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Alternating Projections and DouglasRachford for Sparse Affine Feasibility
"... The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NPcomplete problem that is typically solved numerically via convex heuristics or nicelybehaved nonconvex relaxations. In this work we consider elementary methods base ..."
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The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NPcomplete problem that is typically solved numerically via convex heuristics or nicelybehaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. It has been shown recently that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent DouglasRachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.
New DouglasRachford Algorithmic Structures and Their Convergence Analyses
, 2014
"... In this paper we study new algorithmic structures with DouglasRachford (DR) operators to solve convex feasibility problems. We propose to embed the basic twosetDR algorithmic operator into the StringAveraging Projections (SAP) and into the BlockIterative Projection (BIP) algorithmic structures ..."
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In this paper we study new algorithmic structures with DouglasRachford (DR) operators to solve convex feasibility problems. We propose to embed the basic twosetDR algorithmic operator into the StringAveraging Projections (SAP) and into the BlockIterative Projection (BIP) algorithmic structures, thereby creating new DR algorithmic schemes that include the recently proposed cyclic DouglasRachford algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiplesetDR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasinonexpansive operators and firmly nonexpansive operators. 1 1
Linear Convergence of the ADMM/Douglas Rachford Algorithms for Piecewise LinearQuadratic Functions and Application to Statistical Imaging
, 2015
"... We consider the problem of minimizing the sum of a convex, piecewise linearquadratic function and a convex piecewise linearquadratic function composed with an injective linear mapping. We show that, for such problems, iterates of the alternating directions method of multipliers converge linearly t ..."
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We consider the problem of minimizing the sum of a convex, piecewise linearquadratic function and a convex piecewise linearquadratic function composed with an injective linear mapping. We show that, for such problems, iterates of the alternating directions method of multipliers converge linearly to fixed points from which the solution to the original problem can be computed. Our proof strategy uses duality and strong metric subregularity of the DouglasRachford fixed point mapping. Our analysis does not require strong convexity and yields error bounds to the set of model solutions. We demonstrate an application of this result to exact penalization for signal deconvolution and denoising with multiresolution statistical constraints.
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, 2013
"... In this paper we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrixcompletion problems. These guidelines are ..."
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In this paper we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrixcompletion problems. These guidelines are
Norm Convergence of Realistic Projection and Reflection Methods
, 2014
"... We provide sufficient conditions for norm convergence of various projection and reflection methods, as well as giving limiting examples regarding convergence rates. ..."
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We provide sufficient conditions for norm convergence of various projection and reflection methods, as well as giving limiting examples regarding convergence rates.