Equivalence of two KKT conditions
The KKT conditions are usually defined as follows
begin{align}
nabla_x cal L(x^*) &= nabla f(x^*) - sum_{iincal I}mu_i g(x^*) - sum_{iincal E}lambda_i h(x^*) = 0
\
g(x^*) &leq 0
\
h(x^*) &= 0
\
mu_i &geq 0
\
mu_i g(x^*) &= 0
end{align}Wikipedia
However, I've encountered another set of equations which claim to also be the KKT conditions
begin{align}
frac{partial cal L}{partial x_i}(x^*) &leq 0
\
x_i^*frac{partial cal L}{partial x_i}(x^*) &= 0
\
g_j(x^*) &leq 0
\
x_i^* &geq 0
\
lambda_j &geq 0
\
lambda_j g_j(x^*) &= 0
end{align}
Some paper
However, this features an inequality for $nabla_xcal L(x^*)$ instead of an equality, so is this just a special case or is the equality enforced by conditions in other equations?
karush-kuhn-tucker
asked Dec 6 at 16:32
Frank Vel
2,34942245
add a comment |
The KKT conditions are usually defined as follows
begin{align}
nabla_x cal L(x^*) &= nabla f(x^*) - sum_{iincal I}mu_i g(x^*) - sum_{iincal E}lambda_i h(x^*) = 0
\
g(x^*) &leq 0
\
h(x^*) &= 0
\
mu_i &geq 0
\
mu_i g(x^*) &= 0
end{align}Wikipedia
However, I've encountered another set of equations which claim to also be the KKT conditions
begin{align}
frac{partial cal L}{partial x_i}(x^*) &leq 0
\
x_i^*frac{partial cal L}{partial x_i}(x^*) &= 0
\
g_j(x^*) &leq 0
\
x_i^* &geq 0
\
lambda_j &geq 0
\
lambda_j g_j(x^*) &= 0
end{align}
Some paper
However, this features an inequality for $nabla_xcal L(x^*)$ instead of an equality, so is this just a special case or is the equality enforced by conditions in other equations?
karush-kuhn-tucker
asked Dec 6 at 16:32
Frank Vel
2,34942245
add a comment |
The KKT conditions are usually defined as follows
begin{align}
nabla_x cal L(x^*) &= nabla f(x^*) - sum_{iincal I}mu_i g(x^*) - sum_{iincal E}lambda_i h(x^*) = 0
\
g(x^*) &leq 0
\
h(x^*) &= 0
\
mu_i &geq 0
\
mu_i g(x^*) &= 0
end{align}Wikipedia
However, I've encountered another set of equations which claim to also be the KKT conditions
begin{align}
frac{partial cal L}{partial x_i}(x^*) &leq 0
\
x_i^*frac{partial cal L}{partial x_i}(x^*) &= 0
\
g_j(x^*) &leq 0
\
x_i^* &geq 0
\
lambda_j &geq 0
\
lambda_j g_j(x^*) &= 0
end{align}
Some paper
However, this features an inequality for $nabla_xcal L(x^*)$ instead of an equality, so is this just a special case or is the equality enforced by conditions in other equations?
karush-kuhn-tucker
asked Dec 6 at 16:32
Frank Vel
2,34942245
The KKT conditions are usually defined as follows
begin{align}
nabla_x cal L(x^*) &= nabla f(x^*) - sum_{iincal I}mu_i g(x^*) - sum_{iincal E}lambda_i h(x^*) = 0
\
g(x^*) &leq 0
\
h(x^*) &= 0
\
mu_i &geq 0
\
mu_i g(x^*) &= 0
end{align}Wikipedia
However, I've encountered another set of equations which claim to also be the KKT conditions
begin{align}
frac{partial cal L}{partial x_i}(x^*) &leq 0
\
x_i^*frac{partial cal L}{partial x_i}(x^*) &= 0
\
g_j(x^*) &leq 0
\
x_i^* &geq 0
\
lambda_j &geq 0
\
lambda_j g_j(x^*) &= 0
end{align}
Some paper
However, this features an inequality for $nabla_xcal L(x^*)$ instead of an equality, so is this just a special case or is the equality enforced by conditions in other equations?
karush-kuhn-tucker
karush-kuhn-tucker
asked Dec 6 at 16:32
Frank Vel
2,34942245
asked Dec 6 at 16:32
Frank Vel
2,34942245
asked Dec 6 at 16:32
Frank Vel
2,34942245
asked Dec 6 at 16:32
Frank Vel
2,34942245
asked Dec 6 at 16:32
Frank Vel
2,34942245
2,34942245
add a comment |
add a comment |
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