Cfrgtkky

I'm trying to implement the gradient descent method for the problem of minimising the following function:

$$f(x) = frac{1}{2}(x-m)^{T}A(x-m)-sumlimits_{i=1}^nlogleft(x_i^{2}right),$$

where $x in R^n$ is a vector; $m in R^n$ is a fixed vector; and $A$ is a fixed positive definite matrix.

The only applications of gradient descent I have come across is for linear regression! So, as a starting point for helping me to solve this, I'd like to know in what situations this cost function would be applied. Does anyone out there recognise it?

From the question, we have that f is a convex quadratic function

$$f(x) = frac{1}{2}(x-m)^{mathrm T} mathrm A (x-m)-sumlimits_{i=1}^nlogleft(x_i^{2}right),$$

where $x in R^n$ is a vector; $m in R^n$ is a fixed vector, and $A$ is a fixed positive definite matrix (symmetric and positive semidefinite).

$$nabla f ( x) = frac{mathrm T+1}{2} mathrm A (x-m)^{mathrm T}-sumlimits_{i=1}^nleft(frac{2x}{x_i^{2}ln(10)}right)$$

Using gradient descent with step $mu$,

$$ x_{k+1} = x_k - mu nabla f ( x_k)$$

Choose $mu$ such that $f(x_k) < f(x_{k+1}) $,then do a loop until we find $x^{*}: f(x^{*}_k) - f(x^{*}_{k+1}) sim 0$

This is a general way to take gradient descent for a convex quadratic function in n-dimensional space. Hope it is helpful.