Chapter 1. Introduction

Talk about what’s non-convex optimization. Why study it? Most of practical problems can be modeled as non-convex optimization.

Two examples: the sparse recovery(find the independent fators or when given data item n « facotrs p), the recommond system (low rank matrix compeletion)

THe non-convex optimization can be solved by convex relaxation approch (for example LASSO formulation). note: structured problem lead to absent relaxation gap

Chapter 2. Mathematical tools

convex analysis

1. convex combination

2. convex set
   

3. convex function (must be continugous differentiable?) might not be continugous and differentiable
   https://www.zhihu.com/question/31437665

closed set has the min/max point. open set might not

a. strongly convex / strongly smooth (ss function is not must to be convex?) must to be convex. but might not derivated

strongly convex must be strictly convex
vise via not (https://math.stackexchange.com/questions/2194323/what-is-a-strictly-convex-function-that-is-not-strongly-convex)

b. Liptichitz function / Jensen inequality

convex project (are PPI/II/O require the project function to be norm-2?)

1. first order properties: PPI/ PPII
   

if z not in C, then the angle is not acute angle

1. zeroth order property: PPO
   

projected gradient descent

convergence guarantees

Chapter 3. non-convex projected gradient descent

project sparse vector / low rank matrix

restricted convext/ strong convex/ smooth

convergence condition / speed

Chapter 4. Alternating Minimization

Marginal Convexity / Joint Convexity

update step ?