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Convex Analysisの二,三の進展について

In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the $\ell_1$-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an $\ell_1$-problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several state-of-the-art algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the $\ell_1$-norm fidelity should be the fidelity of choice in compressive sensing.

Alternating Direction Algorithms for $\ell_1$-Problems in Compressive Sensing

Quantification of non-Gaussianity for water diffusion in brain by means of diffusional kurtosis imaging (DKI) is reviewed. Diffusional non-Gaussianity is a consequence of tissue structure that creates diffusion barriers and compartments. The degree of non-Gaussianity is conveniently quantified by the diffusional kurtosis and derivative metrics, such as the mean, axial, and radial kurtoses. DKI is a diffusion-weighted MRI technique that allows the diffusional kurtosis to be estimated with clinical scanners using standard diffusion-weighted pulse sequences and relatively modest acquisition times. DKI is an extension of the widely used diffusion tensor imaging method, but requires the use of at least 3 b-values and 15 diffusion directions. This review discusses the underlying theory of DKI as well as practical considerations related to data acquisition and post-processing. It is argued that the diffusional kurtosis is sensitive to diffusional heterogeneity and suggested that DKI may be useful for investigating ischemic stroke and neuropathologies, such as Alzheimer’s disease and schizophrenia.

MRI quantification of non-Gaussian water diffusion by kurtosis analysis.

We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang.

Generalized approximate message passing for estimation with random linear mixing

Convex Analysis: (pms-28)

The alternating directions method (ADM) is an effective method for solving a class of variational inequalities (VI) when the proximal and penalty parameters in sub-VI problems are properly selected. In this paper, we propose a new ADM method which needs to solve two strongly monotone sub-VI problems in each iteration approximately and allows the parameters to vary from iteration to iteration. The convergence of the proposed ADM method is proved under quite mild assumptions and flexible parameter conditions.

A new inexact alternating directions method for monotone variational inequalities

We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.

A Note on the Alternating Direction Method of Multipliers

In this paper, we aim to prove the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a mild calmness condition, which holds automatically for convex composite piecewise linear-quadratic programming, we establish the global Q-linear rate of convergence for a general semi-proximal ADMM with the dual step-length being taken in (0, (1+51/2)/2). This semi-proximal ADMM, which covers the classic one, has the advantage to resolve the potentially nonsolvability issue of the subproblems in the classic ADMM and possesses the abilities of handling the multi-block cases efficiently. We demonstrate the usefulness of the obtained results when applied to two- and multi-block convex quadratic (semidefinite) programming.

Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming

As the provision of roads has become market driven, many intriguing issues have emerged, such as the strategic interactions among private firms in determining road supply and pricing in a network and, more importantly, the resulting inefficiencies when competition substitutes for government regulation. This paper studies both toll and capacity competition among private asymmetric roads with congestion in a network with parallel links. We find that oligopolistic competition yields higher tolls but lower construction capacities than the socially optimal levels and, as a result, the level of traffic congestion does not decline. Based on some widely used assumptions, we show that the usage or the congestion level of each road controlled by each private firm, represented by the volume–capacity (v/c) ratio, is independent of the other competitors’ choices of capacities and tolls in the oligopolistic market. To quantify the inefficiency of the oligopolistic equilibria via higher tolls and less realized traffic demand, we establish the upper bound of the ratio of the social welfare between the competitive equilibrium and the social optimum. The inefficiency bound depends mainly on the marginal benefit of trip-makers and the realized traffic demand at equilibrium on the private roads. Exact bounds are given by considering both the worst and symmetric cases. Naturally, our results show that the distortion disappears when the oligopoly changes to perfect competition.

Competition and efficiency of private toll roads

The Douglas--Rachford alternating direction method of multipliers (ADMM) has been widely used in various areas. The global convergence of ADMM is well known, while research on its convergence rate ...

Deren Han

Papers

A new inexact alternating directions method for monotone variational inequalities

A Note on the Alternating Direction Method of Multipliers

Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming

Competition and efficiency of private toll roads

Local Linear Convergence of the Alternating Direction Method of Multipliers for Quadratic Programs