What if LICQ does not hold?
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Suppose given a nonlinear optimization programming:
$min_x,y f(x,y)$ st
$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$
and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.
linear-algebra nonlinear-optimization
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up vote
0
down vote
favorite
Suppose given a nonlinear optimization programming:
$min_x,y f(x,y)$ st
$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$
and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.
linear-algebra nonlinear-optimization
There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose given a nonlinear optimization programming:
$min_x,y f(x,y)$ st
$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$
and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.
linear-algebra nonlinear-optimization
Suppose given a nonlinear optimization programming:
$min_x,y f(x,y)$ st
$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$
and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.
linear-algebra nonlinear-optimization
linear-algebra nonlinear-optimization
asked Sep 7 at 9:52
yas are
304
304
There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48
add a comment |Â
There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48
There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48
There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48
add a comment |Â
1 Answer
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LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.
You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).
LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.
You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).
LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
add a comment |Â
up vote
1
down vote
LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.
You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).
LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
add a comment |Â
up vote
1
down vote
up vote
1
down vote
LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.
You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).
LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.
LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.
You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).
LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.
answered Sep 7 at 10:10
daw
22.2k1542
22.2k1542
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
add a comment |Â
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
â yas are
Sep 7 at 10:20
add a comment |Â
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There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
â Eff
Sep 7 at 10:48