Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

Nobuo Yamashita; Yan Gu

arxiv: 1906.12112 · v1 · pith:6DO3Z2H3new · submitted 2019-06-28 · 🧮 math.OC

Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

Yan Gu , Nobuo Yamashita This is my paper

Pith reviewed 2026-05-25 14:02 UTC · model grok-4.3

classification 🧮 math.OC

keywords alternating direction method of multipliersproximal ADMMindefinite proximal termsvariable metricBFGS updateglobal convergenceconvex optimization

0 comments

The pith

Variable-metric ADMM with indefinite proximal terms converges globally when the matrices meet stated conditions, and a BFGS construction satisfies them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a proximal alternating direction method of multipliers allowing the proximal term to be indefinite and to change at each step through a variable metric. It supplies sufficient conditions on those terms that guarantee global convergence for linearly constrained convex optimization problems. The authors also construct a concrete indefinite term using the BFGS update that meets the required conditions. Earlier fixed or semidefinite versions are extended by this variable indefinite approach. A reader cares because the method keeps subproblems easy while potentially allowing faster practical behavior without losing the convergence guarantee.

Core claim

We consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions for the global convergence.

What carries the argument

The sequence of variable-metric indefinite proximal matrices, generated by a BFGS-style update, that obeys the sufficient conditions required for the convergence analysis.

If this is right

Global convergence of the iterates holds for the ADMM even when the proximal terms are allowed to be indefinite.
The BFGS update supplies a practical way to generate proximal terms that obey the convergence conditions.
The method applies directly to linearly constrained convex optimization problems.
Each subproblem remains as easy to solve as in the standard proximal ADMM.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of indefinite proximal construction could be tried in other splitting algorithms that rely on proximal terms.
Numerical tests on specific convex problems could reveal whether the indefinite terms produce measurable speedups compared with semidefinite alternatives.
The sufficient conditions might admit further weakening while still preserving convergence.

Load-bearing premise

The sequence of indefinite proximal matrices must satisfy the specific sufficient conditions on their properties that the convergence proof relies upon.

What would settle it

A convex optimization instance together with a sequence of proximal matrices obeying the stated conditions for which the generated iterates fail to converge to an optimal solution.

read the original abstract

This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions for the global convergence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies new sufficient conditions for global convergence of variable-metric indefinite proximal ADMM plus a BFGS construction claimed to meet them, but the conditions look easy to violate in practice.

read the letter

The core advance is the move from fixed indefinite proximal terms or variable-metric positive-semidefinite ones to a variable-metric indefinite setting, together with explicit sufficient conditions on the sequence of proximal matrices and a concrete BFGS update that is asserted to satisfy those conditions. This is a direct, incremental step from the authors' earlier BFGS work and the fixed-indefinite results cited in the abstract. If the conditions are both checkable and routinely met, the result gives a practical route to retain convergence while allowing the speed-up that indefinite terms sometimes provide. That part is cleanly motivated and stays within standard convex-analysis arguments plus the new matrix conditions. The main weakness is that the sufficient conditions appear to require a uniform lower bound on the smallest eigenvalue of a certain combination involving the metric and the constraint matrix, plus summability on the differences. Standard BFGS updates can let eigenvalues drift when the secant equation holds only approximately or when indefiniteness builds up, exactly as the stress-test note flags. The abstract states that the proposed BFGS term meets the conditions, yet without seeing the full derivation or any numerical check that the implemented matrices stay inside the required bounds, it is not clear whether the guarantee actually covers the algorithm that would be coded. No experiments are described in the abstract, so the claimed practical benefit remains untested in the provided material. The work is aimed at specialists who already follow proximal ADMM variants and quasi-Newton updates for constrained convex problems. It contains enough new technical material on the conditions and the construction to justify sending it to referees, provided they are asked to verify whether the BFGS term reliably satisfies the stated bounds without extra safeguards. I would send it for review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a proximal ADMM with variable metric indefinite proximal terms for linearly constrained convex optimization. It derives sufficient conditions on the sequence of indefinite proximal matrices ensuring global convergence and proposes a BFGS-based construction claimed to satisfy those conditions.

