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

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

for Conjugate Gradient Method, with

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

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

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

for Conjugate Gradient Method, with

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

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

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

for Conjugate Gradient Method, with

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

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

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

for Conjugate Gradient Method, with

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

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