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# ArgMin

 ArgMin[f, x]gives a position xmin at which f is minimized. ArgMin[f, {x, y, ...}]gives a position {xmin, ymin, ...} at which f is minimized. ArgMin[{f, cons}, {x, y, ...}]gives a position at which f is minimized subject to the constraints cons. ArgMin[{f, cons}, {x, y, ...}, dom]gives a position at which f is minimized over the domain dom, typically Reals or Integers.
• ArgMin returns a list of the form {xmin, ymin, ...}.
• ArgMin[..., {x, y, ...}, ...] is effectively equivalent to {x, y, ...}/.Last[Minimize[..., {x, y, ...}, ...].
• cons can contain equations, inequalities or logical combinations of these.
• If f and cons are linear or polynomial, ArgMin will always find a global minimum.
• ArgMin will return exact results if given exact input.
• If ArgMin is given an expression containing approximate numbers, it automatically calls NArgMin.
• If the minimum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, ArgMin will return 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.
Find a minimizer point for a univariate function:
Find a minimizer point for a multivariate function:
Find a minimizer point for a function subject to constraints:
Find a minimizer point as a function of parameters:
Find a minimizer point for a univariate function:
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Find a minimizer point for a multivariate function:
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Find a minimizer point for a function subject to constraints:
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Find a minimizer point 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:
Algebraic minimization:
Bounded transcendental minimization:
Piecewise minimization:
Unconstrained parametric minimization:
Constrained parametric minimization:
Integer linear programming:
Polynomial minimization over the integers:
 Options   (1)
Finding an exact minimum point can take a long time:
With WorkingPrecision->100, we get an approximate minimum point:
 Applications   (3)
Find the lengths of sides of a unit area rectangle with minimal perimeter:
Find the lengths of sides of a unit area triangle with minimal perimeter:
The minimal perimeter triangle is equilateral:
Find a point on a parabola closest to its axis:
Assuming a particular relationship between the and parameters:
Minimize gives both the value of the minimum and the minimizer point:
ArgMin gives an exact global minimizer point:
NArgMin attempts to find a global minimizer point numerically, but may find a local minimizer:
FindArgMin finds a local minimizer point 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, ArgMin may give a point on the boundary:
Here the objective function tends to the minimum value when y tends to infinity:
ArgMin can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
A finite minimum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:
ArgMin 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:
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