## Quadratic Functions

Preliminaries Linear algebra Calculus 1,2 Taylor series Quadratic Functions1 Quadratic function, a type of performance index, is universal. One of its key properties is that it can be represented in a second-order Taylor series precisely. \[ F(\mathbf{x})=\frac{1}{2}\mathbf{x}^TA\mathbf{x}+\mathbf{d}\mathbf{x}+c\tag{1} \] where \(A\) is a symmetric matrix(if it is not symmetric, it can be easily converted into symmetric). And recall the property of gradient: \[ \nabla (\mathbf{h}^T\mathbf{x})=\nabla (\mathbf{x}^T\mathbf{h})=\mathbf{h}\tag{2} \] and \[ \nabla (\mathbf{x}^TQ\mathbf{x})=Q\mathbf{x}+Q^T\mathbf{x}=2Q\mathbf{x}\tag{3} \]...