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单词 Lagrange multiplier
释义
Lagrange multiplier

Physics
  • A method used in solving problems in the calculus of variations. One wants to find the maximum or minimum values of a function f(x,y), where x and y are related by some equation ϕ‎(x,y)=c, where c is a constant. The function F(x,y)=f(x,y)+Lϕ‎(x,y) is formed, with L being a Lagrange multiplier. The partial derivatives of F with respect to x and y are set to zero. Together with the equation ϕ‎(x,y)=c, this enables the maximum or minimum value of f to be determined. The method can be extended to more than two variables.


Mathematics
  • A method of evaluating maxima and minima of a function f, where one or more constraints gi = 0 have to be satisfied. A new function is constructed as

    L=f+λ1g1+λ2g2++λngn.

    Then the partial derivatives of L with respect to the original variables and each λ‎i are taken, and the stationary points found by solving the set of simultaneous equations obtained by setting each derivative to zero.

    For example, to maximize f(x,y) = 3x + 4y subject to g(x,y) = x2 + y2−1=0, we define

    L(x,y,λ)=3x+4y+λ(x2+y21),

    and the equations

    0=Lx=3+2λx,0=Ly=4+2λy,0=Lλ=x2+y21

    have solutions (x,y) = (3/5,4/5), a maximum, and (x,y) = (–3/5, –4/5), a minimum.


Economics
  • A variable introduced to solve a problem involving constrained optimization. Suppose that the function f(x, y) has to be maximized by choice of x and y subject to the constraint that g(x, y) ≤ k. The solution can be found by constructing the Lagrangean function

    L(x,y,λ)=f(x,y)+λ[k-g(x,y)]

    where λ‎ is the Lagrange multiplier. Let fx denote ∂f(x, y)/∂x, etc. The optimal values of x, y, and λ‎ solve the necessary conditions

    Lxfx-λgx=0,Lyfy-λgy=0,λ[k-g(x,y)]=0,andλ0.

    If the constraint is not binding, λ‎ = 0 and the maximum occurs where fx = fy = 0. If the constraint is binding, λ‎ > 0 and the optimum is found by solving the three equations Lx = 0, Ly = 0, and g(x, y) − k = 0. The Lagrange multiplier, λ‎, measures the increase in the objective function (f(x, y)) that is obtained through a marginal relaxation in the constraint (an increase in k). For this reason, the Lagrange multiplier is often termed a shadow price. For example, if f(x, y) is a utility function, which is maximized subject to the constraint that total spending on x and y is less than or equal to income, k, then λ‎ measures the marginal utility of income—the additional utility provided by one more unit of income.


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