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內容簡介

　　《錐優化的基於核函數的內點算法》共分七章，第1章介紹錐優化理論方法的發展曆程，第2章介紹核函數及其性質、由核函數確定的障礙函數的性質，第3～6章分彆介紹中心路徑的概念、錐的代數性質，給齣求解綫性規劃問題、P*(k)綫性互補問題、半正半優化問題、二階錐優化問題的基於核函數的內點算法，分析瞭大步算法、小步內點算法的計算復雜性。
　　《錐優化的基於核函數的內點算法》可作為運籌學專業的本科生、研究生關於內點算法的入門書，同時也可作為研究人員的關於內點算法的參考書。

內頁插圖

目錄

Preface
Chapter 1 Introduction
1.1 Conic optimization problems
1.2 Conic duality
1.3 From the dual cone to the dual problem
1.4 Development of the interior-point methods
1.5 Scope of the book
Chapter 2 Kernel Functions
2.1 Definition of kernel functions and basic properties
2.2 The further conditions of kernel functions
2.3 Properties of kernel functions
2.4 Examples of kernel functions
2.5 Barrier functions based on kernel functions
2.6 Generalization of kernel function
2.6.1 Finite kernel function
2.6.2 Parametric kernel function
Chapter 3 Kernel Function-based Interior-point Algorithm for LO
3.1 The central path for LO
3.2 The search directions for LO
3.3 The generic primal-dual interior-point algorithm for LO
3.4 Analysis of the algorithm
3.4.1 Decrease of the barrier function during an inner iteration
3.4.2 Choice of the step size
3.5 Iteration bounds
3.6 Summary of computation for complexity bound
3.7 Complexity analysis based on kernel functions
3.8 Summary of results
Chapter 4 Kernel Function-based Interior-point Algorithm for P*(k) LCP
4.1 The P*(k)-LCP
4.2 The central path for P*(k)-LCP
4.3 The new search directions for P*(k)-LCP
4.4 The generic primal-dual interior-point algorithm for P*(k)-LCP...
4.5 The properties of the barrier function
4.6 Analysis of the algorithm
4.6.1 Growth behavior of the barrier function
4.6.2 Determining the default step size
4.7 Decrease of the barrier function during an inner iteration
4.8 Complexity of the algorithm
4.8.1 Iteration bound for the large-update methods
4.8.2 Iteration bound for the small-update methods
Chapter 5 Kernel Function-based Interior-point Algorithm for SDO
5.1 Special matrix functions
5.2 The central path for SDO
5.3 The new search directions for SDO
5.4 The generic primal-dual interior-point algorithm for SDO
5.5 The properties of the barrier function
5.6 Analysis of the algorithm
5.6.1 Decrease of the barrier function during an inner iteration
5.6.2 Choice of the step size
5.7 Iteration bounds
5.8 Kernel function-based schemes
5.9 The example
5.10 Numerical results
Chapter 6 Kernel Function-based Interior-point Algorithm for SOCO
6.1 Algebraic properties of second-order cones
6.2 Barrier functions defined on second-order cone
6.3 Rescaling the cone
6.4 The central path for SOCO
6.5 The new search directions for SOCO
6.6 The generic primal-dual interior-point algorithm for SOCO
6.7 Analysis of the algorithm
6.8 The crucial inequality
6.9 Decrease of the barrier function during an inner iteration
6.10 Increase of the barrier function during a μ-update
6.11 Iteration-bounds
6.12 Numerical results
6.13 Some technical lemmas
Appendix Three Technical Lemmas
Reference

精彩書摘

　　A linear optimization problem is the minimization of a linear function over a polyhedral set which can be viewed as the intersection of an affine space and the coneof nonnegative orthant. Many problems can be formulated as, or approximated bya linear optimization problem. There are many versions of interior-point methodsfor linear optimization. But the basic scheme of these methods is to remove theconstraint set and add a multiple of the barrier function to the objective function.Therefore, the barrier-based scheme reduces the constrained problem into a seriesof unconstrained problems, then to "trace" the path formed by the optimal solutions of unconstrained problems. "Trace" means that the optimal solutions ofunconstrained problems can be replaced by a good enough approximation of theoptimal solutions of unconstrained problems. The procedure of the scheme canbe gone on with updating the barrier parameter until the optimal set of linearoptimization problem is reached.

前言/序言

　　錐優化問題是一個凸規劃問題，它的目標函數是綫性函數，約束集是仿射空間和一個錐的交集，它從優化問題可行域的結構推廣瞭綫性規劃問題，為求解非綫性最優化問題提供瞭一種新的框架，錐優化有凸結構和豐富的對偶理論，對偶問題具有對稱的簡潔結構，同時，又有廣泛應用背景，除瞭在傳統學科，在經濟、金融、管理和工程技術等領域亦有廣泛的應用，近年來，錐優化與新興學科有瞭廣泛交叉和應用，如在無綫傳感網絡、信息理論、編碼理論等信息學科找到瞭豐富的應用，20世紀80年代齣現的內點算法推動瞭算法計算復雜性研究的發展，也成為求解錐優化問題的強大工具，迄今為止，錐優化和內點算法已成為數學規劃和優化領域最活躍的研究課題之一。
　　本書根據作者和其閤作者Roos教授、Ghami博士、王國強博士近年來的研究工作，全麵介紹求解綫性規劃、P（K）綫性互補問題、半正定優化、二階錐優化基於核函數的內點算法，核函數的重要性體現在它有簡單的解析錶達式、容易計算的高階導數等良好性質。
錐優化的基於核函數的內點算法 下載 mobi epub pdf txt 電子書

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Chapter 1 Introduction

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3.6 Summary of computation for complexity bound

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2.6.2 Parametric kernel function

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Chapter 3 Kernel Function-based Interior-point Algorithm for LO

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2.6.1 Finite kernel function

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3.4.2 Choice of the step size

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2.3 Properties of kernel functions

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3.2 The search directions for LO

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2.4 Examples of kernel functions

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