MinValue

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:

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.