Significance. If the sufficient conditions are correctly stated and the BFGS construction is shown to meet them, the result would extend the theory of proximal ADMM to indefinite metrics while retaining global convergence, providing a theoretical basis for observed practical speedups with slightly indefinite terms.

major comments (2)

[§3] §3 (convergence theorem): the sufficient conditions require a uniform lower bound on the smallest eigenvalue of a linear combination of the metric and constraint matrix together with a summability condition on successive differences; the manuscript must explicitly verify that the proposed BFGS update preserves these bounds for all iterates, as standard BFGS can produce eigenvalue drift when the secant condition holds only approximately.
[§4] §4 (BFGS construction): the claim that the new indefinite proximal term based on the BFGS update satisfies the conditions of §3 is load-bearing for the main contribution, yet the provided derivation does not include an error analysis or inductive argument confirming that the eigenvalue lower bound remains positive uniformly.

minor comments (2)

[Introduction] The reference to the fixed indefinite proximal term result and to Gu-Yamashita should be expanded with precise citations and a short comparison table of convergence rates.
[§2] Notation for the proximal matrix sequence {B_k} should be introduced once and used consistently; several early equations reuse B without the subscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that the manuscript's main claims rest on the BFGS construction satisfying the convergence conditions, yet the current text does not supply an explicit inductive verification or error analysis. We will revise the paper to include this material.

read point-by-point responses

Referee: [§3] §3 (convergence theorem): the sufficient conditions require a uniform lower bound on the smallest eigenvalue of a linear combination of the metric and constraint matrix together with a summability condition on successive differences; the manuscript must explicitly verify that the proposed BFGS update preserves these bounds for all iterates, as standard BFGS can produce eigenvalue drift when the secant condition holds only approximately.

Authors: We agree that an explicit verification is required. In the revision we will insert a new lemma (with proof) that proceeds by induction on the iterates. The argument exploits the specific form of the indefinite BFGS update together with the fact that the secant condition is enforced only on the range of the constraint matrix; this prevents the eigenvalue drift that can occur in the classical positive-definite BFGS setting. The summability condition on successive differences will also be verified directly from the update formula. revision: yes
Referee: [§4] §4 (BFGS construction): the claim that the new indefinite proximal term based on the BFGS update satisfies the conditions of §3 is load-bearing for the main contribution, yet the provided derivation does not include an error analysis or inductive argument confirming that the eigenvalue lower bound remains positive uniformly.

Authors: We acknowledge the gap. The revised manuscript will contain a dedicated subsection (or appendix) that supplies the missing inductive argument and a uniform lower-bound estimate on the relevant eigenvalue. The proof will be self-contained and will not rely on external results beyond standard properties of the BFGS update under the paper's assumptions. revision: yes

Circularity Check

0 steps flagged

Convergence proof and BFGS construction are independent of each other and of prior self-citations.

full rationale

The paper states sufficient conditions on the sequence of indefinite proximal matrices and separately constructs a BFGS-based term asserted to meet those conditions. No equation reduces the claimed convergence to a fitted parameter or to a definition that presupposes the result. The citation to the authors' prior semidefinite BFGS work supplies only motivation and background; the new indefinite conditions and their verification are developed within the present manuscript without load-bearing reliance on unverified self-referential steps. The derivation therefore remains self-contained against external convex-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The work relies on standard convex optimization assumptions and the existence of the claimed sufficient conditions.

axioms (1)

domain assumption The problem is a linearly constrained convex optimization problem.
Explicitly stated as the setting for which the method is developed.

pith-pipeline@v0.9.0 · 5697 in / 1176 out tokens · 26122 ms · 2026-05-25T14:02:35.915396+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present sufficient conditions on {T_k} for the global convergence of VMIP-ADMM... we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Condition 2.1... (g) ∃ c ∈ (0, 0.5), T_k + 3/2 Σ_g - γ_{k-1/2} T_k^+ - 2T^- + (3/4 - 1/2 c)βBᵀB ⪰ 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 1 internal anchor

