Optimization using Calculus
Optimization of
Functions of Multiple
Variables: Unconstrained
Optimization
1
D Nagesh Kumar, IISc
Optimization Methods: M2L3
Objectives

To study functions of multiple variables, which are more difficult to
analyze owing to the difficulty in graphical representation and
tedious calculations involved in mathematical analysis for
unconstrained optimization.

To study the above with the aid of the gradient vector and the
Hessian matrix.

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To discuss the implementation of the technique through examples
D Nagesh Kumar, IISc
Optimization Methods: M2L3
Unconstrained optimization

If a convex function is to be minimized, the stationary point is the
global minimum and analysis is relatively straightforward as
discussed earlier.

A similar situation exists for maximizing a concave variable function.

The necessary and sufficient conditions for the optimization of
unconstrained function of several variables are discussed.
3
D Nagesh Kumar, IISc
Optimization Methods: M2L3
Necessary condition

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In case of multivariable functions a necessary condition for a
stationary point of the function f(X) is that each partial derivative is
equal to zero. In other words, each element of the gradient vector  x f
defined below must be equal to zero. i.e. the gradient vector of f(X),
at X=X*, defined as follows, must be equal to zero:
 f
* 
(

)
 x
 1

 f
* 
(

)
 x
2
0
x f  






 f

*
(

)


 dxn

D Nagesh Kumar, IISc
Optimization Methods: M2L3
Sufficient condition

For a stationary point X* to be an extreme point, the matrix of second
partial derivatives (Hessian matrix) of f(X) evaluated at X* must be:

positive definite when X* is a point of relative minimum, and

negative definite when X* is a relative maximum point.

When all eigen values are negative for all possible values of X, then
X* is a global maximum, and when all eigen values are positive for
all possible values of X, then X* is a global minimum.
If some of the eigen values of the Hessian at X* are positive and some
negative, or if some are zero, the stationary point, X*, is neither a
local maximum nor a local minimum.

5
D Nagesh Kumar, IISc
Optimization Methods: M2L3
Example
Analyze the function f ( x)   x12  x22  x32  2 x1 x2  2 x1 x3  4 x1  5 x3  2
and classify the stationary points as maxima, minima and points of
inflection
Solution
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D Nagesh Kumar, IISc
Optimization Methods: M2L3
Example …contd.
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D Nagesh Kumar, IISc
Optimization Methods: M2L3
Example …contd.
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D Nagesh Kumar, IISc
Optimization Methods: M2L3
Example …contd.
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D Nagesh Kumar, IISc
Optimization Methods: M2L3
Thank you
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D Nagesh Kumar, IISc
Optimization Methods: M2L3