Lagrange Multipliers

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					                                     Lagrange Multipliers




                                                        Georgia Institute of Technology
Optimization in Engineering Design
                                                        Systems Realization Laboratory
                                                                                      1
                                           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
                                                              Georgia Institute of Technology
      Optimization in Engineering Design
                                                              Systems Realization Laboratory
                                                                                            2
  Finding an Optimum using Lagrange Multipliers

• New problem is:                      minimize L(x, l) = f(x) - l h1(x)

• 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.

• 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.




                                                                   Georgia Institute of Technology
  Optimization in Engineering Design
                                                                   Systems Realization Laboratory
                                                                                                 3
                                              Method

1.     Original problem is rewritten as: minimize L(x, l) = f(x) - l h1(x)
2.     Take derivatives of L(x, l) with respect to xi and set them equal to
       zero.
       •     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)
3.     Express all xi in terms of Langrangian multiplier l
4.     Plug x in terms of l in constraint h1(x) = 0 and solve l.
5.     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)

                                                                       Georgia Institute of Technology
     Optimization in Engineering Design
                                                                       Systems Realization Laboratory
                                                                                                     4
                                       Multiple constraints

• The Lagrangian multiplier method can be used for any
  number of equality constraints.

• 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                 L(x, l) = f(x) - lT h(x)
where
     lT is the transpose vector of Lagrangian multpliers and has length k

                                                          Georgia Institute of Technology
  Optimization in Engineering Design
                                                          Systems Realization Laboratory
                                                                                        5
                                          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:
    minimize                f(x)
    Subject to              gj(x)  0 for j = 1, 2, ..., J
                            hk(x) = 0 for k = 1, 2, ..., K
                               x = (x1, x2, ..., xN)
                                                             Georgia Institute of Technology
 Optimization in Engineering Design
                                                             Systems Realization Laboratory
                                                                                           6

				
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