# Verify if my idea is correct

Hi guys I have been reading Numerical optimization by Nocedal. While I was on the chapter on quasi- Newton methods it was mentioned that $$g_{k+1}^Ts_k=0$$ where $$g_k = Ax_k -b^Tx_k$$ if we use exact line search on a quadratic (The claim is in chapter 7.2 Limited quasi-Newton methods pg 181 in the second edition of the book).I think I understand why but I wanted to check and see if I am correct.

$$\phi(x_k) = x_k^TAx_k-b^Tx_k$$

if we take the directional derivative in direction $$p$$ we get $$\nabla \phi(x_k)p = (Ax_k-b)^Tp$$. Now if we are trying to minimize the function we have the optimality condition that $$\nabla \phi(x_k)p =0$$. This leads me to $$g_k^T p_k =0$$ which will imply that for $$s_k = \alpha_k p_k$$ we have $$g_k^Ts_k = 0$$ (it seems as if I am an index off). So my question is am I making a small mistake or I am really not understanding why we have $$g_{k+1}^Ts_k =0$$ in the book.

• I think you need to define what $g_k$ is as well if anyone is to have a chance of helping you. A page reference in Nocedal might also be useful. – postmortes Jun 16 at 6:24
• – LinAlg Jun 30 at 19:25