A set \(\mathcal{C}\) is closed if it contains its boundary: \[x^k\in\mathcal{C},x^k\rightarrow\bar{x}\Rightarrow\bar{x}\in\mathcal{C}\]
\(g\) is a subgradient of a convex function \(f\) at \(x\in{}dom~f\) if \[f(y)\geq{}f(x)+g^\top(y-x)\quad\text{for all }y\in{}dom~f\]
\[\text{(P) }\min_xc^Tx,s.t.Ax=b,x\geq0\] \[\text{(D) }\min_yb^Ty,s.t.A^Ty+s=c,s\geq0\]
\[c^Tx=b^T\Leftrightarrow{}x^Ts=0\Leftrightarrow{}x_is_i=0\]
\[\min{}f_0(x),s.t.f_i(x)\leq0,h_i(x)=0\] variable \(x\in{}R^n\), domain \(\mathcal{D}\), optimal value \(p^\star\) \[L:R^n\times{}R^m\times{}R^p\rightarrow{}R,dom~L=\mathcal{D}\times{}R^m\times{}R^p\] \[L(x,\lambda,\nu)=f_0(x)+\sum_{i=1}^m\lambda_if_i(x)+\sum_{i=1}^p\nu_ih_i(x)\]
\[\text{minimize }f_0(x)\]
\[\text{subject to }f_i(x)\leq0\quad{}\land{}\quad{}Ax=b\]
\(f:R^n\rightarrow{}R\) is convex if dom\(f\) is a convex set and \[f(\theta{}x+(1-\theta)y)\leq\theta{}f(x)+(1-\theta)f(y)\] for all \(x,y\in\) dom\(f\), \(0\leq\theta\leq1\)