[1]

A TTOUCH , L

H. A TTOUCH , L. M. B RICENO -A RIAS , AND P. L. C OMBETTES , A parallel splitting method for coupled monotone inclusions , SIAM Journal on Control and Optimization, 48 (2010), pp. 32 46– 3270

work page 2010
[2]

B ANERT , R

S. B ANERT , R. I. B OT, AND E. R. C SETNEK , Fixing and extending some recent results on the ADMM algorithm, arXiv preprint arXiv:1612.05057, (2016)

work page arXiv 2016
[3]

H. H. B AUSCHKE , P. L. C OMBETTES , ET AL ., Convex analysis and monotone operator theory in Hilbert spaces, vol. 408, Springer, 2011

work page 2011
[4]

B OYD , N

S. B OYD , N. P ARIKH , E. C HU, B. P ELEATO , AND J. E CKSTEIN , Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends R⃝ in Machine Learning, 3 (2011), pp. 1–122

work page 2011
[5]

C HAMBOLLE AND T

A. C HAMBOLLE AND T. P OCK , A ﬁrst-order primal-dual algorithm for convex problems wit h applications to imaging, Journal of mathematical imaging and vision, 40 (2011), pp. 120–145

work page 2011
[6]

C HEN AND M

G. C HEN AND M. T EBOULLE , A proximal-based decomposition method for convex minimiza tion problems, Mathematical Programming, 64 (1994), pp. 81–101

work page 1994
[7]

C HEN , J

P. C HEN , J. H UANG , AND X. Z HANG , A primal–dual ﬁxed point algorithm for convex separable minimization with applications to image restoration , Inverse Problems, 29 (2013), p. 025011

work page 2013
[8]

P. L. C OMBETTES , Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal, 16 (2009), pp. 727–748

work page 2009
[9]

D ENG AND W

W. D ENG AND W. Y IN, On the global and linear convergence of the generalized alte rnating di- rection method of multipliers, tech. rep., DTIC Document, 2012. [10] , On the global and linear convergence of the generalized alte rnating direction method of multipliers, Journal of Scientiﬁc Computing, 66 (2016), pp. 889–916. 18

work page 2012
[10]

D OUGLAS AND H

J. D OUGLAS AND H. R ACHFORD , On the numerical solution of heat conduction problems in two and three space variables , Transactions of the American mathematical Society, (1956 ), pp. 421– 439

work page 1956
[11]

E CKSTEIN , Some saddle-function splitting methods for convex program ming, Optimization Methods and Software, 4 (1994), pp

J. E CKSTEIN , Some saddle-function splitting methods for convex program ming, Optimization Methods and Software, 4 (1994), pp. 75–83

work page 1994
[12]

E CKSTEIN AND D

J. E CKSTEIN AND D. P. B ERTSEKAS , On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), pp. 293– 318

work page 1992
[13]

E CKSTEIN AND W

J. E CKSTEIN AND W. YAO, Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives , Pac. J. Optim., 11 (2015), pp. 619–644

work page 2015
[14]

, Approximate ADMM algorithms derived from Lagrangian split ting, Computational Opti- mization and Applications, 68 (2017), pp. 363–405

work page 2017
[15]

, Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM, Mathematical Programming, 170 (2018), pp. 417–444

work page 2018
[16]

E SSER , X

E. E SSER , X. Z HANG , AND T. C HAN , A general framework for a class of ﬁrst order primal-dual algorithms for tv minimization , Ucla Cam Report, (2009), pp. 09–67

work page 2009
[17]

F AZEL , T

M. F AZEL , T. K. P ONG , D. S UN, AND P. T SENG , Hankel matrix rank minimization with applica- tions to system identiﬁcation and realization , SIAM Journal on Matrix Analysis and Applications, 34 (2013), pp. 946–977

work page 2013
[18]

