Mixing
How does one minimize/maximize the Lagrangian if its gradient is non-linear?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
If one is trying to maximize(or minimize) the Lagrangian
$$mathcalL(x,y,lambda) = f(x,y) - lambda cdot g(x,y)$$
its fairly straightforward that this is achieved by solving:
$$nabla_x,y,lambda mathcalL(x , y, lambda)=0. $$
From the examples that I have seen, the Lagrangian never has a degree higher than 2, and so its gradient is always linear and so the above equation can be solved as a system of linear equations. The examples I have seen where there are non-linear terms in the above equation is solved by hand.
Is there a go-to method for optimizing with equality constraints for any function where you may not have linear gradients? Is some iterative method (i.e: gradient decent/ascent) the go-to method?
lagrange-multiplier constraints
asked Jul 14 at 21:44
Duncan Frost
1
add a comment |Â
up vote
0
down vote
favorite
If one is trying to maximize(or minimize) the Lagrangian
$$mathcalL(x,y,lambda) = f(x,y) - lambda cdot g(x,y)$$
its fairly straightforward that this is achieved by solving:
$$nabla_x,y,lambda mathcalL(x , y, lambda)=0. $$
From the examples that I have seen, the Lagrangian never has a degree higher than 2, and so its gradient is always linear and so the above equation can be solved as a system of linear equations. The examples I have seen where there are non-linear terms in the above equation is solved by hand.
Is there a go-to method for optimizing with equality constraints for any function where you may not have linear gradients? Is some iterative method (i.e: gradient decent/ascent) the go-to method?
lagrange-multiplier constraints
asked Jul 14 at 21:44
Duncan Frost
1
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If one is trying to maximize(or minimize) the Lagrangian
$$mathcalL(x,y,lambda) = f(x,y) - lambda cdot g(x,y)$$
its fairly straightforward that this is achieved by solving:
$$nabla_x,y,lambda mathcalL(x , y, lambda)=0. $$
From the examples that I have seen, the Lagrangian never has a degree higher than 2, and so its gradient is always linear and so the above equation can be solved as a system of linear equations. The examples I have seen where there are non-linear terms in the above equation is solved by hand.
Is there a go-to method for optimizing with equality constraints for any function where you may not have linear gradients? Is some iterative method (i.e: gradient decent/ascent) the go-to method?
lagrange-multiplier constraints
asked Jul 14 at 21:44
Duncan Frost
1
If one is trying to maximize(or minimize) the Lagrangian
$$mathcalL(x,y,lambda) = f(x,y) - lambda cdot g(x,y)$$
its fairly straightforward that this is achieved by solving:
$$nabla_x,y,lambda mathcalL(x , y, lambda)=0. $$
From the examples that I have seen, the Lagrangian never has a degree higher than 2, and so its gradient is always linear and so the above equation can be solved as a system of linear equations. The examples I have seen where there are non-linear terms in the above equation is solved by hand.
Is there a go-to method for optimizing with equality constraints for any function where you may not have linear gradients? Is some iterative method (i.e: gradient decent/ascent) the go-to method?
lagrange-multiplier constraints
asked Jul 14 at 21:44
Duncan Frost
1
asked Jul 14 at 21:44
Duncan Frost
1
asked Jul 14 at 21:44
Duncan Frost
1
asked Jul 14 at 21:44
Duncan Frost
1
1
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01
add a comment |Â
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2851984%2fhow-does-one-minimize-maximize-the-lagrangian-if-its-gradient-is-non-linear%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
It's more of a field of study. Lots of options.
– jnez71
Jul 14 at 21:57
You might be interested in reading about the augmented Lagrangian method. Many iterative optimization algorithms can be interpreted as methods for solving the KKT conditions.
– littleO
Jul 14 at 22:01