Lec02 Convex sets¶

1.Affine and Convex sets¶

Line:¶

all point x $$ x=\theta x_1+(1-\theta )x_2 $$

Line segment:¶

line that the

\[ 0 \leq \theta \leq1 \]

Affine set:¶

for any two distinct points in the set , exist a line. every affine set can be expressed as solution of system of linear equations.

Convex combination and convex hull¶

convex hull 仿射包，记为aff C。仿射包是包含C的最小仿射集合，或者我们可以理解为壳。

仿射维数与相对内部¶

定义C的仿射维数为其仿射包的维数

相对内部与相对边界

Convex set:¶

for any two distinct points in the set ,exist a line setment.

Convex cone¶

conic combination 锥组合¶

\[ x=\theta_1x_1+\theta_2x_2 \]

with $\theta_1\geq 0, \theta_2\geq 0$

convex cone 凸锥¶

set that contains all conic combination of points in the set.

锥包¶

集合C的锥包为C中元素的所有锥组合的集合。

2.Some important examples¶

2.1Hyperplanes and Halfspaces¶

Hyperplane¶

set of the form $\{x|a^Tx=b\}(a\neq 0)$

Halfpspaces¶

set of the form $\{x|a^Tx\leq b\}(a\neq 0)$

a is a normal vector
hyperplanes are affine and convex; halfspaces are convex

2.2 Euclidean balls and Ellipsoids¶

(Euclidean) Ball¶

with center $x_c$ and radius $r$:

\[B(x_c, r) =\{x |\ ||x - x_c||_2 ≤ r\} = \{x_c + ru\ |\ ||u||_2 ≤ 1\}\]

Ellipsoid:¶

set of the form:

\[ \{x | (x - x_c)^T P^{ -1}(x - x_c) ≤ 1\} \]

with $P ∈ S^n _{++}$(i.e., P symmetric positive definite)

other representation: $\{xc + Au | \ ||u||_2 ≤ 1\}$ with A square and nonsingular

2.3 Norm balls and norm cones¶

norm¶

a function ||·|| that satisfies

$||x||≥ 0;||x|| = 0$ if and only if $x = 0$
$||tx||= |t|||x||$ for $t ∈ R$
$||x+y|| ≤ ||x|| + ||y||$

norm ball¶

with center $x_c$ and radius r:

\[ \{x | ||x - x_c|| ≤ r\} \]

norm cone:¶

\[\{(x, t) | ||x|| ≤ t\}\subseteq R^{n+1}\]

Euclidean norm cone is called secondorder cone

norm balls and cones are convex

2.4 Polyhedra 多面体¶

单纯形¶

Title

Contents

2.5 Positive semidefinite cone¶

some notations

3.Operations that preserve convexity¶

3.1 intersection 交集¶

the intersection of (any number of) convex sets is conve

3.2 affine functions¶

Exmaple

Scaling 放缩 Translation平移 Projection 投影
加法直积部分和
线性矩阵不等式的解是凸集
双曲锥是凸集

3.3 perspective and linear-fractional function¶

Perspective funtcion¶

$P:R^{n+1}→R^n$

\[P (x, t) = x/t, dom P = \{(x, t) | t > 0\}\]

images and inverse images of convex sets under perspective are convex. 这说明了domP 的子集C如果是凸集，那么他的象也是凸集。

透视函数可以理解为，规范化使得最后一维的分量为1，然后舍弃他。

Linear-fractional¶

$f : R^n → R^m:$

\[f(x) = \frac{Ax + b}{c^T x + d},\quad dom \ f = \{x | c^T x + d > 0\}\]

images and inverse images of convex sets under linear-fractional functions are convex

4.Generalized Inequalities¶

4.1Proper cone and generalized inequalities¶

proper cone¶

a convex cone K is a proper cone if

K is closed
K is solid
K is pointed - which means K contains no line

generalized inequalities¶

suppose K is a proper cone,we define

\[ x ≼_K y \iff y-x \in K \]

to be more strict $$ x ≺_K y \iff y-x \in int K $$

Examples

非负象限及分量不等式
半正定锥及矩阵不等式
[0.1]上的多项式锥

Properties

4.2 Minimum and minimal elements¶

minimum elements¶

$x ∈ S$ is the minimum element of S with respect to $≼_K$if

\[y ∈ S ⇒ x ≼_K y\]

minimal element¶

$x ∈ S$ is a minimal element of S with respect to $≼_K$ if

\[ y ∈ S, y ≼_K x ⇒ y = x \]