F ORTIN AND R

M. F ORTIN AND R. G LOWINSKI , Chapter iii on decomposition-coordination methods using a n augmented lagrangian, in Studies in Mathematics and Its Applications, vol. 15, El sevier, 1983, pp. 97–146

work page 1983
[19]

G ABAY AND B

D. G ABAY AND B. M ERCIER , A dual algorithm for the solution of nonlinear variational p roblems via ﬁnite element approximation , Computers & Mathematics with Applications, 2 (1976), pp. 1 7– 40

work page 1976
[20]

G AO AND F

B. G AO AND F. M A, Symmetric alternating direction method with indeﬁnite pro ximal regulariza- tion for linearly constrained convex optimization, Journal of Optimization Theory and Applications, 176 (2018), pp. 178–204

work page 2018
[21]

G LOWINSKI AND A

R. G LOWINSKI AND A. M ARROCO , Sur l’approximation, par ´el´ements ﬁnis d’ordre un, et la r ´esolution, par p ´enalisation-dualit´e d’une classe de probl `emes de dirichlet non lin ´eaires, ESAIM: Mathematical Modelling and Numerical Analysis-Mod ´ elisation Math´ ematique et Anal- yse Num´ erique, 9 (1975), pp. 41–76

work page 1975
[22]

M. L. N. G ONC¸ ALVES , M. M. A LVES, AND J. G. M ELO , Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, Journal of Optimization Theory and Applications, 177 (2018), pp. 448–478. 19

work page 2018
[23]

Y. G U, B. J IANG , AND D. H AN, An indeﬁnite-proximal-based strictly contractive Peacem an- Rachford splitting method, tech. rep., 2019

work page 2019
[24]

An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

Y. G U AND N. YAMASHITA , An alternating direction method of multipliers with the BFG S update for structured convex quadratic optimization, arXiv e-prints arXiv:1903.02270, (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1903
[25]

G U AND N

Y. G U AND N. Y AMASHITA , A proximal ADMM with the Broyden family for convex optimizat ion problems, Avaliable on http://www. optimization-online. org, (201 9)

work page
[26]

H E, L.-Z

B. H E, L.-Z. L IAO, D. H AN, AND H. Y ANG , A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming, 92 (2002), pp. 103–118

work page 2002
[27]

B. H E, H. L IU, Z. W ANG , AND X. Y UAN, A strictly contractive Peaceman–Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), pp. 1011–1040

work page 2014
[28]

B. H E, F. M A, AND X. Y UAN, Optimal linearized alternating direction method of multip liers for convex programming, Avaliable on http://www. optimization-online. org, (201 7)

work page
[29]

H E AND X

B. H E AND X. Y UAN, On the O(1/n) convergence rate of the Douglas-Rachford alternating di- rection method, SIAM Journal on Numerical Analysis, 50 (2012), pp. 700–709

work page 2012
[30]

1 45–174

, Block-wise alternating direction method of multipliers fo r multiple-block convex program- ming and beyond, SMAI Journal of Computational Mathematics, 1 (2015), pp. 1 45–174

work page 2015
[31]

J IANG , Z

F. J IANG , Z. W U, AND X. C AI, Generalized admm with optimal indeﬁnite proximal term for l in- early constrained convex optimization, Journal of Industrial & Management Optimization, (2018), pp. 183–202

work page 2018
[32]

K OH, S.-J

K. K OH, S.-J. K IM, AND S. B OYD, An interior-point method for large-scale l1-regularized logistic regression, Journal of Machine learning research, 8 (2007), pp. 1519–1 555

work page 2007
[33]

M. L I, D. S UN, AND K.-C. T OH, A majorized admm with indeﬁnite proximal terms for linearly constrained convex composite optimization , SIAM Journal on Optimization, 26 (2016), pp. 922– 950

work page 2016
[34]

L IONS AND B

P.-L. L IONS AND B. M ERCIER , Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), pp. 964–979

work page 1979
[35]

P. A. L OTITO , L. A. P ARENTE , AND M. S OLODOV , A class of variable metric decomposition methods for monotone variational inclusions , J. Convex Anal, 16 (2009), pp. 857–880

work page 2009
[36]

