Minimize
Minimize[f,x]
minimizes f with respect to x.
Minimize[f,{x,y,…}]
minimizes f with respect to x, y, ….
Minimize[{f,cons},{x,y,…}]
minimizes f subject to the constraints cons.
Minimize[…,x∈reg]
constrains x to be in the region reg.
Details and Options


- Minimize returns a list of the form {fmin,{x->xmin,y->ymin,…}}.
- cons can contain equations, inequalities, or logical combinations of these.
- The constraints cons can be any logical combination of:
-
lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities {x,y,…}∈reg region specification Exists[x,cond,expr] existential quantifiers - If f and cons are linear or polynomial, Minimize will always find a global minimum.
- Minimize[{f,cons},x∈reg] is effectively equivalent to Minimize[{f,cons∧x∈reg},x].
- For x∈reg, the different coordinates can be referred to using Indexed[x,i].
- Minimize will return exact results if given exact input.
- If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize.
- If the minimum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, Minimize will return the infimum and the closest specifiable point.
- If no domain is specified, all variables are assumed to be real.
- x∈Integers can be used to specify that a particular variable can take on only integer values.
- If the constraints cannot be satisfied, Minimize returns {+Infinity,{x->Indeterminate,…}}.
- Even if the same minimum is achieved at several points, only one is returned.
- N[Minimize[…]] calls NMinimize for optimization problems that cannot be solved symbolically.
- Minimize[f,x,WorkingPrecision->n] uses n digits of precision while computing a result. »
Examples
open allclose allBasic Examples (5)
Scope (21)
Unconstrained univariate polynomial minimization:
Constrained univariate polynomial minimization:
Univariate transcendental minimization:
Univariate piecewise minimization:
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:

The objective function may be unbounded:

There may be no points satisfying the constraints:

Bounded transcendental minimization:
Unconstrained parametric minimization:
Constrained parametric minimization:
Polynomial minimization over the integers:
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 the exact solution takes a long time:
With WorkingPrecision->100, you get an exact minimum value, but it might be incorrect:
Applications (10)
Basic Applications (3)
Geometric Distances (6)
The shortest distance of a point in a region ℛ to a given point p and a point q realizing the shortest distance is given by Minimize[EuclideanDistance[p,q],q∈ℛ]. Find the shortest distance and the nearest point to {1,1} in the unit Disk[]:
Find the shortest distance and the nearest point to {1,3/4} in the standard unit simplex Simplex[2]:
Find the shortest distance and the nearest point to {1,1,1} in the standard unit sphere Sphere[]:
Find the shortest distance and the nearest point to {-1/3,1/3,1/3} in the standard unit simplex Simplex[3]:
The nearest points p∈ and q∈ and their distance can be found through Minimize[EuclideanDistance[p,q],{p∈,q∈}]. Find the nearest points in Disk[{0,0}] and Rectangle[{3,3}] and the distance between them:
Find the nearest points in Line[{{0,0,0},{1,1,1}}] and Ball[{5,5,0},1] and the distance between them:
Geometric Centers (1)
If ℛ⊆n is a region that is full dimensional, then the Chebyshev center is the center of the largest inscribed ball of ℛ. The center and the radius of the largest inscribed ball of ℛ can be found through Minimize[SignedRegionDistance[ℛ,p], p∈ℛ]. Find the Chebyshev center and the radius of the largest inscribed ball for Rectangle[]:
Find the Chebyshev center and the radius of the largest inscribed ball for Triangle[]:
Properties & Relations (6)
Minimize gives an exact global minimum of the objective function:
NMinimize attempts to find a global minimum numerically, but may find a local minimum:
FindMinimum finds local minima 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, Minimize may give a point on the boundary:

Here the objective function tends to the minimum value when y tends to infinity:

Minimize can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
This computes the minimum value:
Use RegionDistance and RegionNearest to compute the distance and the nearest point:
Both can be computed using Minimize:
Use RegionBounds to compute the bounding box:
Possible Issues (1)
Minimize 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 (2003), Minimize, Wolfram Language function, https://reference.wolfram.com/language/ref/Minimize.html (updated 2014).
BibTeX
BibLaTeX
CMS
Wolfram Language. 2003. "Minimize." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Minimize.html.
APA
Wolfram Language. (2003). Minimize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Minimize.html