凸优化应用讲义

1 Preliminary Knowledge

1.1 Nomenclatures

1.2 Convex Sets and Convex Functions

1.3 Convex Optimization

1.3.1 Gradient Descent and Coordinate Descent

1.3.2 Karush-Kuhn-Tucker (KKT) Conditions

1.4 Some Lemmas in Linear Algebra

1.5 A Brief Introduction of CVX Toolbox

Problems

References

2 Support Vector Machines

2.1 Basic SVM

2.2 Soft Margin SVM

2.3 Kernel SVM

2.4 Multi-kernel SVM

2.5 Multi-class SVM

2.6 Decomposition and SMO

2.7 Further Discussions

Problems

References

3 Parameter Estimations

3.1 Maximum Likelihood Estimation

3.2 Measurements with iid Noise

3.3 Expectation Maximization for Mixture Models

3.4 The General Expectation Maximization

3.5 Expectation Maximization for PPCA Model with Missing Data

3.6 K-Means Clustering

Problems

References

4 Norm Approximation and Regularization

4.1 Norm Approximation

4.2 Tikhonov Regularization

4.3 1-Norm Regularization for Sparsity

4.4 Regularization and MAP Estimation

Problems

References

5 Semidefinite Programming and Linear Matrix Inequalities

5.1 Semidefinite Matrix and Semidefinite Programming

5.2 LMI and Classical Linear Control Problems

5.2.1 Stability of Continuous-Time Linear Systems

5.2.2 Stability of Discrete-Time Linear Systems..'

5.2.3 LMI and Algebraic Riccati Equations

5.3 LMI and Linear Systems with Time Delay

Problems

References

6 Convex Relaxation

6.1 Basic Idea of Convex Relaxation

6.2 Max-Cut Problem

6.3 Solving Sudoku Puzzle

Problems

References

7 Geometric Problems

7.1 Distances

7.2 Sizes

7.3 Intersection and Containment

Problems

References

Index

凸优化理论和方法能够解决一大类常见的优化问题。本书介绍了凸优化在支撑向量机、参数估计、范数逼近、控制器设计等问题中的应用，以期读者掌握将实际问题转换（或近似转换）成凸优化问题的基本知识和基本方法，能够灵活使用凸优化理论和方法解决实际问题。本书潜在的读者包括运筹优化方向、机器学习方向、统计方向、控制方向、信号处理方向的研究生和高年级本科生。读者需对凸优化理论和线性代数理论有一定的了解。