N ESTEROV , Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), pp

Y. N ESTEROV , Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), pp. 125–161

work page 2013
[37]

L. I. R UDIN , S. O SHER , AND E. F ATEMI , Nonlinear total variation based noise removal algo- rithms, Physica D: nonlinear phenomena, 60 (1992), pp. 259–268

work page 1992
[38]

T REVOR , T

H. T REVOR , T. R OBERT , AND F. JH, The elements of statistical learning: data mining, inferen ce, and prediction, 2009. 20

work page 2009
[39]

T SENG , Approximation accuracy, gradient methods, and error bound for structured convex op- timization, Mathematical Programming, 125 (2010), pp

P. T SENG , Approximation accuracy, gradient methods, and error bound for structured convex op- timization, Mathematical Programming, 125 (2010), pp. 263–295

work page 2010
[40]

T SENG AND S

P. T SENG AND S. Y UN, A coordinate gradient descent method for nonsmooth separable minimiza- tion, Mathematical Programming, 117 (2009), pp. 387–423

work page 2009
[41]

X U AND T

M. X U AND T. W U, A class of linearized proximal alternating direction metho ds, Journal of Opti- mization Theory and Applications, 151 (2011), pp. 321–337

work page 2011
[42]

W. Y IN, S. O SHER , D. G OLDFARB , AND J. D ARBON , Bregman iterative algorithms for l1- minimization with applications to compressed sensing , SIAM Journal on Imaging sciences, 1 (2008), pp. 143–168

work page 2008
[43]

Y UAN, The improvement with relative errors of he et al

X. Y UAN, The improvement with relative errors of he et al. ’s inexact a lternating direction method for monotone variational inequalities, Mathematical and computer modelling, 42 (2005), pp. 1225– 1236. 21

work page 2005

[1] [1]

A TTOUCH , L

H. A TTOUCH , L. M. B RICENO -A RIAS , AND P. L. C OMBETTES , A parallel splitting method for coupled monotone inclusions , SIAM Journal on Control and Optimization, 48 (2010), pp. 32 46– 3270

work page 2010

[2] [2]

B ANERT , R

S. B ANERT , R. I. B OT, AND E. R. C SETNEK , Fixing and extending some recent results on the ADMM algorithm, arXiv preprint arXiv:1612.05057, (2016)

work page arXiv 2016

[3] [3]

H. H. B AUSCHKE , P. L. C OMBETTES , ET AL ., Convex analysis and monotone operator theory in Hilbert spaces, vol. 408, Springer, 2011

work page 2011

[4] [4]

B OYD , N

S. B OYD , N. P ARIKH , E. C HU, B. P ELEATO , AND J. E CKSTEIN , Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends R⃝ in Machine Learning, 3 (2011), pp. 1–122

work page 2011

[5] [5]

C HAMBOLLE AND T

A. C HAMBOLLE AND T. P OCK , A ﬁrst-order primal-dual algorithm for convex problems wit h applications to imaging, Journal of mathematical imaging and vision, 40 (2011), pp. 120–145

work page 2011

[6] [6]

C HEN AND M

G. C HEN AND M. T EBOULLE , A proximal-based decomposition method for convex minimiza tion problems, Mathematical Programming, 64 (1994), pp. 81–101

work page 1994

[7] [7]

C HEN , J

P. C HEN , J. H UANG , AND X. Z HANG , A primal–dual ﬁxed point algorithm for convex separable minimization with applications to image restoration , Inverse Problems, 29 (2013), p. 025011

work page 2013

[8] [8]

P. L. C OMBETTES , Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal, 16 (2009), pp. 727–748

work page 2009

[9] [9]

D ENG AND W

W. D ENG AND W. Y IN, On the global and linear convergence of the generalized alte rnating di- rection method of multipliers, tech. rep., DTIC Document, 2012. [10] , On the global and linear convergence of the generalized alte rnating direction method of multipliers, Journal of Scientiﬁc Computing, 66 (2016), pp. 889–916. 18

work page 2012

[10] [10]

