NMaximize

NMaximize[f,x]

maximizes f numerically with respect to x.

NMaximize[f,{x,y,…}]

maximizes f numerically with respect to x, y, ….

NMaximize[{f,cons},{x,y,…}]

maximizes f numerically subject to the constraints cons.

NMaximize[…,x∈reg]

constrains x to be in the region reg.

Details and Options

  • NMaximize returns a list of the form {fmax,{x->xmax,y->ymax,…}}.
  • cons can contain equations, inequalities or logical combinations of these.
  • The constraints cons can be any logical combination of:
  • lhs==rhsequations
    lhs>rhs or lhs>=rhs inequalities
    {x,y,…}∈regregion specification
  • NMaximize[{f,cons},x∈reg] is effectively equivalent to NMaximize[{f,cons∧x∈reg},x].
  • For x∈reg, the different coordinates can be referred to using Indexed[x,i].
  • NMaximize always attempts to find a global maximum of f subject to the constraints given.
  • By default, all variables are assumed to be real.
  • x∈Integers can be used to specify that a variable can take on only integer values.
  • If f and cons are linear, NMaximize can always find global maxima, over both real and integer values.
  • Otherwise, NMaximize may sometimes find only a local maximum.
  • If NMaximize determines that the constraints cannot be satisfied, it returns {-Infinity,{x->Indeterminate,…}}.
  • The following options can be given:
  • AccuracyGoalAutomaticnumber of digits of final accuracy sought
    EvaluationMonitorNoneexpression to evaluate whenever f is evaluated
    MaxIterations100maximum number of iterations to use
    MethodAutomaticmethod to use
    PrecisionGoalAutomaticnumber of digits of final precision sought
    StepMonitorNoneexpression to evaluate whenever a step is taken
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • The default settings for AccuracyGoal and PrecisionGoal are WorkingPrecision/2.
  • The settings for AccuracyGoal and PrecisionGoal specify the number of digits to seek in both the value of the position of the maximum, and the value of the function at the maximum.
  • NMaximize continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
  • Possible settings for the Method option include "NelderMead", "DifferentialEvolution", "SimulatedAnnealing", and "RandomSearch".

Examples

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Basic Examples  (3)

Find the global maximum of an unconstrained problem:

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Extract the maximizing argument:

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Find the global maximum of problems with constraints:

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Maximize a function over a geometric region:

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Plot it:

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Scope  (9)

Options  (6)

Properties & Relations  (1)

Possible Issues  (3)

See Also

NMinimize  Maximize  FindMaximum  NArgMax  NMaxValue  LinearProgramming  BayesianMaximization  SpherePoints

Tutorials

Introduced in 2003
(5.0)
| Updated in 2014
(10.0)