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[,xreg]
constrains x to be in the region reg.

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

Details and OptionsDetails and Options

  • Minimize returns a list of the form .
  • cons can contain equations, inequalities, or logical combinations of these.
  • The constraints cons can be any logical combination of:
  • lhs==rhsequations
    lhs!=rhsinequations
    or 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},xreg] is effectively equivalent to Minimize[{f,consxreg},x].
  • For , 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.
  • xIntegers 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. »

ExamplesExamplesopen allclose all

Basic Examples  (5)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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Introduced in 2003
(5.0)
| Updated in 2014
(10.0)