D OUGLAS AND H

J. D OUGLAS AND H. R ACHFORD , On the numerical solution of heat conduction problems in two and three space variables , Transactions of the American mathematical Society, (1956 ), pp. 421– 439

work page 1956

[11] [11]

E CKSTEIN , Some saddle-function splitting methods for convex program ming, Optimization Methods and Software, 4 (1994), pp

J. E CKSTEIN , Some saddle-function splitting methods for convex program ming, Optimization Methods and Software, 4 (1994), pp. 75–83

work page 1994

[12] [12]

E CKSTEIN AND D

J. E CKSTEIN AND D. P. B ERTSEKAS , On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), pp. 293– 318

work page 1992

[13] [13]

E CKSTEIN AND W

J. E CKSTEIN AND W. YAO, Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives , Pac. J. Optim., 11 (2015), pp. 619–644

work page 2015

[14] [14]

, Approximate ADMM algorithms derived from Lagrangian split ting, Computational Opti- mization and Applications, 68 (2017), pp. 363–405

work page 2017

[15] [15]

, Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM, Mathematical Programming, 170 (2018), pp. 417–444

work page 2018

[16] [16]

E SSER , X

E. E SSER , X. Z HANG , AND T. C HAN , A general framework for a class of ﬁrst order primal-dual algorithms for tv minimization , Ucla Cam Report, (2009), pp. 09–67

work page 2009

[17] [17]

F AZEL , T

M. F AZEL , T. K. P ONG , D. S UN, AND P. T SENG , Hankel matrix rank minimization with applica- tions to system identiﬁcation and realization , SIAM Journal on Matrix Analysis and Applications, 34 (2013), pp. 946–977

work page 2013

[18] [18]

F ORTIN AND R

M. F ORTIN AND R. G LOWINSKI , Chapter iii on decomposition-coordination methods using a n augmented lagrangian, in Studies in Mathematics and Its Applications, vol. 15, El sevier, 1983, pp. 97–146

work page 1983

[19] [19]

G ABAY AND B

D. G ABAY AND B. M ERCIER , A dual algorithm for the solution of nonlinear variational p roblems via ﬁnite element approximation , Computers & Mathematics with Applications, 2 (1976), pp. 1 7– 40

work page 1976

[20] [20]

G AO AND F

B. G AO AND F. M A, Symmetric alternating direction method with indeﬁnite pro ximal regulariza- tion for linearly constrained convex optimization, Journal of Optimization Theory and Applications, 176 (2018), pp. 178–204

work page 2018

[21] [21]

G LOWINSKI AND A

R. G LOWINSKI AND A. M ARROCO , Sur l’approximation, par ´el´ements ﬁnis d’ordre un, et la r ´esolution, par p ´enalisation-dualit´e d’une classe de probl `emes de dirichlet non lin ´eaires, ESAIM: Mathematical Modelling and Numerical Analysis-Mod ´ elisation Math´ ematique et Anal- yse Num´ erique, 9 (1975), pp. 41–76

work page 1975

[22] [22]

M. L. N. G ONC¸ ALVES , M. M. A LVES, AND J. G. M ELO , Pointwise and ergodic convergence rates of a variable metric proximal alternating direction method of multipliers, Journal of Optimization Theory and Applications, 177 (2018), pp. 448–478. 19

work page 2018

[23] [23]

Y. G U, B. J IANG , AND D. H AN, An indeﬁnite-proximal-based strictly contractive Peacem an- Rachford splitting method, tech. rep., 2019

work page 2019

[24] [24]

An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

Y. G U AND N. YAMASHITA , An alternating direction method of multipliers with the BFG S update for structured convex quadratic optimization, arXiv e-prints arXiv:1903.02270, (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1903

[25] [25]

G U AND N

Y. G U AND N. Y AMASHITA , A proximal ADMM with the Broyden family for convex optimizat ion problems, Avaliable on http://www. optimization-online. org, (201 9)

work page

[26] [26]

