Our problem is to find the stationary value of ##a_{ik}q_iq_k## under the auxiliary condition the we move on the surface ##b_{ik}q_iq_j = 1##.
We drop the auxiliary condition and minimize ##a_{ik}q_iq_k - \frac 1 {\lambda} b_{ik}q_iq_k##.
Ok, so let's try to solve it. Do you agree with me that if ##x_i = q_i##, ##f(x_i) = a_{ik}q_iq_k##, ##g(x_i) = b_{ik}q_iq_k## and ##c = 1## then the system becomes:wrobel said:I can solve the problem independently on that book.
it is not a matter of convention, ##\lambda## must be situated as I saiddRic2 said:Note that I used 1λ\frac 1 {\lambda} as a multiplier to stick to the convention introduced by the author of the book)
what about ##\lambda=0##? :)dRic2 said:You can always define a new variable λ′=1λ\lambda' = \frac 1 {\lambda} any get the same result
Lagrange multipliers are a mathematical tool used in optimization to find the maximum or minimum value of a function subject to a set of constraints. They involve creating a new function, called the Lagrangian, by adding the constraints to the original function using a set of unknown variables called multipliers. The optimal solution can then be found by taking the partial derivatives of the Lagrangian and setting them equal to zero.
Lagrange multipliers should be used when a function needs to be optimized subject to a set of constraints. This can include problems in economics, engineering, and physics, among others. Lagrange multipliers are particularly useful when the constraints are nonlinear or when there are multiple constraints that cannot be easily combined into a single equation.
The method of undetermined multipliers is a specific application of Lagrange multipliers, where the constraints are equations that must be satisfied exactly. In this case, the Lagrangian can be simplified and the optimal solution can be found by solving a system of equations. The method of undetermined multipliers is often used in engineering and physics problems.
One limitation of Lagrange multipliers is that they can only be used for constrained optimization problems, where the constraints are in the form of equations. They also require the constraints to be differentiable and continuous. Additionally, Lagrange multipliers can be computationally intensive for problems with a large number of constraints or variables.
No, Lagrange multipliers cannot be used for non-differentiable functions, as they rely on taking the partial derivatives of the Lagrangian to find the optimal solution. If the function is not differentiable, then the partial derivatives do not exist and the method cannot be applied. In this case, alternative methods such as subgradients or convex optimization may be used.