Proximal Splitting Methods in Nonsmooth Convex Optimization

[1]  W. Fenchel.  On Conjugate Convex Functions , 1949, Canadian Journal of Mathematics.

[2]  W. Fenchel.  Convex cones, sets, and functions , 1953 .

[3]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[4]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[5]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[6]  R. Rockafellar,et al.  On the maximal monotonicity of subdifferential mappings. , 1970 .

[7]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[8]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[9]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[10]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[11]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[12]  Gregory B. Passty.  Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .

[13]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[14]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[15]  D. Gabay.  Applications of the method of multipliers to variational inequalities , 1983 .

[16]  Y. Nesterov.  A method for unconstrained convex minimization problem with the rate of convergence o(1/k^2) , 1983 .

[17]  M. Teboulle,et al.  Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming , 1986 .

[18]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[19]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

[20]  P. Tseng.  Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

[21]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[22]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[23]  G. Gonnet,et al.  On Lambert's W Function , 1993 .

[24]  J. Rodriguez,et al.  Problem (1) , 1994 .

[25]  Stanley Osher,et al.  Total variation based image restoration with free local constraints , 1994, Proceedings of 1st International Conference on Image Processing.

[26]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[27]  H. Attouch.  A General Duality Principle for the Sum of Two Operators 1 , 1996 .

[28]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[29]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[30]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[31]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[32]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[33]  Felipe Alvarez,et al.  On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces , 2000, SIAM J. Control. Optim..

[34]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[35]  P. L. Combettes,et al.  Quasi-Fejérian Analysis of Some Optimization Algorithms , 2001 .

[36]  H. Attouch,et al.  An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping , 2001 .

[37]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[38]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[39]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[40]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[41]  R. Rockafellar,et al.  Deviation Measures in Risk Analysis and Optimization , 2002 .

[42]  Bingsheng He,et al.  A new inexact alternating directions method for monotone variational inequalities , 2002, Math. Program..

[43]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[44]  P. L. Combettes,et al.  Solving monotone inclusions via compositions of nonexpansive averaged operators , 2004 .

[45]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[46]  ANTONIN CHAMBOLLE,et al.  An Algorithm for Total Variation Minimization and Applications , 2004, Journal of Mathematical Imaging and Vision.

[47]  O. SIAMJ.,et al.  PROX-METHOD WITH RATE OF CONVERGENCE O(1/t) FOR VARIATIONAL INEQUALITIES WITH LIPSCHITZ CONTINUOUS MONOTONE OPERATORS AND SMOOTH CONVEX-CONCAVE SADDLE POINT PROBLEMS∗ , 2004 .

[48]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2004 .

[49]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[50]  J. Hiriart-Urruty,et al.  Fundamentals of Convex Analysis , 2004 .

[51]  Heinz H. Bauschke,et al.  The asymptotic behavior of the composition of two resolvents , 2005, Nonlinear Analysis: Theory, Methods & Applications.

[52]  Heinz H. Bauschke,et al.  A new proximal point iteration that converges weakly but not in norm , 2005 .

[53]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[54]  Yurii Nesterov,et al.  Excessive Gap Technique in Nonsmooth Convex Minimization , 2005, SIAM J. Optim..

[55]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[56]  Regina Sandra Burachik,et al.  A new geometric condition for Fenchel's duality in infinite dimensional spaces , 2005, Math. Program..

[57]  Hans-Jakob Lüthi,et al.  Convex risk measures for portfolio optimization and concepts of flexibility , 2005, Math. Program..

[58]  Bernhard Schölkopf,et al.  Combining a Filter Method with SVMs , 2006, Feature Extraction.

[59]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[60]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

[61]  Yurii Nesterov,et al.  Smoothing Technique and its Applications in Semidefinite Optimization , 2004, Math. Program..

[62]  Patrick L. Combettes,et al.  Proximal Thresholding Algorithm for Minimization over Orthonormal Bases , 2007, SIAM J. Optim..

