Minimizing Certain Convex Functions

Warga, J.

doi:10.1137/0111043

Cited by 87 publications

(45 citation statements)

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“…Note that, by Lemma 5.6 (or simply from (38) and (40) , i = 1, 2. By taking the limits for n → ∞ in (37)- (40) we obtain…”

Section: ) If and Only If There Exists Anmentioning

confidence: 99%

“…In particular, we stress very clearly that well-known approaches as in [9,14,35,36] are not directly applicable to this problem, because either they address additive problems or smooth convex minimizations, which is not the case of total variation minimization. We emphasize that the successful convergence of such alternating algorithms is far from being obvious for nonsmooth and nonadditive problems, as many counterexamples can be constructed, see for instance [38]. Moreover, for total variation minimization, the interesting solutions may be discontinuous, e.g., along curves in 2D.…”

Section: Difficulty Of the Problemmentioning

confidence: 99%

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A convergent overlapping domain decomposition method for total variation minimization

2010

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In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles. Mathematics Subject Classification (2000)

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“…Note that, by Lemma 5.6 (or simply from (38) and (40) , i = 1, 2. By taking the limits for n → ∞ in (37)- (40) we obtain…”

Section: ) If and Only If There Exists Anmentioning

confidence: 99%

Section: Difficulty Of the Problemmentioning

confidence: 99%

A convergent overlapping domain decomposition method for total variation minimization

2010

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“…Associated with distributed solutions approaches arising from these two motivations are the Gauss-Seidel approaches and the Jacobi approaches. Gauss-Seidel decomposition/coordination techniques such as the alternating direction method of multipliers (ADMM) [29,24,4,17,36,9] and the block coordinate descent (BCD) method [58,45,6,31,5,4,30,56] perform one subproblem computation at a time based on the knowledge of the most up-to-date information passed from the other subproblems. The potential for parallelization of the solution computation (treated distinctly from distribution) depends on the nature of coupling among the subproblems.…”

Section: Distributed Versus Parallel Optimizationmentioning

confidence: 99%

Distributed Computation of Pareto Sets

Dandurand¹,

Wiecek²

2015

SIAM J. Optim.

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The needs of multidisciplinary engineering design have motivated the development of distributed solution approaches to computing efficient solutions to decomposable multiobjective optimization problems (MOPs). The decomposition is necessary due to the assumption that the overall MOP is not solvable since access to its solution space is subproblem-restricted. The state-of-the-art analyses for distributed solution approaches such as the alternating direction method of multipliers (ADMM) and the block coordinate descent (BCD) method were developed in the context of problem decomposition originating in a single objective setting and are not immediately applicable to MOPs. Applying certain scalarization techniques well-suited for nonconvex MOPs, the decomposable MOP is reformulated into a single objective problem (SOP) but the decomposability is not preserved and the SOP is not suitable for the application of ADMM. Furthermore, coupling between the subproblems makes BCD in its current form likewise inadequate. To address these challenges to distributed multiobjective optimization, existing theory is extended for (1) iterative augmented Lagrangian coordination techniques and (2) the block coordinate descent method. For the former, each augmented Lagrangian subproblem is subject to convex constraints and is solved approximately for solutions satisfying a necessary condition for optimality (but not necessarily optimal). For the latter, existing convergence proofs under generalized convexity assumptions are redeveloped so that new insights and convergence results become evident. An integration of the above two convergence analyses yields convergence results for a distributed solution approach that addresses coupling between subproblems and does not require convergence of the generated sequence of solution-multiplier pairs. Based on this study, a multi-objective decomposition algorithm is developed for the distributed generation of efficient solutions to nonconvex decomposable MOPs.

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“…This algorithm has a long history in applied mathematics and has its roots in the Gauss-Siedel method for solving linear systems (Warge (1963); Ortega and Rheinbold (1970); Tseng (2001)). The CDA optimizes an objective function by working on one coordinate (or a block of coordinates) at a time, iteratively cycling through all coordinates until convergence is reached.…”

Section: Introductionmentioning

confidence: 99%

Majorization minimization by coordinate descent for concave penalized generalized linear models

Jiang

Huang

2013

Stat Comput

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Recent studies have demonstrated theoretical attractiveness of a class of concave penalties in variable selection, including the smoothly clipped absolute deviation and minimax concave penalties. The computation of the concave penalized solutions in high-dimensional models, however, is a difficult task. We propose a majorization minimization by coordinate descent (MMCD) algorithm for computing the concave penalized solutions in generalized linear models. In contrast to the existing algorithms that use local quadratic or local linear approximation to the penalty function, the MMCD seeks to majorize the negative log-likelihood by a quadratic loss, but does not use any approximation to the penalty. This strategy makes it possible to avoid the computation of a scaling factor in each update of the solutions, which improves the efficiency of coordinate descent. Under certain regularity conditions, we establish theoretical convergence property of the MMCD. We implement this algorithm for a penalized logistic regression model using the SCAD and MCP penalties. Simulation studies and a data example demonstrate that the MMCD works sufficiently fast for the penalized logistic regression in high-dimensional settings where the number of covariates is much larger than the sample size.

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Minimizing Certain Convex Functions

Cited by 87 publications

References 0 publications

A convergent overlapping domain decomposition method for total variation minimization

A convergent overlapping domain decomposition method for total variation minimization

Distributed Computation of Pareto Sets

Majorization minimization by coordinate descent for concave penalized generalized linear models

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