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# FindMaxValue

 FindMaxValue[f, x]gives the value at a local maximum of f. FindMaxValue[f, {x, x0}]gives the value at a local maximum of f, found by a search starting from the point x=x0. FindMaxValue[f, {{x, x0}, {y, y0}, ...}]gives the value at a local maximum of a function of several variables. FindMaxValue[{f, cons}, {{x, x0}, {y, y0}, ...}] gives the value at a local maximum subject to the constraints cons. FindMaxValue[{f, cons}, {x, y, ...}]starts from a point within the region defined by the constraints.
• If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.
• cons can contain equations, inequalities or logical combinations of these.
• FindMaxValue first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
• FindMaxValue[f, {x, x0, x1}] searches for a local maximum in f using x0 and x1 as the first two values of x, avoiding the use of derivatives.
• FindMaxValue[f, {x, x0, xmin, xmax}] searches for a local maximum, stopping the search if x ever gets outside the range xmin to xmax.
• Except when f and cons are both linear, the results found by FindMaxValue may correspond only to local, but not global, maxima.
• By default, all variables are assumed to be real.
• For linear f and cons, can be used to specify that a variable can take on only integer values.
Find a maximum value of a univariate function:
Find a maximum value of a multivariate function:
Find the maximum value of a function subject to constraints:
Find a maximum value of a univariate function:
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Find a maximum value of a multivariate function:
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Find the maximum value of a function subject to constraints:
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 Scope   (6)
With different starting points, get the values of different local maxima:
Value at a local maximum of a two-variable function starting from x=2, y=2:
Value at a local maximum constrained within a disk:
Starting point does not have to be provided:
For linear objective and constraints, integer constraints can be imposed:
Or constraints can be specified:
 Options   (7)
This enforces convergence criteria and :
This enforces convergence criteria and :
Setting a high WorkingPrecision makes the process convergent:
Plot convergence to the local minimum:
Use a given gradient; the Hessian is computed automatically:
In this case the default derivative-based methods have difficulties:
Direct search methods which do not require derivatives can be helpful in these cases:
NMaximize also uses a range of direct search methods:
Steps taken by FindMaxValue in finding the minimum of a function:
Set the working precision to ; by default AccuracyGoal and PrecisionGoal are set to :
FindMaximum gives both the value of the maximum and the minimizing argument:
FindArgMax gives the location of the maximum as a list:
FindMaxValue gives the value at the maximum:
If the constraint region is empty, the algorithm will not converge:
If the maximum value is not finite, the algorithm will not converge:
Integer linear programming algorithm is only available for machine-number problems:
Sometimes providing a suitable starting point can help the algorithm to converge:
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