# 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
• 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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