!!! Example 集合符号语言且建立在$R_+^2$上的一个例子 - 首先，用简单的集合符号我们可以让 $x\in S$，目标是如何判断x是否是最小元/极小元。 - 1.如果 $S\subseteq x+K$，那么x是最小元 - 2.如果 $(x-K)\cap S = \{x\}$，那么x是极小元 - 示意图如下 -

5.Separating and supporting hyperplane¶

5.1 Separating hyperplane theorem¶

if C and D are disjoint convex sets, then there exists $a \neq 0$, b such that

\[a^T x ≤ b \ for\ x ∈ C, a^T x ≥ b \ for\ x ∈ D\]

then we can say the hyperplane $\{x | a^T x = b\}$ separates C and D

question Deeper learning - 超平面分离定理的证明、严格分离、逆定理……

5.2 Supporting hyperplane theorem¶

hyperplane¶

suppose $x_0$ is a point on the boundary of C,if for any $x\in C$, $$ a^Tx\leq a^Tx_0 $$ then we can say hyperplane $$ {x|a^Tx=aTx_0} $$ is a supporting hyperplane of C at point $x_0$

Supporting hyperplane theorem¶

if C is convex, then there exists a supporting hyperplane at every boundary point of C

6. Dual cones and generalized inequalities¶

Dual cone $K^*$¶

\[ K^*=\{y | y^T x ≥ 0 \ for \ all \ x ∈ K\} \]

Example

$K = R^n_+: K^∗ = R^n _+$

$K = S^n_+: K^∗ = S^n _+$ $K = \{(x, t) | ||x||_2 ≤ t\}: K^∗ = \{(x, t) | ||x||_2 ≤ t\}$ $K = \{(x, t) | ||x||_1 ≤ t\}: K^∗ = \{(x, t) | ||x||_∞ ≤ t\}$ - first three examples are self-dual cones

一些性质

对偶锥总是凸的，即使K不是凸锥
如果 $K_1 \subseteq K_2$,那么 $K_2^* \subseteq K_1^*$
如果K有非空内部，那么$K^*$ 是尖的
如果K 的闭包是尖的，那么$K^*$ 必有非空内部
$K^{**}$ 是K的凸包的闭包，因此如果K是凸的且是闭的，那么$K^{**}$ = K
这些最终说明了如果K是一个proper cone 那么它的对偶锥也是，进一步有$K^{**}$ = K

Dual generalized inequalities¶

\[ y ≽_{K^*} 0 \iff y^T x ≥ 0 \ for\ all \ x ≽_K 0 \]

Minimum and minimal elements via dual inequalities¶

mininum element¶

x is minimum element of S iff for all $λ ≻_{K^∗} 0$, x is the unique minimizer of $λ^T z$ over S

从几何上来看，这样意味着对于任意 $λ ≻_{K^∗} 0$，超平面 $\{z|\ \lambda ^T(z-x)=0\}$是x在S的一个严格支撑超平面

minimal elements¶

if x minimizes $λ^T z$ over S for some $λ ≻_{K^∗} 0$, then x is minimal.

Lec02 Convex sets¶

1.Affine and Convex sets¶

Line:¶

Line segment:¶

Affine set:¶

Convex combination and convex hull¶

仿射维数与相对内部¶

Convex set:¶

Convex cone¶

conic combination 锥组合¶

convex cone 凸锥¶

锥包¶

2.Some important examples¶

2.1Hyperplanes and Halfspaces¶

Hyperplane¶

Halfpspaces¶

2.2 Euclidean balls and Ellipsoids¶

(Euclidean) Ball¶

Ellipsoid:¶

2.3 Norm balls and norm cones¶

norm¶

norm ball¶

norm cone:¶

2.4 Polyhedra 多面体¶

单纯形¶

2.5 Positive semidefinite cone¶

3.Operations that preserve convexity¶

3.1 intersection 交集¶

3.2 affine functions¶

3.3 perspective and linear-fractional function¶

Perspective funtcion¶

Linear-fractional¶

4.Generalized Inequalities¶

4.1Proper cone and generalized inequalities¶

proper cone¶

generalized inequalities¶

4.2 Minimum and minimal elements¶

minimum elements¶

minimal element¶

5.Separating and supporting hyperplane¶

5.1 Separating hyperplane theorem¶

5.2 Supporting hyperplane theorem¶

hyperplane¶

Supporting hyperplane theorem¶

6. Dual cones and generalized inequalities¶

Dual cone \(K^*\)¶

Dual generalized inequalities¶

Minimum and minimal elements via dual inequalities¶

mininum element¶

minimal elements¶