D. Lee, and M.~Razaviyayn.
\newblock Convergence to second-order stationarity for constrained non-convex
optimization.
\newblock {\em arXiv preprint arXiv:1810.02024}, 2018.
\bibitem{obozinski2011group}
G.~Obozinski, L.~Jacob, and J.-P. Vert.
\newblock Group lasso with overlaps: the latent group lasso approach.
\newblock {\em arXiv preprint arXiv:1110.0413}, 2011.
\bibitem{parikh2014proximal}
N.~Parikh, S.~Boyd, et~al.
\newblock Proximal algorithms.
\newblock {\em Foundations and Trends in Optimization}, 1(3):127--239, 2014.
\bibitem{park2016provable}
D.~Park, A.~Kyrillidis, S.~Bhojanapalli, C.~Caramanis, and S.~Sanghavi.
\newblock Provable burer-monteiro factorization for a class of norm-constrained
matrix problems.
\newblock {\em arXiv preprint arXiv:1606.01316}, 2016.
\bibitem{pataki1998rank}
G.~Pataki.
\newblock On the rank of extreme matrices in semidefinite programs and the
multiplicity of optimal eigenvalues.
\newblock {\em Mathematics of operations research}, 23(2):339--358, 1998.
\bibitem{Peng2007}
J.~Peng and Y.~Wei.
\newblock Approximating {K}--means--type clustering via semidefinite
programming.
\newblock {\em SIAM J. Optim.}, 18(1):186--205, 2007.
\bibitem{powell1969method}
M.~J. Powell.
\newblock A method for nonlinear constraints in minimization problems.
\newblock {\em Optimization}, pages 283--298, 1969.
\bibitem{raghavendra2008optimal}
P.~Raghavendra.
\newblock Optimal algorithms and inapproximability results for every csp?
\newblock In {\em Proceedings of the fortieth annual ACM symposium on Theory of
computing}, pages 245--254. ACM, 2008.
\bibitem{rockafellar1993lagrange}
R.~T. Rockafellar.
\newblock Lagrange multipliers and optimality.
\newblock {\em SIAM review}, 35(2):183--238, 1993.
\bibitem{singer2011angular}
A.~Singer.
\newblock Angular synchronization by eigenvectors and semidefinite programming.
\newblock {\em Applied and computational harmonic analysis}, 30(1):20, 2011.
\bibitem{singer2011three}
A.~Singer and Y.~Shkolnisky.
\newblock Three-dimensional structure determination from common lines in
cryo-em by eigenvectors and semidefinite programming.
\newblock {\em SIAM journal on imaging sciences}, 4(2):543--572, 2011.
\bibitem{song2007dependence}
L.~Song, A.~Smola, A.~Gretton, and K.~M. Borgwardt.
\newblock A dependence maximization view of clustering.
\newblock In {\em Proceedings of the 24th international conference on Machine
learning}, pages 815--822. ACM, 2007.
\bibitem{tepper2018clustering}
M.~Tepper, A.~M. Sengupta, and D.~Chklovskii.
\newblock Clustering is semidefinitely not that hard: Nonnegative sdp for
manifold disentangling.
\newblock {\em Journal of Machine Learning Research}, 19(82), 2018.
\bibitem{tran2018adaptive}
Q.~Tran-Dinh, A.~Alacaoglu, O.~Fercoq, and V.~Cevher.
\newblock An adaptive primal-dual framework for nonsmooth convex minimization.
\newblock {\em arXiv preprint arXiv:1808.04648}, 2018.
\bibitem{tran2018smooth}
Q.~Tran-Dinh, O.~Fercoq, and V.~Cevher.
\newblock A smooth primal-dual optimization framework for nonsmooth composite
convex minimization.
\newblock {\em SIAM Journal on Optimization}, 28(1):96--134, 2018.
\bibitem{waldspurger2018rank}
I.~Waldspurger and A.~Waters.
\newblock Rank optimality for the burer-monteiro factorization.
\newblock {\em arXiv preprint arXiv:1812.03046}, 2018.
\bibitem{xiao2017/online}
H.~Xiao, K.~Rasul, and R.~Vollgraf.
\newblock Fashion-mnist: a novel image dataset for benchmarking machine
learning algorithms, 2017.
\bibitem{xu2017inexact}
Y.~Xu.
\newblock Iteration complexity of inexact augmented lagrangian methods for
constrained convex programming.
\newblock {\em arXiv preprint arXiv:1711.05812v2}, 2017.
\bibitem{xu2017globally}
Y.~Xu and W.~Yin.
\newblock A globally convergent algorithm for nonconvex optimization based on
block coordinate update.
\newblock {\em Journal of Scientific Computing}, 72(2):700--734, 2017.
\bibitem{yang2015sdpnal}
L.~Yang, D.~Sun, and K.-C. Toh.
\newblock Sdpnal+: a majorized semismooth newton-cg augmented lagrangian method
for semidefinite programming with nonnegative constraints.
\newblock {\em Mathematical Programming Computation}, 7(3):331--366, 2015.
\bibitem{yurtsever2015universal}
A.~Yurtsever, Q.~T. Dinh, and V.~Cevher.
\newblock A universal primal-dual convex optimization framework.
\newblock In {\em Advances in Neural Information Processing Systems}, pages
3150--3158, 2015.
\bibitem{yurtsever2018conditional}
A.~Yurtsever, O.~Fercoq, F.~Locatello, and V.~Cevher.
\newblock A conditional gradient framework for composite convex minimization
with applications to semidefinite programming.
\newblock {\em arXiv preprint arXiv:1804.08544}, 2018.
\bibitem{zhao1998semidefinite}
Q.~Zhao, S.~E. Karisch, F.~Rendl, and H.~Wolkowicz.
\newblock Semidefinite programming relaxations for the quadratic assignment
problem.
\newblock {\em Journal of Combinatorial Optimization}, 2(1):71--109, 1998.
\end{thebibliography}

Event Timeline