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
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