Convex Optimization Note(1): Affine Set and Convex Set
1. Convex Sets
1.1 Affine Set
Definition:
Property:
Solution Set of Linear Equation
Affine Set(线性方程的解集是affine set,同时affine set都能写成线性方程的解)Subspace:
where (无论选择哪个 ,subspace 始终能穿过原点)
1.2 Affine Hull
Definition: All affine combinations of points in set
Property: Smallest affine set that contains
Example: Affine set of a circle
1.3 Convex Set
Definition: Affine set with
Property:
works for infinite sums and integral
pass through expectation
1.4 Cone(non-negative homogeneous)
Definition:
1.5 Simplex
Definition: Convex hull of
affine independent: 直观理解与线性相关定义类似,数学表达上对于
,如果 线性无关,则这k+1个向量仿射无关。值得一提的是对于二维平面,要证明四个及以上向量仿射无关就需要证明三个及以上二维向量线性无关,而三个二维向量必定线性相关,因此不存在四个及以上仿射相关的二维向量,因此 中的simplex只可能是三角形(而不会是四边形或更多边形)。
1.6 Positive Semi-definite Cone
Definition: Set of Semi-definite matrix
Property: This set is a convex cone. (任意两个半正定矩阵线性组合,系数为正的情况下,结果仍为半正定矩阵)
2. Operation that preserves convexity
Intersection of convex sets is still convex
Affine Function projects convex sets to convex sets
Affine Function: 等同于线性变换
Perspective Function -> Linear-Fractional Function
to
3. General Inequality
3.1 Proper Cone
Convex; Closed; Solid(non-emplty interior); Pointed(不超过半平面)
3.2 Inequality
Let
4. Seperating and supporting hyperplane
Seperating hyperplane: For two convex sets that are disjoint(no intersection), there exists a hyperplane that seperates these two sets.
Converse theorm: If at least one convex set is open, existing seperating hyperplane means disjoint sets.
Supporting hyperplane: Seperating hyperplane on a boundary point
5. Dual Cone
Definition: For a cone
几何上可以理解为与所有目标锥内的向量夹角都不超过90度的向量构成的锥
Property:
is the minimum element of set with respect to , it means that for , is the minimizer of over