学习教材主要是Boyd的《Convex Optimization》
Chapter 2 convex sets，主要是convex sets的一些数学定义，以及一些常见的convex sets。
2.1 Affine and convex sets
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lines: y=θx1+(1−θ)x2=x2+θ(x1−x2)y=\theta x_1+(1-\theta)x_2=x_2+\theta (x_1-x_2)y=θx1+(1−θ)x2=x2+θ(x1−x2), θ∈R\theta\in\mathbb{R}θ∈R, x1,x2∈Rn,x1≠x2x_1,x_2\in\mathbb{R^n},x_1\neq x_2x1,x2∈Rn,x1=x2
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affine sets: x1,x2∈Cx_1,x_2\in Cx1,x2∈C, θ∈R\theta\in\mathbb{R}θ∈R, θx1+(1−θ)x2∈C\theta x_1+(1-\theta)x_2\in Cθx1+(1−θ)x2∈C
- affine hull: affC={ θ1x1+⋯+θkxk∣x1,…,xk∈C,θ1+⋯+θk=1}\mathbf{aff} C=\{\theta_1 x_1+\dots+\theta_k x_k|x_1,\ldots,x_k\in C, \theta_1+\dots+\theta_k=1\}affC={ θ1x1+⋯+θkxk∣x1,…,xk∈C,θ1+⋯+θk=1}, affine dimension of CCC is the dimension of its affine hull
- relative interior: relintC={ x∈C∣B(x,r)∩affC⊆C for some r>0}\mathbf{relint}C=\{x\in C|B(x,r)\cap\mathbf{aff}C\subseteq C\;\text{for some}\;r>0\}relintC={ x∈C∣B(x,r)∩affC⊆Cfor somer>0}, B(x,r)={ y∣∥y−x∥≤r}B(x,r)=\{y|\Vert y-x\Vert\leq r\}B(x,r)={ y∣∥y−x∥≤r}
- relative boundary: clC\relintC\mathbf{cl}C\backslash \mathbf{relint}CclC\relintC,
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convex sets: x1,x2∈Cx_1,x_2\in Cx1,x2∈C, 0≤θ≤10\leq\theta\leq 10≤θ≤1, θx1+(1−θ)x2∈C\theta x_1+(1-\theta)x_2\in Cθx1+(1−θ)x2∈C
- convex hull: convC={ θ1x1+⋯+θkxk∣xi∈C,θi≥0,i=1,…,k,θ1+⋯+θk=1}\mathbf{conv}C=\{\theta_1x_1+\dots+\theta_kx_k|x_i\in C, \theta_i\geq 0, i=1,\ldots,k,\theta_1+\dots+\theta_k=1\}convC={ θ1x1+⋯+
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