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:

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Minimize a multivariate function:

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Minimize a function subject to constraints:

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A minimization problem containing parameters:

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Minimize a function over a geometric region:

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Plot it:

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Scope  (21)

Options  (1)

Applications  (10)

Properties & Relations  (6)

Possible Issues  (1)

See Also

MinValue  ArgMin  Maximize  NMinimize  FindMinimum  Min  MinimalBy  D  FindInstance  LinearProgramming  LeastSquares  RegionDistance

Tutorials

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