Lagrange Multiplier

Lagrange Multiplier

In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) provides a strategy for finding the local maxima and minima of a function subject to equality constraints.

For instance (see Figure 1), consider the optimization problem

maximize
subject to

We need both and to have continuous first partial derivatives. We introduce a new variable called a Lagrange multiplier and study the Lagrange function defined by

where the term may be either added or subtracted. If is a maximum of for the original constrained problem, then there exists such that is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of are zero, i.e. ). However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist.

Read more about Lagrange Multiplier:  Introduction, Handling Multiple Constraints, Interpretation of The Lagrange Multipliers, Sufficient Conditions

Other articles related to "lagrange multiplier, lagrange multipliers":

Lagrange Multiplier - Applications - Control Theory
... In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in ...
First Class Constraint - Second Class Constraints - An Example: A Particle Confined To A Sphere
... coordinates that manifestly solve the constraint or one can use a Lagrange multiplier ... reasons, instead, consider the problem in Cartesian coordinates with a Lagrange multiplier term ... where the last term is the Lagrange multiplier term enforcing the constraint ...