Mixing
ADMM - proximal vs. augmented lagrangian interpretation: stuck on algebraic step :(
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I'm trying to understand conciliation of the proximal formulation of ADMM with its Augmented Lagrangian (method of multipliers) interpretation. In particular, I'm struggling with page 156 here (https://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf)
where it says "pull the linear terms into the quadratic ones". To be more specific:
$mathbf y^{T}mathbf x + (rho/2) ||mathbf x- mathbf z||_2^2$
should become
$(rho/2) ||mathbf x- mathbf z + (1/rho)mathbf y||_2^2$
I cannot make sense of that passage. Can someone detail how the equivalence is achieved? Am I missing something obvious?
linear-algebra convex-optimization
asked Jan 25 at 8:38
Lester JackLester Jack
11
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add a comment |
$begingroup$
I'm trying to understand conciliation of the proximal formulation of ADMM with its Augmented Lagrangian (method of multipliers) interpretation. In particular, I'm struggling with page 156 here (https://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf)
where it says "pull the linear terms into the quadratic ones". To be more specific:
$mathbf y^{T}mathbf x + (rho/2) ||mathbf x- mathbf z||_2^2$
should become
$(rho/2) ||mathbf x- mathbf z + (1/rho)mathbf y||_2^2$
I cannot make sense of that passage. Can someone detail how the equivalence is achieved? Am I missing something obvious?
linear-algebra convex-optimization
asked Jan 25 at 8:38
Lester JackLester Jack
11
$endgroup$
2
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We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
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– littleO
Jan 25 at 9:07
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26
add a comment |
$begingroup$
I'm trying to understand conciliation of the proximal formulation of ADMM with its Augmented Lagrangian (method of multipliers) interpretation. In particular, I'm struggling with page 156 here (https://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf)
where it says "pull the linear terms into the quadratic ones". To be more specific:
$mathbf y^{T}mathbf x + (rho/2) ||mathbf x- mathbf z||_2^2$
should become
$(rho/2) ||mathbf x- mathbf z + (1/rho)mathbf y||_2^2$
I cannot make sense of that passage. Can someone detail how the equivalence is achieved? Am I missing something obvious?
linear-algebra convex-optimization
asked Jan 25 at 8:38
Lester JackLester Jack
11
$endgroup$
I'm trying to understand conciliation of the proximal formulation of ADMM with its Augmented Lagrangian (method of multipliers) interpretation. In particular, I'm struggling with page 156 here (https://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf)
where it says "pull the linear terms into the quadratic ones". To be more specific:
$mathbf y^{T}mathbf x + (rho/2) ||mathbf x- mathbf z||_2^2$
should become
$(rho/2) ||mathbf x- mathbf z + (1/rho)mathbf y||_2^2$
I cannot make sense of that passage. Can someone detail how the equivalence is achieved? Am I missing something obvious?
linear-algebra convex-optimization
linear-algebra convex-optimization
asked Jan 25 at 8:38
Lester JackLester Jack
11
asked Jan 25 at 8:38
Lester JackLester Jack
11
edited Jan 25 at 8:56
Lester Jack
asked Jan 25 at 8:38
Lester JackLester Jack
11
asked Jan 25 at 8:38
Lester JackLester Jack
11
asked Jan 25 at 8:38
Lester JackLester Jack
11
11
2
$begingroup$
We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
$endgroup$
– littleO
Jan 25 at 9:07
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26
add a comment |
2
$begingroup$
We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
$endgroup$
– littleO
Jan 25 at 9:07
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26
2
2
$begingroup$
We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
$endgroup$
– littleO
Jan 25 at 9:07
$begingroup$
We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
$endgroup$
– littleO
Jan 25 at 9:07
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26
add a comment |
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$begingroup$
We are completing the square, a useful trick in optimization. If you expand the second expression, it is the same as the first expression, except for a term which does not depend on $x$. This extra term is irrelevant if we are minimizing with respect to $x$.
$endgroup$
– littleO
Jan 25 at 9:07
$begingroup$
@littleO, that's it. Thank you!
$endgroup$
– Lester Jack
Jan 25 at 9:26