[63]  Nikolas P. Galatsanos,et al.  Variational Bayesian Image Restoration Based on a Product of $t$-Distributions Image Prior , 2008, IEEE Transactions on Image Processing.

[64]  S. Simons.  From Hahn-Banach to monotonicity , 2008 .

[65]  Liu Guoyin,et al.  On Extension of Fenchel Duality and its Application , 2008 .

[66]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[67]  P. L. Combettes,et al.  Iterative construction of the resolvent of a sum of maximal monotone operators , 2009 .

[68]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[69]  Benar Fux Svaiter,et al.  General Projective Splitting Methods for Sums of Maximal Monotone Operators , 2009, SIAM J. Control. Optim..

[70]  Sorin-Mihai Grad,et al.  Duality in Vector Optimization , 2009, Vector Optimization.

[71]  Heinz H. Bauschke,et al.  The Baillon-Haddad Theorem Revisited , 2009, 0906.0807.

[72]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[73]  Wotao Yin,et al.  Alternating direction augmented Lagrangian methods for semidefinite programming , 2010, Math. Program. Comput..

[74]  Ernie Esser,et al.  Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman , 2009 .

[75]  Patrick L. Combettes,et al.  A Parallel Splitting Method for Coupled Monotone Inclusions , 2009, SIAM J. Control. Optim..

[76]  José M. Bioucas-Dias,et al.  Fast Image Recovery Using Variable Splitting and Constrained Optimization , 2009, IEEE Transactions on Image Processing.

[77]  R. Boţ,et al.  Conjugate Duality in Convex Optimization , 2010 .

[78]  J. Borwein,et al.  Convex Functions: Constructions, Characterizations and Counterexamples , 2010 .

[79]  Marc Teboulle,et al.  Gradient-based algorithms with applications to signal-recovery problems , 2010, Convex Optimization in Signal Processing and Communications.

[80]  Patrick L. Combettes,et al.  Proximal Algorithms for Multicomponent Image Recovery Problems , 2011, Journal of Mathematical Imaging and Vision.

[81]  Junfeng Yang,et al.  Alternating Direction Algorithms for 1-Problems in Compressive Sensing , 2009, SIAM J. Sci. Comput..

[82]  Patrick L. Combettes,et al.  A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality , 2010, SIAM J. Optim..

[83]  Radu Ioan Bot,et al.  Optimization problems in statistical learning: Duality and optimality conditions , 2011, Eur. J. Oper. Res..

[84]  H. Attouch,et al.  Alternating proximal algorithms for linearly constrained variational inequalities: Application to do , 2011 .

[85]  Radu Ioan Bot,et al.  Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures , 2011, Math. Methods Oper. Res..

[86]  B. Mordukhovich,et al.  Applications of variational analysis to a generalized Heron problem , 2011, 1106.0088.

[87]  L. Ljung,et al.  Just Relax and Come Clustering! : A Convexification of k-Means Clustering , 2011 .

[88]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[89]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[90]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[91]  P. L. Combettes,et al.  Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators , 2011, Set-Valued and Variational Analysis.

[92]  S. Setzer,et al.  Infimal convolution regularizations with discrete ℓ1-type functionals , 2011 .

[93]  Francis R. Bach,et al.  Clusterpath: an Algorithm for Clustering using Convex Fusion Penalties , 2011, ICML.

[94]  R. Boţ,et al.  Iterative regularization with a general penalty term—theory and application to L1 and TV regularization , 2012 .

[95]  Boris S. Mordukhovich,et al.  Solving a Generalized Heron Problem by Means of Convex Analysis , 2012, Am. Math. Mon..

[96]  R. Boţ,et al.  A variable smoothing algorithm for solving convex optimization problems , 2012, 1207.3254.

[97]  R. Boţ,et al.  On the acceleration of the double smoothing technique for unconstrained convex optimization problems , 2012, 1205.0721.

[98]  Francesco Orabona,et al.  PRISMA: PRoximal Iterative SMoothing Algorithm , 2012, ArXiv.

