內容簡介
錐最佳化的基於核函式的內點算法本書根據作者和其合作者Roos教授、Ghami博士、王國強博士近年來的研究工作，全面介紹求解線性規劃、P*(k)線性互補問題、半正定最佳化、二階錐最佳化基於核函式的內點算法。核函式的重要性體現在它有簡單的解析表達式、容易計算的高階導數等良好性質。由核函式確定的障礙函式繼承了這些良好性質，準確刻畫了錐最佳化問題的中心路徑，基於障礙函式設計的原始對偶內點算法，並有程式化的分析方法，使得內點算法的計算複雜性分析變得十分容易。
目錄
Preface
Chapter 1 Introduction
1.1conicoptimization 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
