凸优化笔记-个人整理，易于理解
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2023-07-27
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凸优化笔记
目录
Ⅰ. Introduction ............................................................................................................................. 3
1. mathematical optimization（一般的优化问题） ....................................................... 3
2.convex optimization ........................................................................................................ 3
3.一个例子：照明 ............................................................................................................. 4
4.一些补充 ......................................................................................................................... 5
Ⅱ. Convex sets ............................................................................................................................ 5
1.convex set vs affine set .................................................................................................... 5
2.常见的凸集 ..................................................................................................................... 6
a.凸锥 ......................................................................................................................... 6
b.重要简单的凸集 ..................................................................................................... 6
c.超平面（Hyperplane）和半空间（Half space） ................................................. 6
d.欧几里得球（Euclidean ball）与椭球（ellipsoid） ............................................ 6
e.范数球（norm ball）与范数锥（norm cone） ..................................................... 7
f.多面体（polyhedra） .............................................................................................. 7
g.半正定锥 ................................................................................................................. 7
3. Operations that preserve convexity(保凸运算)(对于凸集) ......................................... 7
a.交集 intersection .................................................................................................... 7
b.仿射 ........................................................................................................................ 7
c.透视（perspective）和线性分式函数（line-fractional） ................................... 7
Ⅲ.凸函数 ................................................................................................................................... 8
1.定义(包括定义，条件 都可以用来判凸依据) ........................................................... 8
a.定义一 ..................................................................................................................... 8
b.定义二 ..................................................................................................................... 8
c.扩展值延伸（extend-value extension） ................................................................ 9
d.一阶条件（定义三） ............................................................................................. 9
f.二阶条件（定义四） .............................................................................................. 9
2.常见的凸（凹）函数 ................................................................................................... 10
a.Example on R: ....................................................................................................... 10
b.Example on : .................................................................................................. 10
3.下水平集（sublevel set）与上境图（epigraph） ..................................................... 10
4.Jensen 不等式 .............................................................................................................. 12
5.保凸运算（对于函数） .............................................................................................. 13
a.非负加权和（nonnegative weighted sums） ....................................................... 13
b.复合仿射映射（composition with affine function） .......................................... 13
c.逐点最大和上确界（pointwise maximum and supremum） .............................. 13
d.复合函数（composition） ................................................................................... 13
e.最小化（minimization） ...................................................................................... 14
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