Nonlinear Opt.: Lagrangian condition 2
Non-linear Optimiaziation: Langrange conditions The langrange method is a mathematical method to solve none-linare optimization problems with restrictions. With the aid of the Lagrangian multipliers (lambda i) the objective function and the restrictions can be combined to the Langragian function. Through the partial derivations with respect to the given variables (xi) there is a system of equations generated. You can locate the minima/maxima of the initial problem by solving the system of equations.
Theory: formal
Given: optimization problem Z = f(x1,…., xn) min/max ! Restrictions hi(x1,…,xn)= b with b ԑ R
xj ≥ 0 with j = 1,…, n f(x1,…,xn), gi(x1,…,xn) have to be contiuously differentiable 1 step: Formulate the Langrangian function L(x,λ) with the langrangian multipliers: L(x1,…,xn; λ1,…,λm)= f(x1,…,xn) - ∑_(i=1)^m▒〖λ_i*g_i(x_1,…,x_n)〗 2 step: Differentiate the Langrangian function with respect to (x1,…,xn): ∂L/(∂x_j )=∂f/(∂x_j ) (x)-∑_(i=1)^m▒〖λ ∂g(i)/∂x(j) 〗(x) 3 step: Formulate the Langrangian conditions: ∂L/(∂x(1))=⋯!=0 …. … ∂L/(∂x(n))=⋯!=0 4 step: Solve the system of equations
Example:
To illustrate the problem of optimization we use a production function g(r1,r2) with the factors r1 and r2 , and the variable output x. The prices for the factor inputs p1 and p2 is given. In the end we want to calculate the optimal relation of factor input, due to the variable output. As a result we get the cost function due to the input- and output variables. The production function is given by: X=g(r1,r2) = √(r1*r2) The factor prices: P1 = 8 P2 = 2 Step 1: Formulate the Langragian function with the Langragian multipliers. L(r1,r2,λ) = 8*r1 + 2*r2 + λ*(x - √(r1*r2)
f(rn) g(rn)
λ: Langragian multiplier
Step 2/Step 3: Formulate the necessary condition, by building the partial derivations and set them equal to zero:
δL/δr1=8- λ*0,5*√(r2/r1) = 0 ( 1 ) δL/δr2=2- λ*0,5*√(r1/r2) = 0 (2)
δL/δλ=x-√(r1*r2) = 0 (3)
Step 4: Solve the system of equation. By solving the first two equtions, you get the optimal relation between the factor inputs r1 and r2: 8 = λ * 0,5 * √(r2/r1) 2 = λ * 0,5*√(r1/r2)
You get the optimal relation:
4*r1 = 1*r1 If you insert this realtion into the production function, you get the optimal measure of the factor inputs: x = √(r1*4*r1) and x = √(0,25*r2*r2)
r1* = 0,5*x r2* = 2*x
The cost functions:
K(x)=0,5*x*p1+2*x*p2 K(x)=(0,5*p1+2*p2)*x
Sources:
Koschig, Robert: Das Optimierungsverfahren mit Lagrange-Multiplikatoren. Stand 09/2012. http://massmatics.de/de/files/2012/09/Lagrangeoptimierung-v1.0.pdf (abgerufen am 24.06.2013) Wendt, O., OR-Skript SS13, (2013) Corsten, H., Gössinger, R. (2009): Produktionswirtschaft - Einführung in das industrielle Produktionsmanagement, 12. Auflage, Oldenburg Verlag, München, S. 136 ff.
