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

Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparseness-inducing regularization term. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

/pdf/gradient-projection-for-sparse-reconstruction-application-to-jxqvrma0ll.pdf

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Conjugate gradient methods are widely used for unconstrained optimization, especially large scale problems. The strong Wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods. This paper presents a new version of the conjugate gradient method, which converges globally, provided the line search satisfies the standard Wolfe conditions. The conditions on the objective function are also weak, being similar to those required by the Zoutendijk condition.

A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property

Conjugate gradient methods are a class of important methods for unconstrained optimization, especially when the dimension is large. This paper proposes a new conjugacy condition, which considers an inexact line search scheme but reduces to the old one if the line search is exact. Based on the new conjugacy condition, two nonlinear conjugate gradient methods are constructed. Convergence analysis for the two methods is provided. Our numerical results show that one of the methods is very efficient for the given test problems.

New Conjugacy Conditions and Related Nonlinear Conjugate Gradient Methods

This paper studies projected Barzilai-Borwein (PBB) methods for large-scale box-constrained quadratic programming. Recent work on this method has modified the PBB method by incorporating the Grippo-Lampariello-Lucidi (GLL) nonmonotone line search, so as to enable global convergence to be proved. We show by many numerical experiments that the performance of the PBB method deteriorates if the GLL line search is used. We have therefore considered the question of whether the unmodified method is globally convergent, which we show not to be the case, by exhibiting a counter example in which the method cycles. A new projected gradient method (PABB) is then considered that alternately uses the two Barzilai-Borwein steplengths. We also give an example in which this method may cycle, although its practical performance is seen to be superior to the PBB method. With the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods, we examine other recent work on nonmonotone line searches, and propose a new adaptive variant with some attractive features. Further numerical experiments show that the PABB method with the adaptive line search is the best BB-like method in the positive definite case, and it compares reasonably well against the GPCG algorithm of More and Toraldo. In the indefinite case, the PBB method with the adaptive line search is shown on some examples to find local minima with better solution values, and hence may be preferred for this reason.

/pdf/projected-barzilai-borwein-methods-for-large-scale-box-3maqkglmvr.pdf

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

The conjugate gradient method is a powerful solution scheme for solving unconstrained optimization problems, especially for large-scale problems. However, the convergence rate of the method without restart is only linear. In this paper, we will consider an idea contained in [16] and present a new restart technique for this method. Given an arbitrary descent direction d

 t
 and the gradient g

 t
 , our key idea is to make use of the BFGS updating formula to provide a symmetric positive definite matrix P

 t
 such that d

 t
 =−P

 t
 
g

 t
 , and then define the conjugate gradient iteration in the transformed space. Two conjugate gradient algorithms are designed based on the new restart technique. Their global convergence is proved under mild assumptions on the objective function. Numerical experiments are also reported, which show that the two algorithms are comparable to the Beale–Powell restart algorithm.

On restart procedures for the conjugate gradient method

Combined with non-monotone line search, the Barzilai and Borwein (BB) gradient method has been successfully extended for solving unconstrained optimization problems and is competitive with conjugate gradient methods. In this paper, we establish the R-linear convergence of the BB method for any-dimensional strongly convex quadratics. One corollary of this result is that the BB method is also locally R-linear convergent for general objective functions, and hence the stepsize in the BB method will always be accepted by the non-monotone line search when the iterate is close to the solution.

Yu-Hong Dai

Papers

A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property

New Conjugacy Conditions and Related Nonlinear Conjugate Gradient Methods

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

On restart procedures for the conjugate gradient method

R-linear convergence of the Barzilai and Borwein gradient method