Optimization

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Optimization
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General Problem
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One Independent Variable
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One Independent Variable (contd)

• slope0 is necessary but not sufficient condition
  for a maximum

f(x)0 too, but it is a point of inflexion
Minimum, f(x)0 too
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One Independent Variable
x0
f(x)
Slope of f(x) is zero
x
x0
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One Independent Variable
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Two Independent Variables
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Two Independent Variable
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Constrained Optimization
contours
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Constrained Optimization (contd)
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Constrained Optimization (contd)
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Lagranges method

• Laganges method introduces a third variable
  called the Lagrangian multiplier (?) that allows
  us to treat the problem as if an unconstrained
  one.

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Lagranges method (contd)
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Lagranges method
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The Farmers Problem
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The Farmers Problem (contd)
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Envelope theorem
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Envelope theorem (Contd)

• The total derivative of the maximum value of y
  with respect to a is just equal to the partial
  derivative of y with respect to a, evaluated at
  the optimal choice of the xis.
• This is called envelope theorem!

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Envelope theorem (contd)
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Envelope theorem (contd)