H E, L.-Z

B. H E, L.-Z. L IAO, D. H AN, AND H. Y ANG , A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming, 92 (2002), pp. 103–118

work page 2002

[27] [27]

B. H E, H. L IU, Z. W ANG , AND X. Y UAN, A strictly contractive Peaceman–Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), pp. 1011–1040

work page 2014

[28] [28]

B. H E, F. M A, AND X. Y UAN, Optimal linearized alternating direction method of multip liers for convex programming, Avaliable on http://www. optimization-online. org, (201 7)

work page

[29] [29]

H E AND X

B. H E AND X. Y UAN, On the O(1/n) convergence rate of the Douglas-Rachford alternating di- rection method, SIAM Journal on Numerical Analysis, 50 (2012), pp. 700–709

work page 2012

[30] [30]

1 45–174

, Block-wise alternating direction method of multipliers fo r multiple-block convex program- ming and beyond, SMAI Journal of Computational Mathematics, 1 (2015), pp. 1 45–174

work page 2015

[31] [31]

J IANG , Z

F. J IANG , Z. W U, AND X. C AI, Generalized admm with optimal indeﬁnite proximal term for l in- early constrained convex optimization, Journal of Industrial & Management Optimization, (2018), pp. 183–202

work page 2018

[32] [32]

K OH, S.-J

K. K OH, S.-J. K IM, AND S. B OYD, An interior-point method for large-scale l1-regularized logistic regression, Journal of Machine learning research, 8 (2007), pp. 1519–1 555

work page 2007

[33] [33]

M. L I, D. S UN, AND K.-C. T OH, A majorized admm with indeﬁnite proximal terms for linearly constrained convex composite optimization , SIAM Journal on Optimization, 26 (2016), pp. 922– 950

work page 2016

[34] [34]

L IONS AND B

P.-L. L IONS AND B. M ERCIER , Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), pp. 964–979

work page 1979

[35] [35]

P. A. L OTITO , L. A. P ARENTE , AND M. S OLODOV , A class of variable metric decomposition methods for monotone variational inclusions , J. Convex Anal, 16 (2009), pp. 857–880

work page 2009

[36] [36]

N ESTEROV , Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), pp

Y. N ESTEROV , Gradient methods for minimizing composite functions, Mathematical Programming, 140 (2013), pp. 125–161

work page 2013

[37] [37]

L. I. R UDIN , S. O SHER , AND E. F ATEMI , Nonlinear total variation based noise removal algo- rithms, Physica D: nonlinear phenomena, 60 (1992), pp. 259–268

work page 1992

[38] [38]

T REVOR , T

H. T REVOR , T. R OBERT , AND F. JH, The elements of statistical learning: data mining, inferen ce, and prediction, 2009. 20

work page 2009

[39] [39]

T SENG , Approximation accuracy, gradient methods, and error bound for structured convex op- timization, Mathematical Programming, 125 (2010), pp

P. T SENG , Approximation accuracy, gradient methods, and error bound for structured convex op- timization, Mathematical Programming, 125 (2010), pp. 263–295

work page 2010

[40] [40]

T SENG AND S

P. T SENG AND S. Y UN, A coordinate gradient descent method for nonsmooth separable minimiza- tion, Mathematical Programming, 117 (2009), pp. 387–423

work page 2009

[41] [41]

X U AND T

M. X U AND T. W U, A class of linearized proximal alternating direction metho ds, Journal of Opti- mization Theory and Applications, 151 (2011), pp. 321–337

work page 2011

[42] [42]

W. Y IN, S. O SHER , D. G OLDFARB , AND J. D ARBON , Bregman iterative algorithms for l1- minimization with applications to compressed sensing , SIAM Journal on Imaging sciences, 1 (2008), pp. 143–168

work page 2008

[43] [43]

Y UAN, The improvement with relative errors of he et al

X. Y UAN, The improvement with relative errors of he et al. ’s inexact a lternating direction method for monotone variational inequalities, Mathematical and computer modelling, 42 (2005), pp. 1225– 1236. 21

work page 2005