Built-in Mathematica Symbol

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[{f, cons}, {x, y, ...}, dom]
gives the minimum value of f over the domain dom, typically Reals or Integers.

MORE INFORMATION

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.

If f and cons are linear or polynomial, MinValue will always find a global minimum.

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.

EXAMPLES

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Basic Examples (4)

Find the minimum value of a univariate function:

Find the minimum value of a multivariate function:

Find the minimum value of a function subject to constraints:

Find the minimum value as a function of parameters:

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:

In[1]:=

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Find the minimum value as a function of parameters:

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Scope (15)

Unconstrained univariate polynomial minimization:

Constrained univariate polynomial minimization:

Univariate transcendental minimization:

Univariate piecewise minimization:

Multivariate linear constrained minimization:

Linear-fractional constrained minimization:

Unconstrained polynomial minimization:

Constrained polynomial optimization can always be solved:

The minimum value may not be attained:

The objective function may be unbounded:

There may be no points satisfying the constraints:

Algebraic minimization:

Bounded transcendental minimization:

Piecewise minimization:

Unconstrained parametric minimization:

Constrained parametric minimization:

Integer linear programming:

Polynomial minimization over the integers:

Options (1)

Finding the exact minimum takes a long time due to high-degree algebraic number operations:

With WorkingPrecision->100, we get an exact minimum value, but it might be incorrect:

Applications (3)

Find the minimal perimeter among rectangles with a unit area:

Find the minimal perimeter among triangles with a unit area:

Find the distance to a parabola from a point on its axis:

Assuming a particular relationship between the

and

parameters:

Properties & Relations (3)

Minimize gives both the value of the minimum and the minimizer point:

For strict polynomial inequality constraints MinValue may be much faster than Minimize:

MinValue gives an exact global minimum value of the objective function:

NMinValue attempts to find a global minimum numerically, but may find a local minimum:

FindMinValue finds local minima depending on the starting point:

MinValue can solve linear programming problems:

LinearProgramming can be used to solve the same problem given in matrix notation:

Possible Issues (1)

MinValue requires that all functions present in the input be real-valued:

Values for which the equation is satisfied but the square roots are not real are disallowed: