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# 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[{f, cons}, {x, y, ...}, dom]minimizes with variables over the domain dom, typically Reals or Integers.
• Minimize returns a list of the form {fmin, {x->xmin, y->ymin, ...}}.
• cons can contain equations, inequalities or logical combinations of these.
• If f and cons are linear or polynomial, Minimize will always find a global minimum.
• 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.
• can be used to specify that a particular variable can take on only integer values.
• Even if the same minimum is achieved at several points, only one is returned.
Minimize a univariate function:
Minimize a multivariate function:
Minimize a function subject to constraints:
A minimization problem containing parameters:
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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 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 solution 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:
The minimal perimeter triangle is equilateral:
Find the distance to a parabola from a point on its axis:
Assuming a particular relationship between the and parameters:
Minimize gives an exact global minimum of the objective function:
NMinimize attempts to find a global minimum numerically, but may find a local minimum:
FindMinimum finds local minima depending on the starting point:
The minimum point satisfies the constraints, unless messages say otherwise:
The given point minimizes the distance from the point :
When the minimum is not attained, Minimize may give a point on the boundary:
Here the objective function tends to the minimum value when y tends to infinity:
Minimize can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
This computes the minimum value:
Minimize 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: