ArgMax [f,x]
gives a position xmax at which f is maximized.
ArgMax [f,{x,y,…}]
gives a position {xmax,ymax,…} at which f is maximized.
ArgMax [{f,cons},{x,y,…}]
gives a position at which f is maximized subject to the constraints cons.
ArgMax […,x∈rdom]
constrains x to be in the region or domain rdom.
ArgMax
ArgMax [f,x]
gives a position xmax at which f is maximized.
ArgMax [f,{x,y,…}]
gives a position {xmax,ymax,…} at which f is maximized.
ArgMax [{f,cons},{x,y,…}]
gives a position at which f is maximized subject to the constraints cons.
ArgMax […,x∈rdom]
constrains x to be in the region or domain rdom.
Details and Options
- ArgMax finds the global maximum of f subject to the constraints given.
- ArgMax is typically used to find the largest possible values given constraints. In different areas, this may be called the best strategy, best fit, best configuration and so on.
- If f and cons are linear or polynomial, ArgMax will always find a global maximum.
- The constraints cons can be any logical combination of:
-
lhs==rhs equationslhs>rhs, lhs≥rhs, lhs<rhs, lhs≤rhs inequalities (LessEqual ,…)lhsrhs, lhsrhs, lhsrhs, lhsrhs vector inequalities (VectorLessEqual ,…){x,y,…}∈rdom region or domain specification
- ArgMax [{f,cons},x∈rdom] is effectively equivalent to ArgMax [{f,cons∧x∈rdom},x].
- For x∈rdom, the different coordinates can be referred to using Indexed [x,i].
- Possible domains rdom include:
-
Reals real scalar variableIntegers integer scalar variableℛ vector variable restricted to the geometric region
- By default, all variables are assumed to be real.
- ArgMax will return exact results if given exact input. With approximate input, it automatically calls NArgMax .
- If the maximum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, ArgMax will return the closest specifiable point.
- Even if the same maximum is achieved at several points, only one is returned.
- If the constraints cannot be satisfied, ArgMax returns {Indeterminate ,Indeterminate ,…}.
- N [ArgMax[…]] calls NArgMax for optimization problems that cannot be solved symbolically.
Examples
open all close allBasic Examples (5)
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 as a function of parameters:
Find a maximizer point for a function over a geometric region:
Plot it:
Scope (36)
Basic Uses (7)
Maximize over the unconstrained reals:
If the single variable is not given in a list, the result is a value at which the maximum is attained:
Maximize subject to constraints :
Constraints may involve arbitrary logical combinations:
An unbounded problem:
An infeasible problem:
The supremum value may not be attained:
Use a vector variable and a vector inequality:
Univariate Problems (7)
Unconstrained univariate polynomial maximization:
Constrained univariate polynomial maximization:
Exp-log functions:
Analytic functions over bounded constraints:
Periodic functions:
Combination of trigonometric functions with commensurable periods:
Piecewise functions:
Unconstrained problems solvable using function property information:
Multivariate Problems (9)
Multivariate linear constrained maximization:
Linear-fractional constrained maximization:
Unconstrained polynomial maximization:
Constrained polynomial optimization can always be solved:
The maximum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:
Quantified polynomial constraints:
Algebraic maximization:
Bounded transcendental maximization:
Piecewise maximization:
Convex maximization:
Maximize concave objective function such that is positive semidefinite and :
Plot the region and the minimizing point:
Parametric Problems (4)
Parametric linear optimization:
Coordinates of the minimizer point are continuous functions of parameters:
Parametric quadratic optimization:
Coordinates of the minimizer point are continuous functions of parameters:
Unconstrained parametric polynomial maximization:
Constrained parametric polynomial maximization:
Optimization over Integers (3)
Univariate problems:
Integer linear programming:
Polynomial maximization over the integers:
Optimization over Regions (6)
Maximize over a region:
