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I am studying optimization methods and I was able to understand and derive the search direction

$$p_k = r_{k-1} + \beta p_{k-1}$$

$$\beta = -\frac{p_{k-1}^TAr_{k-1}}{p_{k-1}^TAp_{k-1}}.$$

In this expression $$p_k$$ is the search direction and $$r_k$$ the residual. Nevertheless, in a paper [1] I found the following expression ($$d$$ is the direction, $$g$$ is the gradient and $$y_{k-1} = (g_k-g_{k-1}))$$:

$$d_1 = -Hg_1,\\ d_k = -Hg_k + \frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}}d_{k-1} .$$

I am not able to understand if this is the same as $$p_k$$. In the paper the author writes that the basic CG is with $$H$$ being the identity matrix.

Is $$\frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}} = \beta$$?

[1] A Relationship between the BFGS and Conjugate Gradient Algorithms and Its Implications for New Algorithms, L. Nazareth.

I am studying optimization methods and I was able to understand and derive the search direction

$$p_k = r_{k-1} + \beta p_{k-1}$$

$$\beta = -\frac{p_{k-1}^TAr_{k-1}}{p_{k-1}^TAp_{k-1}}.$$

In this expression $$p_k$$ is the search direction and $$r_k$$ the residual. Nevertheless, in a paper I found the following expression ($$d$$ is the direction, $$g$$ is the gradient and $$y_{k-1} = (g_k-g_{k-1}))$$:

$$d_1 = -Hg_1,\\ d_k = -Hg_k + \frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}}d_{k-1} .$$

I am not able to understand if this is the same as $$p_k$$. In the paper the author writes that the basic CG is with $$H$$ being the identity matrix.

Is $$\frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}} = \beta$$?

I am studying optimization methods and I was able to understand and derive the search direction

$$p_k = r_{k-1} + \beta p_{k-1}$$

$$\beta = -\frac{p_{k-1}^TAr_{k-1}}{p_{k-1}^TAp_{k-1}}.$$

In this expression $$p_k$$ is the search direction and $$r_k$$ the residual. Nevertheless, in a paper [1] I found the following expression ($$d$$ is the direction, $$g$$ is the gradient and $$y_{k-1} = (g_k-g_{k-1}))$$:

$$d_1 = -Hg_1,\\ d_k = -Hg_k + \frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}}d_{k-1} .$$

I am not able to understand if this is the same as $$p_k$$. In the paper the author writes that the basic CG is with $$H$$ being the identity matrix.

Is $$\frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}} = \beta$$?

[1] A Relationship between the BFGS and Conjugate Gradient Algorithms and Its Implications for New Algorithms, L. Nazareth.

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# Search direction for CG method

I am studying optimization methods and I was able to understand and derive the search direction

$$p_k = r_{k-1} + \beta p_{k-1}$$

$$\beta = -\frac{p_{k-1}^TAr_{k-1}}{p_{k-1}^TAp_{k-1}}.$$
In this expression $$p_k$$ is the search direction and $$r_k$$ the residual. Nevertheless, in a paper I found the following expression ($$d$$ is the direction, $$g$$ is the gradient and $$y_{k-1} = (g_k-g_{k-1}))$$:
$$d_1 = -Hg_1,\\ d_k = -Hg_k + \frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}}d_{k-1} .$$
I am not able to understand if this is the same as $$p_k$$. In the paper the author writes that the basic CG is with $$H$$ being the identity matrix.
Is $$\frac{y_{k-1}^THg_k}{y_{k-1}^Td_{k-1}} = \beta$$?