# 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 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},xreg] is effectively equivalent to Minimize[{f,consxreg},x].
• For xreg, 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. »

# 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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## Possible Issues(1)

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