How to | Do Constrained Nonlinear Optimization
An important subset of optimization problems is constrained nonlinear optimization, where the function is not linear and the parameter values are constrained to certain regions.
Mathematica is capable of solving these as well as a variety of other optimization problems.
A simple optimization problem is to find the largest value of
such that the point
is within 1 unit of the origin. The first argument of
Maximize has the function and the constraint in a list; the second argument lists the variables:
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This output is a list whose first element is the maximum value found; the second element
is a list of rules for the values of the independent variables that give that maximum. The notation
is the short form for
Part
:
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You can check to see if this solution meets the constraint by substituting the solution into the original problem. To do this, use
following the original problem. The
symbol is the short form of
ReplaceAll:
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Many optimization problems involve finding the smallest value of some function:
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You can see this result as a decimal approximation with
N:
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If your goal is numerical results, it is more efficient to use the numerical versions
NMinimize and
NMaximize from the start. Note that
NMinimize and
NMaximize use numeric algorithms and may give results which are not global optima:
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Maximize and
Minimize symbolically analyze the expression to optimize, giving proven global optima. If you have an expression that cannot be analyzed by symbolic techniques,
NMaximize and
NMinimize will be more useful and efficient.
Here is a simple expression that
Maximize and
Minimize cannot work with because
NIntegrate does not evaluate if the parameter
is not a number:
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Here is a function for this expression that you can use with
NMinimize.
NumericQ on the argument protects the function from evaluating
NIntegrate when
is not a number. The
is the short form for
PatternTest:
You can use
NMinimize and
NMaximize to perform this optimization numerically. The second argument gives
two starting values for finding slopes:
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NMinimize,
NMaximize,
Minimize, and
Maximize seek global minima or maxima. The sometimes faster functions
FindMinimum and
FindMaximum seek local minima or maxima:
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A common optimization problem is to find parameters that minimize the error between a curve and data. This example calls
FindFit to find a quadratic fit to these data points:
The arguments to
FindFit are the data, the expression, its list of parameters, and the variable. This solves the problem:
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You can add constraints. For example, suppose that all the parameters have to be positive; put those constraints in a list with the expression:
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This compares the two fits. They are very close except when
is small:
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