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f:Rn→Rf:\R^n\to \Rf:Rn→R continuous differentiable and convex
C⊆RnC\subseteq \R^nC⊆Rn convex
directional derivative is non-negative in all feasible directions
Example. C={x∈Rn∣Ax=b}C= \{x\in \R^n \mid Ax=b \}C={x∈Rn∣Ax=b}
if z∈Cz\in Cz∈C, and x∈Cx\in Cx∈C
Ax=bAx=bAx=b
Az=bAz=bAz=b
A(z−x)=Az−Ax=b−b=0A(z-x)=Az-Ax=b-b=0A(z−x)=Az−Ax=b−b=0
range(A)⊕null(AT)=Rmrange(A) \oplus null(A^T)=\R^mrange(A)⊕null(AT)=Rm
range(AT)⊕null(A)=Rnrange(A^T)\oplus null(A)=\R^nrange(AT)⊕null(A)=Rn
x∗x^*x∗ is a minimizer only if
{∇f(x∗)=ATy for some y∈RmAx∗=b\left \{ \begin{aligned}&\nabla f(x^*)=A^Ty\text{ for some } y\in\R^m \\ &Ax^*=b\end{aligned}\right.{∇f(x∗)=ATy for some y∈RmAx∗=b, where AAA is a m×nm\times nm×n matrix.
−∇f(x∗)∈NC(x∗),x∗∈C⇔−∇f(x∗)∈range(AT),Ax∗=b⇔∃y∗ s.t. −∇f(x∗)=ATy\begin{aligned}&-\nabla f(x^*)\in N_C(x^*), x^*\in C \\ \Leftrightarrow & -\nabla f(x^*)\in range(A^T),Ax^*=b \\ \Leftrightarrow & \exist y^* \text{ s.t. } -\nabla f(x^*)=A^Ty\end{aligned}⇔⇔−∇f(x∗)∈NC(x∗),x∗∈C−∇f(x∗)∈range(AT),Ax∗=b∃y∗ s.t. −∇f(x∗)=ATy