MinValue

MinValue[f,x]

gives the minimum value of f with respect to x.

MinValue[f,{x,y,…}]

gives the minimum value of f with respect to x, y, ….

MinValue[{f,cons},{x,y,…}]

gives the minimum value of f subject to the constraints cons.

MinValue[…,x∈reg]

constrains x to be in the region reg.

MinValue[…,…,dom]

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

Details and Options

  • MinValue[…] is effectively equivalent to First[Minimize[…]].
  • MinValue gives the infimum of values of f. It may not be attained for any values of x, y, ….
  • 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, MinValue will always find a global minimum.
  • MinValue[{f,cons},x∈reg] is effectively equivalent to MinValue[{f,cons∧x∈reg},x].
  • For x∈reg, the different coordinates can be referred to using Indexed[x,i].
  • MinValue will return exact results if given exact input.
  • If MinValue is given an expression containing approximate numbers, it automatically calls NMinValue.
  • 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, MinValue returns Infinity.
  • N[MinValue[…]] calls NMinValue for optimization problems that cannot be solved symbolically.

Examples

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Basic Examples  (5)

Find the minimum value of a univariate function:

Find the minimum value of a multivariate function:

Find the minimum value of a function subject to constraints:

Find the minimum value as a function of parameters:

Find the minimum value of a function over a geometric region:

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:

Find the minimum value of a function over a geometric 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 minimum radius of a disk that contains the given three points:

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 minimum takes a long time:

With WorkingPrecision->100, the result is an exact minimum value, but it might be incorrect:

Applications  (9)

Basic Applications  (3)

Find the minimal perimeter among rectangles with a unit area:

Find the minimal perimeter among triangles with a unit area:

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

Assuming a particular relationship between the and parameters:

Geometric Distances  (6)

The distance of a point p to a region ℛ is given by MinValue[EuclideanDistance[p,q],q∈ℛ]. Find the distance of {1,1} to the unit Disk[]:

Plot it:

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

Plot it:

Find the distance of the point {1,1,1} to the standard unit sphere Sphere[]:

Plot it:

Find the distance of the point {-1/3,1/3,1/3} to the standard unit simplex Simplex[3]:

Plot it:

The distance between regions and can be found through MinValue[EuclideanDistance[p,q],{p∈,q∈}]. Find the distance between Disk[{0,0}] and Rectangle[{3,3}]:

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

Properties & Relations  (5)

Minimize gives both the value of the minimum and the minimizer point:

MinValue gives an exact global minimum value of the objective function:

NMinValue attempts to find a global minimum numerically, but may find a local minimum:

FindMinValue finds local minima depending on the starting point:

MinValue can solve linear programming problems:

LinearProgramming can be used to solve the same problem given in matrix notation:

Use RegionDistance to compute the minimum distance from a point to a region:

Compute the distance using MinValue:

Use RegionBounds to compute the bounding box:

Use MaxValue and MinValue to compute the same bounds:

Possible Issues  (1)

MinValue 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 2008
 (7.0)
 |
Updated in 2014
 (10.0)