다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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Convex Analysisの二,三の進展について

The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.

http://www.optimization-online.org/DB_FILE/2008/01/1889.pdf

Probing the Pareto Frontier for Basis Pursuit Solutions

Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (lscr 2) error term added to a sparsity-inducing (usually lscr1) regularizater. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (i.e., separable in the unknowns) plus the original sparsity-inducing regularizer; our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. Under mild conditions (namely convexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard lscr2-lscr1 case, our framework yields efficient solution techniques for other regularizers, such as an lscrinfin norm and group-separable regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard lscr2-lscr1 problem, as well as being efficient on problems with other separable regularization terms.

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Sparse Reconstruction by Separable Approximation

Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: 1) multi-modality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; 2) longitudinal studies, where temporal structural or anatomical changes are investigated; and 3) population modeling and statistical atlases used to study normal anatomical variability. In this paper, we attempt to give an overview of deformable registration methods, putting emphasis on the most recent advances in the domain. Additional emphasis has been given to techniques applied to medical images. In order to study image registration methods in depth, their main components are identified and studied independently. The most recent techniques are presented in a systematic fashion. The contribution of this paper is to provide an extensive account of registration techniques in a systematic manner.

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Deformable Medical Image Registration: A Survey

A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed With exact line search, our method reduces to a nonlinear version of the Hestenes--Stiefel conjugate gradient scheme For any (inexact) line search, our scheme satisfies the descent condition ${\bf g}_k^{\sf T} {\bf d}_k \le -\frac{7}{8} \|{\bf g}_k\|^2$ Moreover, a global convergence result is established when the line search fulfills the Wolfe conditions A new line search scheme is developed that is efficient and highly accurate Efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer High accuracy is achieved by using a convergence criterion, which we call the ``approximate Wolfe'' conditions, obtained by replacing the sufficient decrease criterion in the Wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion Numerical comparisons are given with both L-BFGS and conjugate gradient methods using the unconstrained optimization problems in the CUTE library

A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search

This paper reviews the development of dierent versions of nonlinear conjugate gradient methods, with special attention given to global convergence properties.

/pdf/a-survey-of-nonlinear-conjugate-gradient-methods-115o6h95fm.pdf

A survey of nonlinear conjugate gradient methods

A new nonmonotone line search algorithm is proposed and analyzed. In our scheme, we require that an average of the successive function values decreases, while the traditional nonmonotone approach of Grippo, Lampariello, and Lucidi [SIAM J. Numer. Anal., 23 (1986), pp. 707--716] requires that a maximum of recent function values decreases. We prove global convergence for nonconvex, smooth functions, and R-linear convergence for strongly convex functions. For the L-BFGS method and the unconstrained optimization problems in the CUTE library, the new nonmonotone line search algorithm used fewer function and gradient evaluations, on average, than either the monotone or the traditional nonmonotone scheme.

/pdf/a-nonmonotone-line-search-technique-and-its-application-to-hcnywt45sq.pdf

A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization

This paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a certain non-differentiable (but convex) component. In order to solve these problems, we propose a randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are taken at each iteration depending on the total budget of stochastic samples allowed. The RSPG algorithm also employs a general distance function to allow taking advantage of the geometry of the feasible region. Complexity of this algorithm is established in a unified setting, which shows nearly optimal complexity of the algorithm for convex stochastic programming. A post-optimization phase is also proposed to significantly reduce the variance of the solutions returned by the algorithm. In addition, based on the RSPG algorithm, a stochastic gradient free algorithm, which only uses the stochastic zeroth-order information, has been also discussed. Some preliminary numerical results are also provided.

/pdf/mini-batch-stochastic-approximation-methods-for-nonconvex-5z5m9rgl67.pdf

Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization

Recently, a new nonlinear conjugate gradient scheme was developed which satisfies the descent condition gTkdk ≤ −7/8 VgkV2 and which is globally convergent whenever the line search fulfills the Wolfe conditions. This article studies the convergence behavior of the algorithm; extensive numerical tests and comparisons with other methods for large-scale unconstrained optimization are given.

/pdf/algorithm-851-cg-descent-a-conjugate-gradient-method-with-13a39qnkfc.pdf

Hongchao Zhang

Papers

A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search

A survey of nonlinear conjugate gradient methods

A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization

Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization

Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent