# 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[,xreg]

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==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, MinValue will always find a global minimum.
• MinValue[{f,cons},xreg] is effectively equivalent to MinValue[{f,consxreg},x].
• For xreg, 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.
• xIntegers 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:

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Find the minimum value of a multivariate function:

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Find the minimum value of a function subject to constraints:

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Find the minimum value as a function of parameters:

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

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