高等数学微积分课件 6.6.4.1拉格朗日乘法1

• Suppose that we fix l = l* and the unconstrained minimum of L(x; l) occurs at x = x* and x* satisfies h1(x*) = 0, then x* minimizes f(x) subject to h1(x) = 0. • Trick is to find appropriate value for Lagrangian multiplier l.
• Suppose we have a classical problem formulation with k equality constraints minimize f(x1, x2, ..., xn) Subject to h1(x1, x2, ..., xn) = 0 ...... hk(x1, x2, ..., xn) = 0 This can be converted in minimize where L(x, l) = f(x) - lT h(x)
Georgia Institute of Technology Systems Realization Laboratory
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Optimization in Engineering Design
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Multiple constraints • The Lagrangian multiplier method can be used for any number of equality constraints.
lT is the transpose vector of Lagrangian multpliers and has length k
Optimization in Engineering Design
Georgia Institute of Technology Systems Realization Laboratory
Georgia Institute of Technology Systems Realization Laboratory
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Method
1. 2. Original problem is rewritten as: minimize L(x, l) = f(x) - l h1(x) Take derivatives of L(x, l) with respect to xi and set them equal to zero.
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In closing • Lagrangian multipliers are very useful in sensitivity analyses (see Section 5.3) • Setting the derivatives of L to zero may result in finding a saddle point. Additional checks are always useful. • Lagrangian multipliers require equalities. So a conversion of inequalities is necessary. • Kuhn and Tucker extended the Lagrangian theory to include the general classical single-objective nonlinear programming problem:
3. 4. 5. •
Express all xi in terms of Langrangian multiplier l Plug x in terms of l in constraint h1(x) = 0 and solve l. Calculate x by using the just found value for l. Note that the n derivatives and one constraint equation result in n+1 equations for n+1 variables! (See example 5.3)
minimize Subject to f(x) gj(x) 0 for j = 1, 2, ..., J hk(x) = 0 for k = 1, 2, ..., K x = (x1, x2, ..., xN)
Georgia Institute of Technology Systems Realization Laboratory
Optimization in Engineering Design
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Lagrange Multipliers
Optimization in Engineering Design
Georgia Institute of Technology Systems Realization Laboratory
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Lagrange Multipliers
• The method of Lagrange multipliers gives a set of necessary conditions to identify optimal points of equality constrained optimization problems. • This is done by converting a constrained problem to an equivalent unconstrained problem with the help of certain unspecified parameters known as Lagrange multipliers. • The classical problem formulation minimize f(x1, x2, ..., xn) Subject to h1(x1, x2, ..., xn) = 0 can be converted to minimize L(x, l) = f(x) - l h1(x) where L(x, v) is the Lagrangian function l is an unspecified positive or negative constant called the Lagrangian Multiplier
• If there are n variables (i.e., x1, ..., xn) then you will get n equations with n + 1 unknowns (i.e., n variables xi and one Lagrangian multiplier l)
• This can be done by treating l as a variable, finding the unconstrained minimum of L(x, l) and adjusting l so that h1(x) = 0 is satisfied.
Optimization in Engineering Design
Optimization in Engineering Design
Georgia Institute of Technology Systems Realization Laboratory
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Finding an Optimum using Lagrange Multipliers
• New problem is: minimize L(x, l) = f(x) - l h1(x)