Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems

Artacho, Francisco J. Aragón; Borwein, Jonathan M.; Tam, Matthew K.

doi:10.1007/s10957-013-0488-0

Cited by 52 publications

(26 citation statements)

References 22 publications

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“…The rate of convergence has recently been shown to be linear when A and B are affine subspaces with nonempty intersection [19,Theorem 4.6] and R-linear when A and B are general convex sets whose relative interiors intersect [29,Theorem 4.14]. The Douglas-Rachford algorithm has proven to be extremely effective empirically for many types of problems, see, for example, [1,4,16]. The Douglas-Rachford algorithm has been applied to nonconvex problems (even though the convergence guarantee is not known) in physics applications; see, e.g., [6].…”

Section: General Background On Projection Methodsmentioning

confidence: 99%

“…the number of steps of DR that it took to find the max-rank P; 2. the minimum/maximum/mean number of iterations for the steps in finding P; 1 3. the maximum of the cosine of the angles between three successive iterates; 2 4. the value of the maximum rank found. 3…”

Section: Heuristic For Finding Max-rank Feasible Solutions Using Dr Amentioning

confidence: 99%

“…Let 1 ≤ i, j,ĩ,j ≤ nm with In other words, we order the 2-tuples (i, j) in I by grouping all those with the same intra-block index ( p, q) and block indices s < t together, for some p < q, followed by those tuples with intra-block index (q, p) and block indices s < t. As an example, when m = 2 and n = 3, the following list gives the first few entries of I: (1,4), (1,6), (3,6), (2,3), (2,5), (4,5), (1,2), (3,4), . .…”

Section: Choice Of Orthonormal Basis Of H Smentioning

confidence: 99%

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Projection methods for quantum channel construction

Pelejo

Voronin

et al. 2015

Quantum Inf Process

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We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states {ρ 1 , . . . , ρ k } to another such set {ρ 1 , . . . ,ρ k }. In other words, we must find a completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices, possibly of different dimension. Using the theory of completely positive linear maps, one can formulate the problem as an instance of a positive semidefinite feasibility problem with highly structured constraints. The nature of the constraints makes projection-based algorithms very appealing when the number of variables is huge and standard interior-point methods for semidefinite programming are not applicable. We provide empirical evidence to this effect. We moreover present heuristics for finding both high-rank and low-rank solutions. Our experiments are based on the method of alternating projections and the Douglas-Rachford reflection method.

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Section: General Background On Projection Methodsmentioning

confidence: 99%

Section: Heuristic For Finding Max-rank Feasible Solutions Using Dr Amentioning

confidence: 99%

Section: Choice Of Orthonormal Basis Of H Smentioning

confidence: 99%

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Projection methods for quantum channel construction

Pelejo

Voronin

et al. 2015

Quantum Inf Process

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“…Examples include combinatorial optimization [3,1,13], low-rank matrix reconstruction [4], boolean satisfiability [16], sphere packing [16,13], matrix completion [1], image reconstruction [10], and road design [8]. Intriguingly, success in such settings is not uniform [3], and thus the theory is in need of significant enhancement.…”

Section: Introductionmentioning

confidence: 99%

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

2015

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In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a wellquasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

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“…Each iteration of these methods, employes some combination of (nearest point) projections onto the constraint sets. Their sustained popularity, even in settings without convexity, is due to their relative simplicity and ease-of-implementation, in addition to observed good performance [21,23,1,2].…”

Section: Introductionmentioning

confidence: 99%

Norm convergence of realistic projection and reflection methods

2014

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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. 1 converge in norm. A number of variants and extensions to this example have since been published [31,27,8], some of which cover the case of non-intersecting sets (infeasible problems). These examples consider two sets, the first being either a closed subspace of finite codimension or one of its half-spaces, and the second a convex cone "built-up" from three dimensional "building blocks". For non-convex sets, the question of convergence is more difficult, and currently result focus on the finite dimensional setting [28,12,24,25,13].In light of these examples, it is natural ask what compatibility conditions on the two sets are required to ensure norm convergence. This is further motivated by the pleasing physical interpretation of norm convergence as the "error" becoming arbitrarily small [10].When the constraint sets satisfies certain regularity properties, norm convergence of the method of alternating projections can be guaranteed [6,7], and in some cases a linear rate of converge can also be assured. These regularity conditions are most easily invoked in the analysis of the method of alternating projections. This is because each iteration of the method produces a point contained within one of the two constraint sets for which the regularity properties can be invoked. On the other hand, the Douglas-Rachford method generates points that need not lie within the sets, making it more difficult to analyze. Consequently less is known of its behaviour. Further, to the authors' knowledge no explicit Hundal-like counter-example is known for the Douglas-Rachford algorithm. For recent progress, on convex Douglas-Rachford methods see [25,5].An important practicable instance of the feasibility problem occurs when the space is a Hilbert lattice, one of the sets is the positive Hilbert cone, and the other is a closed affine subspace with finite codimension. Problems of this kind arise, for example, in the so called 'moment problem' (see [6]). Applied to this type of feasibility problem, Bauschke and Borwein proved that the method of alternating projection converges in norm whenever the affine subspace has codimension one [6]. The same was conjectured to stay true for any finite codimension, but remains a stubbornly open problem.The goal of this paper is two-fold. First, to formulate unified sufficient conditions for norm convergence of fundamental projection and reflection methods when applied to feasibility problems with finite codimensional affine space and convex cone constraints, and second, to give examples and counter-examples regarding the convergence rate of these methods.The remainder of the paper is organized as followed: in Section 2 we recall definitions and important theory for our analysis; in Section 3 we formulate sufficient conditions for norm convergence, which we then specialize to Hilbert cones. Finally, in Section 4 we give various examples and...

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Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems

Cited by 52 publications

References 22 publications

Projection methods for quantum channel construction

Projection methods for quantum channel construction

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

Norm convergence of realistic projection and reflection methods

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