Convex FunctionsΒΆ
Definition. Convex function: A function \(f:X_D \rightarrow X_R\) with \(X_D\subseteq\RR^n\) and \(X_R\subseteq\RR\) is a convex function if for any \(x_1\) and \(x_2\) in \(X_D\) and \(\lambda_1 \geq 0\), \(\lambda_2 \geq 0\) such that \(\lambda_1+\lambda_2=1\), we have: \begin{equation} f(\lambda_1x_1+\lambda_2x_2)\leq \lambda_1f(x_1)+\lambda_2f(x_2) \end{equation}