Numerical Optimization
| FindMinimum[f,{x,x0}] | search for a local minimum of f, starting at  |
| FindMinimum[f,x] | search for a local minimum of f |
| FindMinimum[f,{{x,x0},{y,y0},...}] |
| search for a local minimum in several variables |
| FindMinimum[{f,cons},{{x,x0},{y,y0},...}] |
| search for a local minimum subject to the constraints cons starting at  ,  , ... |
| FindMinimum[{f,cons},{x,y,...}] | search for a local minimum subject to the constraints cons |
| FindMaximum[f,x], etc. | search for a local maximum |
Searching for local minima and maxima.
This finds the value of
which minimizes
, starting at
.
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The last element of the list gives the value at which the minimum is achieved.
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Like
FindRoot,
FindMinimum and
FindMaximum work by starting from a point, then progressively searching for a minimum or maximum. But since they return a result as soon as they find anything, they may give only a local minimum or maximum of your function, not a global one.
This curve has two local minima.
| Out[3]= |  |
Starting at
, you get the local minimum on the right.
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This gives the local minimum on the left, which in this case is also the global minimum.
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You can specify variables without initial values.
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You can specify a constraint.
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| NMinimize[f,x] | try to find the global minimum of f |
| NMinimize[f,{x,y,...}] | try to find the global minimum over several variables |
| NMaximize[f,x] | try to find the global maximum of f |
| NMaximize[f,{x,y,...}] | try to find the global maximum over several variables |
Finding global minima and maxima.
This immediately finds the global minimum.
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NMinimize and
NMaximize are numerical analogs of
Minimize and
Maximize. But unlike
Minimize and
Maximize they usually cannot guarantee to find absolute global minima and maxima. Nevertheless, they typically work well when the function
f is fairly smooth, and has a limited number of local minima and maxima.
| NMinimize[{f,cons},{x,y,...}] | try to find the global minimum of f subject to constraints cons |
| NMaximize[{f,cons},{x,y,...}] | try to find the global maximum of f subject to constraints cons |
Finding global minima and maxima subject to constraints.
With the constraint
,
NMinimize will give the local minimum on the right.
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This finds the minimum of
within the unit circle.
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In this case
Minimize can give an exact result.
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But in this case it cannot.
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This gives a numerical approximation, effectively using
NMinimize.
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If both the objective function
f and the constraints
cons are linear in all variables, then minimization and maximization correspond to a
linear programming problem. Sometimes it is convenient to state such problems not in terms of explicit equations, but instead in terms of matrices and vectors.
Linear programming in matrix form.
Here is a linear programming problem in equation form.
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Here is the corresponding problem in matrix form.
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You can specify a mixture of equality and inequality constraints by making the list
be a sequence of pairs
. If
is
, then the
constraint is
. If
is
then it is
, and if
is
then it is
.
This makes the first inequality use
.
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In
LinearProgramming
, you can make
be a list of pairs
representing lower and upper bounds on the
.
In doing large linear programming problems, it is often convenient to give the matrix
as a
SparseArray object.
which minimizes 
subject to the constraints 
and 
and 
