Built-in Mathematica Symbol

ArgMax

ArgMax[f, x]
gives a position x_max at which f is maximized.

ArgMax[f, {x, y, ...}]
gives a position {x_max, y_max, ...} at which f is maximized.

ArgMax[{f, cons}, {x, y, ...}]
gives a position at which f is maximized subject to the constraints cons.

ArgMax[{f, cons}, {x, y, ...}, dom]
gives a position at which f is maximized over the domain dom, typically Reals or Integers.

MORE INFORMATION

ArgMax[..., vars, ...] is effectively equivalent to vars/.Last[Maximize[..., vars, ...].

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.

xIntegers can be used to specify that a particular variable can take on only integer values.

If the constraints cannot be satisfied, ArgMax returns {Indeterminate, Indeterminate, ...}.

EXAMPLES

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Basic Examples (4)

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:

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Find a maximizer point for a multivariate function:

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Find a maximizer point for a function subject to constraints:

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Find a maximizer point as a function of parameters:

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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->200, we 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:

Properties & Relations (4)

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:

Possible Issues (2)

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: