NArgMax

NArgMax[f,x]

gives a position xmax at which f is numerically maximized.

NArgMax[f,{x,y,…}]

gives a position {xmax,ymax,…} at which f is numerically maximized.

NArgMax[{f,cons},{x,y,…}]

gives a position at which f is numerically maximized subject to the constraints cons.

NArgMax[…,x∈reg]

constrains x to be in the region reg.

Details and Options

  • NArgMax returns a list of the form {xmin,ymin,…}.
  • NArgMax[…,{x,y,…}] is effectively equivalent to {x,y,…}/.Last[NMaximize[…,{x,y,…},…].
  • 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
  • NArgMax[{f,cons},x∈reg] is effectively equivalent to NArgMax[{f,cons∧x∈reg},x].
  • For x∈reg, the different coordinates can be referred to using Indexed[x,i].
  • NArgMax 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, NArgMax can always find global maxima, over both real and integer values.
  • Otherwise, NArgMax may sometimes find only a local maximum.
  • If NArgMax determines that the constraints cannot be satisfied, it returns {Indeterminate,…}.
  • NArgMax takes the same options as NMaximize.

Examples

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Basic Examples  (4)

Find a maximizer point for a univariate function:

Find a maximizer point for a multivariate function:

Find a maximizer point for a function subject to constraints:

Find a maximizer point for a function over a geometric region:

Plot it:

Scope  (9)

Or constraints can be specified:

Use NArgMax for linear objective and constraints:

Integer constraints can be imposed:

Maximize over a region:

Plot it:

Find points in two regions realizing the maximum distance:

Plot it:

Find the maximum such that the rectangle and ellipse still intersect:

Plot it:

Find the maximum for which contains the given three points:

Plot it:

Use to specify that is a vector in :

Find points in two regions realizing the maximum distance:

Plot it:

Applications  (1)

Find the lengths of sides of a unit perimeter rectangle with the maximal area:

Properties & Relations  (1)

NMaximize gives both the value of the maximum and the maximizer point:

NArgMax gives the maximizer point:

NMaxValue gives the maximum:

Possible Issues  (2)

The objective function may be unbounded:

There may be no points satisfying the constraints:

Wolfram Research (2008), NArgMax, Wolfram Language function, https://reference.wolfram.com/language/ref/NArgMax.html (updated 2014).

Text

Wolfram Research (2008), NArgMax, Wolfram Language function, https://reference.wolfram.com/language/ref/NArgMax.html (updated 2014).

BibTeX

@misc{reference.wolfram_2020_nargmax, author="Wolfram Research", title="{NArgMax}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/NArgMax.html}", note=[Accessed: 16-April-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_nargmax, organization={Wolfram Research}, title={NArgMax}, year={2014}, url={https://reference.wolfram.com/language/ref/NArgMax.html}, note=[Accessed: 16-April-2021 ]}

CMS

Wolfram Language. 2008. "NArgMax." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/NArgMax.html.

APA

Wolfram Language. (2008). NArgMax. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NArgMax.html