This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ArgMax

 ArgMaxgives a position at which f is maximized. ArgMaxgives a position at which f is maximized. ArgMaxgives a position at which f is maximized subject to the constraints cons. ArgMaxgives a position at which f is maximized over the domain dom, typically Reals or Integers.
• cons can contain equations, inequalities, or logical combinations of these.
• If f and cons are linear or polynomial, ArgMax will always find a global maximum.
• ArgMax will return exact results if given exact input.
• If ArgMax is given an expression containing approximate numbers, it automatically calls NArgMax.
• If the maximum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, ArgMax will return the closest specifiable point.
• If no domain is specified, all variables are assumed to be real.
• Integers can be used to specify that a particular variable can take on only integer values.
• If the constraints cannot be satisfied, ArgMax returns .
Find a maximizer point for a univariate function:
Find a maximizer point for a multivariate function:
Find a maximizer point for a function subject to constraints:
Find a maximizer point as a function of parameters:
Find a maximizer point for a univariate function:
 Out[1]=

Find a maximizer point for a multivariate function:
 Out[1]=

Find a maximizer point for a function subject to constraints:
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Find a maximizer point as a function of parameters:
 Out[1]=
 Scope   (15)
Unconstrained univariate polynomial maximization:
Constrained univariate polynomial maximization:
Univariate transcendental maximization:
Univariate piecewise maximization:
Multivariate linear constrained maximization:
Linear-fractional constrained maximization:
Unconstrained polynomial maximization:
Constrained polynomial optimization can always be solved:
Algebraic maximization:
Bounded transcendental maximization:
Piecewise maximization:
Unconstrained parametric maximization:
Constrained parametric maximization:
Integer linear programming:
Polynomial maximization over the integers:
 Options   (1)
Finding an exact maximum point can take a long time:
With WorkingPrecision, you get an approximate maximum point:
 Applications   (3)
Find the lengths of sides of a unit perimeter rectangle with the maximal area:
Find the lengths of sides of a unit perimeter triangle with the maximal area:
Find the time at which a projectile reaches the maximum height:
Maximize gives both the value of the maximum and the maximizer point:
ArgMax gives an exact global minimizer point:
NArgMax attempts to find a global maximizer numerically, but may find a local maximizer:
FindArgMax finds a local maximizer point depending on the starting point:
The maximum point satisfies the constraints, unless messages say otherwise:
The given point maximizes the distance from the point .
When the maximum is not attained, ArgMax may give a point on the boundary:
Here the objective function tends to the maximum value when y tends to infinity:
ArgMax can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
The maximum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:
ArgMax 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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