ArgMin
ArgMin[f,x]
gives a position xmin at which f is minimized.
ArgMin[f,{x,y,…}]
gives a position {xmin,ymin,…} at which f is minimized.
ArgMin[{f,cons},{x,y,…}]
gives a position at which f is minimized subject to the constraints cons.
ArgMin[…,x∈rdom]
constrains x to be in the region or domain rdom.
Details and Options
![](Files/ArgMin.en/details_1.png)
![](Files/ArgMin.en/details_2.png)
- ArgMin finds the global minimum of f subject to the constraints given.
- ArgMin is typically used to find the smallest 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, ArgMin will always find a global minimum.
- The constraints cons can be any logical combination of:
-
lhs==rhs equations lhs>rhs, lhs≥rhs, lhs<rhs, lhs≤rhs inequalities (LessEqual,…) lhsrhs, lhsrhs, lhsrhs, lhsrhs vector inequalities (VectorLessEqual,…) Exists[…], ForAll[…] quantified conditions {x,y,…}∈rdom region or domain specification - ArgMin[{f,cons},x∈rdom] is effectively equivalent to ArgMin[{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 variable Integers integer scalar variable Vectors[n,dom] vector variable in Matrices[{m,n},dom] matrix variable in ℛ vector variable restricted to the geometric region - By default, all variables are assumed to be real.
- ArgMin will return exact results if given exact input. With approximate input, it automatically calls NArgMin.
- If the minimum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, ArgMin 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, ArgMin returns {Indeterminate,Indeterminate,…}.
- N[ArgMin[…]] calls NArgMin for optimization problems that cannot be solved symbolically.
![](Files/ArgMin.en/Image_1.gif)
Examples
open allclose allBasic Examples (5)
Scope (36)
Basic Uses (7)
Univariate Problems (7)
Multivariate Problems (9)
Multivariate linear constrained minimization:
Linear-fractional constrained minimization:
Unconstrained polynomial minimization:
Constrained polynomial optimization can always be solved:
The minimum value may not be attained:
![](Files/ArgMin.en/12.gif)
The objective function may be unbounded:
![](Files/ArgMin.en/13.gif)
There may be no points satisfying the constraints:
![](Files/ArgMin.en/14.gif)
Quantified polynomial constraints:
Bounded transcendental minimization:
Minimize convex objective function such that
is positive semidefinite and
:
Parametric Problems (4)
Optimization over Integers (3)
Optimization over Regions (6)
Find the minimum distance between two regions:
Find the minimum such that the triangle and ellipse still intersect:
Find the disk of minimum radius that contains the given three points:
Using Circumsphere gives the same result directly:
Use to specify that
is a vector in
:
Options (1)
WorkingPrecision (1)
Finding an exact minimum point can take a long time:
With WorkingPrecision->100, you get an approximate minimum point:
Applications (10)
Basic Applications (3)
Find the lengths of sides of a unit area rectangle with minimal perimeter:
Find the lengths of sides of a unit area triangle with minimal perimeter:
The minimal perimeter triangle is equilateral:
Find a point on a parabola closest to its axis:
Assuming a particular relationship between the and
parameters:
Geometric Distances (6)
The point q in a region ℛ that is nearest to a given point p is given by ArgMin[Norm[p-q],q∈ℛ]. Find the nearest point in Disk[] to {1,1}:
Find the nearest point to {1,2} in the standard unit simplex Simplex[2]:
Find the nearest point to {1,1,1} in the standard unit sphere Sphere[]:
Find the nearest point to {-1,1,1} in the standard unit simplex Simplex[3]:
The the nearest points p∈ and q∈ can be found through ArgMin[Norm[p-q],{p∈,q∈}]. Find the nearest points in Disk[{0,0}] and Rectangle[{3,3}]:
Find the nearest points in Line[{{0,0,0},{1,1,1}}] and Ball[{5,5,0},1]:
Geometric Centers (1)
If ℛ⊆n is a region that is full dimensional, then the Chebyshev center is the point p∈ℛ that minimizes SignedRegionDistance[ℛ,p], i.e. the negation of the distance to the complement region. Find the Chebyshev center for Disk[]:
Find the Chebyshev center for Rectangle[]:
Properties & Relations (6)
Minimize gives both the value of the minimum and the minimizer point:
ArgMin gives an exact global minimizer point:
NArgMin attempts to find a global minimizer point numerically, but may find a local minimizer:
FindArgMin finds a local minimizer point depending on the starting point:
The minimum point satisfies the constraints, unless messages say otherwise:
The given point minimizes the distance from the point {2,}:
When the minimum is not attained, ArgMin may give a point on the boundary:
![](Files/ArgMin.en/25.gif)
Here the objective function tends to the minimum value when y tends to infinity:
![](Files/ArgMin.en/26.gif)
ArgMin can solve linear optimization problems:
LinearOptimization can be used to solve the same problem:
Use RegionNearest to compute a nearest point in the given region:
It can be computed using ArgMin:
Possible Issues (2)
A finite minimum value may not be attained:
![](Files/ArgMin.en/27.gif)
The objective function may be unbounded:
![](Files/ArgMin.en/28.gif)
There may be no points satisfying the constraints:
![](Files/ArgMin.en/29.gif)
ArgMin 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:
Text
Wolfram Research (2008), ArgMin, Wolfram Language function, https://reference.wolfram.com/language/ref/ArgMin.html (updated 2021).
CMS
Wolfram Language. 2008. "ArgMin." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArgMin.html.
APA
Wolfram Language. (2008). ArgMin. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArgMin.html