Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Lagrange multipliers/gradient definition?
I do not fully understand the 'definition' of the Lagrange multipliers. I do understand that a maximum occurs when the constraint and the objective function are tangent to eachother. However, I do not understand why this implies that the gradient of f = labda x gradient of g. Why is it not true that the gradient of f IS EQUAL to the gradient of g? Doesn't the fact that the level curves are parallel imply that the derivatives are equal and thus that the gradients are equal (and not a multiple of eachother)?
1 Answer
- mrcausalityLv 49 years agoFavorite Answer
The gradient is a vector concept that has both magnitude and direction. At a critical point, a constraint contour intersects tangentially with an objective function contour. This is equivalent to having their gradient directions parallel. There is no condition on the relative *magnitude* of the gradient vectors. Indeed, the lagrange multiplier *is* the scaling between the two. Put another (heuristic) way, the gradient measures the density of function contours. There is no reason that the constraint and objective function should have the same density of contours. We need only have them intersect tangentially, which only speaks to direction, not magnitude.
