Lagrange Multiplier Method. Uwe A. Schneider

Transcription

1 Lagrange Multiplier Method Uwe A. Schneider

2 Joseph Louis Lagrange 25 January April 1813 mathematician and astronomer born in Turin, Piedmont, Italy, lived in Prussia and France contributions to all fields of analysis, to number theory, and to classical and celestial mechanics

3 Lagrange Multiplier Method Allows analytical solution of constrained optimization problems Simple example with two variables and one constraint xy, ( xy) max f, st.. g x, y c

4 A Necessary Condition for Optimality If the constraint is binding (i.e. the solution lies on the constraint line), the constraint must meet the contour lines of the objective function (objective isoclines) tangentially If the condition would not be true, one could move forward or backward along the constraint and move to a higher contour line. But then the original point could not be an optimal point The tangency condition is equivalent to requiring that the gradient vectors of the objective function and the constraints are parallel (gradients are orthogonal to contour lines) This condition is not a sufficient condition for an optimal point

5 Contour lines of objective Constraint g(x,y) b Slightly modified from:

6 Parallel Gradient Vectors f = λ xy, xy, f g f f : =, xy, x y g g g : =, xy, x y Parallelism of two vectors does not require them to be of the same size! Hence, a multiplier λ is used to allow for different sizes of the two gradient vectors.

7 Two Cases a) Constraint is binding b) Constraint is not binding f = λ * xy, x*, y* xy, x*, y* g x*, y* c g f = xy, x*, y* = 0 g x*, y* < c λ* = 0 Capturing both cases f = λ * xy, x*, y* xy, x*, y* ( g( x y ) c) *, * λ* = 0 g x*, y* c g The asterisk (*) denotes optimal value

8 The Lagrange (Λ) Function An auxiliary function, which helps to derive the first order necessary conditions for constrained optimization problems (,, ) (, ) (, ) Λ = xyλ f xy λ g xy c (,, ) (, ) (, ) Λ = + xyλ f xy λ g xy c Deriving the first order necessary conditions from the Lagrange is generally easier than remembering the conditions directly

9 The Lagrange Differentiating the Lagrange with respect to all variables and the Lagrange Multiplier(s) yields the first order necessary conditions Λ= xy,, 0 xyλ,, f = λ g = c xy g

10 Maximization A maximization problem, where the variables are restricted from below, i.e. where variables have to assume nonnegative values, requires f f 0 f = 0, x* 0 f f xy, x x* y y * x*, y* x* = 0 y* = 0 x* 0, y* 0 0 x x* x* = 0 x x*

11 Minimization A minimization problem, where the variables are restricted from below, i.e. where variables have to assume nonnegative values, requires f 0 f f xy, x x* y y * x*, y* x* = 0 y* = 0 x* 0, y* 0 f 0 x x* x* = 0 f x x* = 0, x* 0

12 Constrained Maximization (,, ) (, ) (, ) Λ = + xyλ f xy λ g xy c f λ g xy, x*, y* xy, x*, y* 0 Λ 0 xyλ,, ( g( x y ) c) *, * λ* = 0 ( *, *) g x y c f x* = 0, f y* = 0 x x* y y* x* 0, y* 0

13 Constrained Minimization (,, ) (, ) (, ) Λ = xyλ f xy λ g xy c f λ g xy, x*, y* xy, x*, y* 0 Λ 0 xyλ,, ( g( x y ) c) *, * λ* = 0 ( *, *) g x y c f x* = 0, f y* = 0 x x* y y* x* 0, y* 0

14 Lagrange Rule Convert all inequalities to constraints For maximization problems, use a positive sign for Lagrange multiplier For minimization problems, use a negative sign for Lagrange multiplier If you violate this rule, the inequality signs may be inconsistent but generally there is no problem

15 Shadow Prices (SP) SP f = c Shadow prices indicate how the objective function changes if the right hand side of a constraint is marginally changed If g is a resource constraint and c is the endowment of a resource, the shadow price gives the marginal value of the resource, i.e. how much would the objective function f increase if an additional unit of the resource was available

16 LM versus SP (,, λ) (, ) λ (, ) Λ xy = f xy+ g xy c Λ = c λ (,, λ) (, ) λ (, ) Λ xy = f xy g xy c Λ = c λ Economists often prefer the negative sign for the LM because then LM = SP.

17 Extensions Multiple Constraints Simply use one LM for each individual constraint Dynamic Optimization Hamiltonian technique: in principle similar to Lagrange technique but instead of finding extreme points on a function, one searches for extreme functions (also called extremals)

18 2 max st.. x+ y 20 Example Λ = 22 6x+ 16y+ λ = 0 x Λ = 16x + λ = 0 y x x + xy Λ= x x + xy + λ x + y Λ = x+ y 20 0 λ 22 6x+ 16y+ λ = x + λ = 0 x+ y 20 = x x 16x= 0 38x = 342 x* = 9, y* = 11, λ* = 304 f( x*, y*) = 1419