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.

Minimize[…,…,dom]

constrains variables to the domain dom, typically Reals or Integers.

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==rhsequations
    lhs!=rhsinequations
    lhs>rhs or lhs>=rhs inequalities
    {x,y,…}∈regregion 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

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Basic Examples  (5)

Minimize a univariate function:

Minimize a multivariate function:

Minimize a function subject to constraints:

A minimization problem containing parameters:

Minimize a function over a geometric region:

Plot it:

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:

Algebraic minimization:

Bounded transcendental minimization:

Piecewise minimization:

Unconstrained parametric minimization:

Constrained parametric minimization:

Integer linear programming:

Polynomial minimization over the integers:

Minimize over a region:

Plot it:

Find the minimum distance between two regions:

Plot it:

Find the minimum such that the triangle and ellipse still intersect:

Plot it:

Find the disk of minimum radius that contains the given three points:

Plot it:

Using Circumsphere gives the same result directly:

Use to specify that is a vector in :

Find the minimum distance between two regions:

Plot it:

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)

Find the minimal perimeter among rectangles with a unit area:

Find the minimal perimeter among triangles with a unit area:

The minimal perimeter triangle is equilateral:

Find the distance to a parabola from a point on its axis:

Assuming a particular relationship between the and parameters:

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[]:

Plot it:

Find the shortest distance and the nearest point to {1,3/4} in the standard unit simplex Simplex[2]:

Plot it:

Find the shortest distance and the nearest point to {1,1,1} in the standard unit sphere Sphere[]:

Plot it:

Find the shortest distance and the nearest point to {-1/3,1/3,1/3} in the standard unit simplex Simplex[3]:

Plot it:

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:

Plot it:

Find the nearest points in Line[{{0,0,0},{1,1,1}}] and Ball[{5,5,0},1] and the distance between them:

Plot it:

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:

Use Maximize and Minimize to compute the same bounds:

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:

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