Plot it:
Find points in two regions realizing the maximum distance:
Plot it:
Find the maximum such that the triangle 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:
Options (1)
WorkingPrecision (1)
Finding an exact maximum point can take a long time:
With WorkingPrecision->200, you get an approximate maximum point:
Applications (15)
Basic Applications (3)
Find the lengths of sides of a unit perimeter rectangle with the maximal area:
Find the lengths of sides of a unit perimeter triangle with the maximal area:
Find the time at which a projectile reaches the maximum height:
Geometric Distances (9)
The point q in a region ℛ that is farthest from a given point p is given by ArgMax [{Norm [p-q],q∈ℛ},q]. Find the farthest point in Disk [] from {1,1}:
Plot it:
Find the farthest point from {1,2} in the standard unit simplex Simplex [2]:
Plot it:
Find the farthest point from {1,1,1} in the standard unit sphere Sphere []:
Plot it:
Find the farthest point from {-1,1,1} in the standard unit simplex Simplex [3]:
Plot it:
The diameter of a region ℛ is given by the distance between the farthest points in ℛ, which can be computed through ArgMax [Norm [p-q],{q∈ℛ,p∈ℛ}]. Find the diameter of Circle []:
The diameter:
Plot it:
Find the diameter of the standard unit simplex Simplex [2]:
The diameter:
Plot it:
Find the diameter of the standard unit cube Cuboid []:
The diameter:
Plot it:
The farthest points p∈ and q∈ can be found through ArgMax [Norm [p-q],{p∈,q∈}]. Find the farthest points in Disk [{0,0}] and Rectangle [{3,3}]:
The farthest distance:
Plot it:
Find the farthest points in Line [{{0,0,0},{1,1,1}}] and Ball [{5,5,0},1]:
The farthest distance:
Plot it:
Geometric Centers (3)
If ℛ⊆n is a region that is full dimensional, then the Chebyshev center is the point p∈ℛ that maximizes -SignedRegionDistance [ℛ,p], i.e. the distance to the complement region. Find the Chebyshev center for Disk []:
Find the Chebyshev center for Rectangle []:
The analytic center of a region defined by inequalities ℛ=ImplicitRegion [f1[x]≥0∧⋯∧fm[x]≥0,x] is given by ArgMax [{Log [f1[x]⋯ fm[x]],x∈ℛ},x]. Find the analytic center for Triangle [{{0,0},{1,0},{0,1}}]:
From the conditions above you have an inequality representation:
The analytic center:
Plot it:
Find the analytic center for Cylinder []:
Hence you get the inequality representation:
And the analytic center:
Plot it:
Properties & Relations (4)
Maximize gives both the value of the maximum and the maximizer point:
ArgMax gives an exact global maximizer point:
NArgMax attempts to find a global maximizer numerically, but may find a local maximizer:
FindArgMax finds a local maximizer point depending on the starting point:
The maximum point satisfies the constraints, unless messages say otherwise:
The given point maximizes the distance from the point {2,}:
When the maximum is not attained, ArgMax may give a point on the boundary:
Here the objective function tends to the maximum value when y tends to infinity:
ArgMax can solve linear optimization problems:
LinearOptimization can be used to solve the same problem by negating the objective:
Possible Issues (2)
The maximum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:
ArgMax requires that all functions present in the input be real valued:
Values for which the equation is satisfied but the square roots are not real are disallowed:
Related Guides
Text
Wolfram Research (2008), ArgMax, Wolfram Language function, https://reference.wolfram.com/language/ref/ArgMax.html (updated 2021).
CMS
Wolfram Language. 2008. "ArgMax." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArgMax.html.
APA
Wolfram Language. (2008). ArgMax. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArgMax.html
BibTeX
@misc{reference.wolfram_2025_argmax, author="Wolfram Research", title="{ArgMax}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArgMax.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_argmax, organization={Wolfram Research}, title={ArgMax}, year={2021}, url={https://reference.wolfram.com/language/ref/ArgMax.html}, note=[Accessed: 04-January-2026]}