Lagrangian: $min_{mathbf{X} in mathbb{R}^{N times K}} left|mathbf{Y}-mathbf{X}right|_F^2$ s.t....
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Let us say that the optimization problem can be posed in the matrix form as given below
$min_{mathbf{X} in mathbb{R}^{N times K}} left|mathbf{Y}-mathbf{X}right|_F^2$ s.t. $mathbf{A}mathbf{X} = mathbf{B}$,
where $mathbf{Y} in mathbb{R}^{N times K}$, $mathbf{A} in mathbb{R}^{M times N}$, and $mathbf{B} in mathbb{R}^{M times K}$.
Question:
Without vectorizing the formulation, can the Lagrangian be defined as
$L(mathbf{X},mathbf{Lambda}) = left|mathbf{Y}-mathbf{X}right|_F^2 + {rm trace}left(mathbf{Lambda}^T left(mathbf{A}mathbf{X} - mathbf{B} right) right)$ ?
If not, then how to construct a Lagrangian in the matrix form? Thank you.
EDIT:
See this How to set up Lagrangian optimization with matrix constrains .
optimization convex-optimization nonlinear-optimization lagrange-multiplier
asked Nov 23 at 13:19
user550103
6711315
add a comment |
up vote
0
down vote
favorite
Let us say that the optimization problem can be posed in the matrix form as given below
$min_{mathbf{X} in mathbb{R}^{N times K}} left|mathbf{Y}-mathbf{X}right|_F^2$ s.t. $mathbf{A}mathbf{X} = mathbf{B}$,
where $mathbf{Y} in mathbb{R}^{N times K}$, $mathbf{A} in mathbb{R}^{M times N}$, and $mathbf{B} in mathbb{R}^{M times K}$.
Question:
Without vectorizing the formulation, can the Lagrangian be defined as
$L(mathbf{X},mathbf{Lambda}) = left|mathbf{Y}-mathbf{X}right|_F^2 + {rm trace}left(mathbf{Lambda}^T left(mathbf{A}mathbf{X} - mathbf{B} right) right)$ ?
If not, then how to construct a Lagrangian in the matrix form? Thank you.
EDIT:
See this How to set up Lagrangian optimization with matrix constrains .
optimization convex-optimization nonlinear-optimization lagrange-multiplier
asked Nov 23 at 13:19
user550103
6711315
1
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
Thank you for the clarification.
– user550103
Nov 23 at 13:35
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let us say that the optimization problem can be posed in the matrix form as given below
$min_{mathbf{X} in mathbb{R}^{N times K}} left|mathbf{Y}-mathbf{X}right|_F^2$ s.t. $mathbf{A}mathbf{X} = mathbf{B}$,
where $mathbf{Y} in mathbb{R}^{N times K}$, $mathbf{A} in mathbb{R}^{M times N}$, and $mathbf{B} in mathbb{R}^{M times K}$.
Question:
Without vectorizing the formulation, can the Lagrangian be defined as
$L(mathbf{X},mathbf{Lambda}) = left|mathbf{Y}-mathbf{X}right|_F^2 + {rm trace}left(mathbf{Lambda}^T left(mathbf{A}mathbf{X} - mathbf{B} right) right)$ ?
If not, then how to construct a Lagrangian in the matrix form? Thank you.
EDIT:
See this How to set up Lagrangian optimization with matrix constrains .
optimization convex-optimization nonlinear-optimization lagrange-multiplier
asked Nov 23 at 13:19
user550103
6711315
Let us say that the optimization problem can be posed in the matrix form as given below
$min_{mathbf{X} in mathbb{R}^{N times K}} left|mathbf{Y}-mathbf{X}right|_F^2$ s.t. $mathbf{A}mathbf{X} = mathbf{B}$,
where $mathbf{Y} in mathbb{R}^{N times K}$, $mathbf{A} in mathbb{R}^{M times N}$, and $mathbf{B} in mathbb{R}^{M times K}$.
Question:
Without vectorizing the formulation, can the Lagrangian be defined as
$L(mathbf{X},mathbf{Lambda}) = left|mathbf{Y}-mathbf{X}right|_F^2 + {rm trace}left(mathbf{Lambda}^T left(mathbf{A}mathbf{X} - mathbf{B} right) right)$ ?
If not, then how to construct a Lagrangian in the matrix form? Thank you.
EDIT:
See this How to set up Lagrangian optimization with matrix constrains .
optimization convex-optimization nonlinear-optimization lagrange-multiplier
optimization convex-optimization nonlinear-optimization lagrange-multiplier
asked Nov 23 at 13:19
user550103
6711315
asked Nov 23 at 13:19
user550103
6711315
edited Nov 23 at 13:49
asked Nov 23 at 13:19
user550103
6711315
asked Nov 23 at 13:19
user550103
6711315
asked Nov 23 at 13:19
user550103
6711315
6711315
1
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
Thank you for the clarification.
– user550103
Nov 23 at 13:35
add a comment |
1
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
Thank you for the clarification.
– user550103
Nov 23 at 13:35
1
1
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
Thank you for the clarification.
– user550103
Nov 23 at 13:35
Thank you for the clarification.
– user550103
Nov 23 at 13:35
add a comment |
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1
The Lagrangian is exactly as you have defined.
– Alex Silva
Nov 23 at 13:33
Thank you for the clarification.
– user550103
Nov 23 at 13:35