Lagrange multipliers
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- tags
- Math, Calculus, Optimization
- source
- (Bishop 2006)
Lagrange multipliers (sometimes referred to as undetermined multipliers) are used to find Stationary Points of a function \(f\) of several variables subject to one or more constraints. Given a function \(f(\xvec)\) with domain \(\numfield^D\) we want to find the maximum (or any Stationary Point) subject to the constraint relating the elements of \(\xvec\) in the form
\[g(\xvec) = 0.\]
Given a D-dimensional vector space, the constraint \(g(\xvec)=0\) represents a (D-1)-dimensional surface in the space. Any point on the constraint surface will have a gradient \(\nabla g(\xvec)\) which is orthogonal to the surface.
Next we seek a point \(\xvec^*\) on the constraint surface such that \(f(\xvec)\) is maximized. This point must have the property that the gradient \(\nabla f(\xvec^*)\) is also orthogonal to the surface. Thus \(\nabla f(\xvec^*)\) and \(\nabla g(\xvec*)\) are parallel at this point meaning there exists some parameter \(\lambda\)
\[\nabla f + \lambda \nabla g = 0.\]
This brings us to the Lagrangian function.
The Lagrangian function is defined by \[L(\xvec, \lambda) \defeq f(\xvec) + \lambda g(\xvec)\]
Finding a Stationary Point of \(L(\xvec, \lambda)\) with respect to both \(\xvec\) and \(\lambda\) will find the value \(\xvec^*\) satisfying the constraint.