[99]  Shiqian Ma,et al.  Fast Multiple-Splitting Algorithms for Convex Optimization , 2009, SIAM J. Optim..

[100]  Yurii Nesterov,et al.  Double Smoothing Technique for Large-Scale Linearly Constrained Convex Optimization , 2012, SIAM J. Optim..

[101]  R. Boţ,et al.  Employing different loss functions for the classification of images via supervised learning , 2014 .

[102]  R. Boţ,et al.  Solving Systems of Monotone Inclusions via Primal-dual Splitting Techniques , 2013 .

[103]  Jalal M. Fadili,et al.  Iteration-Complexity of a Generalized Forward Backward Splitting Algorithm , 2013, ICASSP 2014.

[104]  Gabriele Steidl,et al.  Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems , 2013, SSVM.

[105]  Laurent Condat,et al.  A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms , 2012, Journal of Optimization Theory and Applications.

[106]  Radu Ioan Bot,et al.  Convergence Analysis for a Primal-Dual Monotone + Skew Splitting Algorithm with Applications to Total Variation Minimization , 2012, Journal of Mathematical Imaging and Vision.

[107]  Radu Ioan Bot,et al.  A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems , 2012, Computational Optimization and Applications.

[108]  Radu Ioan Bot,et al.  A Primal-Dual Splitting Algorithm for Finding Zeros of Sums of Maximal Monotone Operators , 2012, SIAM J. Optim..

[109]  Patrick L. Combettes,et al.  An Algorithm for Splitting Parallel Sums of Linearly Composed Monotone Operators, with Applications to Signal Recovery , 2013, 1305.5828.

[110]  R. Boţ,et al.  Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators , 2013, 1306.3191.

[111]  Radu Ioan Bot,et al.  A Douglas-Rachford Type Primal-Dual Method for Solving Inclusions with Mixtures of Composite and Parallel-Sum Type Monotone Operators , 2012, SIAM J. Optim..

[112]  Bang Công Vu,et al.  A splitting algorithm for dual monotone inclusions involving cocoercive operators , 2011, Advances in Computational Mathematics.

[113]  Patrick L. Combettes,et al.  Systems of Structured Monotone Inclusions: Duality, Algorithms, and Applications , 2012, SIAM J. Optim..

[114]  Mohamed-Jalal Fadili,et al.  A Generalized Forward-Backward Splitting , 2011, SIAM J. Imaging Sci..

[115]  Richard G. Baraniuk,et al.  Fast Alternating Direction Optimization Methods , 2014, SIAM J. Imaging Sci..

[116]  R. Boţ,et al.  A Tseng’s Type Penalty Scheme for Solving Inclusion Problems Involving Linearly Composed and Parallel-Sum Type Monotone Operators , 2014 .

[117]  Ernö Robert Csetnek,et al.  Recent Developments on Primal–Dual Splitting Methods with Applications to Convex Minimization , 2014 .

[118]  Eric C. Chi,et al.  Splitting Methods for Convex Clustering , 2013, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[119]  Nelly Pustelnik,et al.  Epigraphical Projection and Proximal Tools for Solving Constrained Convex Optimization Problems: Part I , 2012, ArXiv.

[120]  Bang Công Vu,et al.  A Splitting Algorithm for Coupled System of Primal–Dual Monotone Inclusions , 2014, Journal of Optimization Theory and Applications.

[121]  Radu Ioan Bot,et al.  On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems , 2013, Mathematical Programming.

[122]  Radu Ioan Bot,et al.  Convex risk minimization via proximal splitting methods , 2013, Optim. Lett..

[123]  Radu Ioan Bot,et al.  Inertial Douglas-Rachford splitting for monotone inclusion problems , 2014, Appl. Math. Comput..

[124]  Radu Ioan Bot,et al.  An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems , 2014, Numerical Algorithms.

[125]  Jonas Schmitt.  Portfolio Selection Efficient Diversification Of Investments , 2016 .