Step size for steepest decent method

Question

I need to find the step size $α_k$ when considering function $Q(x)=1/2x^TAx+b^Tx$. I know that the gradient(g) is $Ax-b$ and I should minimize the function $Φ_k(α)=Q(x^k-αg(x^k))$. I am stuck after putting the values into $Φ_k(α)$ and even that I am not sure of doing correctly. Can you help me with the $Φ_k(α)$ function and if possible give some directions on what to do next? Here is a more clear view of the problem

Answer 1

The gradient is $Ax\color{red}+b$.

If you differentiate $\Phi_k(\alpha)$ with respect to $\alpha$ using chain rule and set it to $0$, we should get

$$g(x^k)^T[A(x^k - \alpha g(x^k))+b]=0$$

$$g(x^k)^T[g(x^k) - \alpha Ag(x^k)]=0$$

I will leave the task of isolating $\alpha$ to you.