First-order necessary conditions for a quadratic function

[retspool]

Hey So i need to write down the first order necessary condition (F.O.C). and i need to find out where does a stationary point exist

I know how to solve for the F.O.C when i am given an equation in the standard quadratic polynomial form but here the equation is

f(x) = 1/2 xT Qx - cTx

where T stands for transpose

so its read as 1/2 times x transpose times Q times x - c Transpose times x

Also i need to show that the Newton's method will determine the minimizer of the function in one iteration, regardless of the starting point.
where Q is a positive definite MatrixAny suggestions or help will be appreciatied

Thanks

 

[lippyka]

The first order necessary condition (F.O.C) is that the derivative of f(x) with respect to x must be equal to zero. This can be expressed as:∂f/∂x = 0 For the given equation, this is equivalent to solving the following system of equations: Qx - cT = 0This system of equations can be solved using linear algebra techniques such as matrix inversion or Gaussian elimination. Once the solution to the system of equations has been found, a stationary point exists at the point x which satisfies the system of equations. To show that Newton's method will determine the minimizer of the function in one iteration, regardless of the starting point, one must consider the properties of Newton's method. Newton's method is an iterative method for finding the roots of a function. It uses a first-order Taylor series approximation of the function to determine the next point in the sequence. The algorithm converges quadratically if the initial guess is close enough to the true minimum, and the Hessian matrix of the function is positive definite. Since the Hessian matrix of f(x) is Q, which is positive definite, Newton's method will converge quadratically to the minimum of the function in one iteration, regardless of the starting point.