Week 1: Convex Sets

1 Definition of convex set
2 Operations preserving convexity
- 2.1 Affine transformation (shift, scale, rotate)
- 2.2 Intersection
3 Examples
- 3.1 Hyperplanes
- 3.2 Halfspaces
- 3.3 Convex hull of x1…xnx_1 \dots x_nx1…xn
- 3.4 Conic combination of x1…xnx_1 \dots x_nx1…xn
- 3.5 Affine combination of x1…xnx_1 \dots x_nx1…xn
- 3.6 Ellipse, norm balls
- 3.7 Polyhedra
- 3.8 All positive semidefinite (symmetric) matrices
- 3.9 Level sets

1 Definition of convex set

A set CCC is convex if the line segment between any two points in CCC lies in CCC, i.e., if for any x1,x2∈Cx_1, x_2 \in Cx1,x2∈C and any θ\thetaθ with 0≤θ≤10 ≤ \theta ≤ 10≤θ≤1, we have:
θx1+(1−θ)x2∈C\theta x_1+(1-\theta)x_2\in Cθx1+(1−θ)x2∈C

2 Operations preserving convexity

2.1 Affine transformation (shift, scale, rotate)

C′={Ax+b∣x∈C}C'=\{Ax+b|x\in C\}C′={Ax+b∣x∈C} C′C'C′ is a convex set ⇔\Leftrightarrow⇔ CCC is a convex set.

2.2 Intersection

C′={x∣x∈C1,x∈C2}C'=\{x|x\in C_1, x\in C_2\}C′={x∣x∈C1,x∈C2} C′C'C′ is a convex set ⇔\Leftrightarrow⇔ C1C_1C1, C2C_2C2 are convex sets.

3 Examples

3.1 Hyperplanes

{x:aTx=b},a≠0\{x:a^Tx=b\}, a\neq0{x:aTx=b},a=0

3.2 Halfspaces

{x:aTx≤b},a≠0\{x:a^Tx\leq b\}, a\neq0{x:aTx≤b},a=0

3.3 Convex hull of x1…xnx_1 \dots x_nx1…xn

{x:x=∑i=1nθixi,θi≥0,∑i=1nθi=1}\{x:x=\sum_{i=1}^n \theta_ix_i, \theta_i\geq 0, \sum_{i=1}^n \theta_i=1\}{x:x=i=1∑nθixi,θi≥0,i=1∑nθi=1}

3.4 Conic combination of x1…xnx_1 \dots x_nx1…xn

{x:x=∑i=1nθixi,θi≥0}\{x:x=\sum_{i=1}^n \theta_ix_i,\theta_i\geq 0\}{x:x=i=1∑nθixi,θi≥0}

3.5 Affine combination of x1…xnx_1 \dots x_nx1…xn

{x:x=∑i=1nθixi,θ∈R}\{x:x=\sum_{i=1}^n \theta_ix_i, \theta \in \mathbb{R}\}{x:x=i=1∑nθixi,θ∈R}

3.6 Ellipse, norm balls

{x:(x−c)TM(x−c)≤1,M≥0}\{x:(x-c)^TM(x-c)\leq 1, M\geq 0 \}{x:(x−c)TM(x−c)≤1,M≥0} {x:∣∣x∣∣≤u}\{x:||x||\leq u\}{x:∣∣x∣∣≤u}

3.7 Polyhedra

{x:Ax≤b,cx=d}\{x:Ax\leq b, cx=d\}{x:Ax≤b,cx=d}

3.8 All positive semidefinite (symmetric) matrices

3.9 Level sets

fff is a convex function ⇒\Rightarrow⇒ any level set {x:f(x)≤c}\{x:f(x)\leq c\}{x:f(x)≤c} is a convex set.

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Optimization Week 1: Convex Sets

Week 1: Convex Sets

1 Definition of convex set

2 Operations preserving convexity

2.1 Affine transformation (shift, scale, rotate)

2.2 Intersection

3 Examples

3.1 Hyperplanes

3.2 Halfspaces

3.3 Convex hull of x1…xnx_1 \dots x_nx1…xn

3.4 Conic combination of x1…xnx_1 \dots x_nx1…xn

3.5 Affine combination of x1…xnx_1 \dots x_nx1…xn

3.6 Ellipse, norm balls

3.7 Polyhedra

3.8 All positive semidefinite (symmetric) matrices

3.9 Level sets

Optimization Week 1: Convex Sets相关推荐

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Optimization Week 1: Convex Sets

Week 1: Convex Sets

1 Definition of convex set

2 Operations preserving convexity

2.1 Affine transformation (shift, scale, rotate)

2.2 Intersection

3 Examples

3.1 Hyperplanes

3.2 Halfspaces

3.3 Convex hull of x1…xnx_1 \dots x_nx1​…xn​

3.4 Conic combination of x1…xnx_1 \dots x_nx1​…xn​

3.5 Affine combination of x1…xnx_1 \dots x_nx1​…xn​

3.6 Ellipse, norm balls

3.7 Polyhedra

3.8 All positive semidefinite (symmetric) matrices

3.9 Level sets

Optimization Week 1: Convex Sets相关推荐

最新文章

热门文章

3.3 Convex hull of x1…xnx_1 \dots x_nx1…xn

3.4 Conic combination of x1…xnx_1 \dots x_nx1…xn

3.5 Affine combination of x1…xnx_1 \dots x_nx1…xn