Built-in Mathematica Symbol

FindArgMax

FindArgMax[f, x]
gives the position x_max of a local maximum of f.

FindArgMax[f, {x, x₀}]
gives the position x_max of a local maximum of f, found by a search starting from the point x=x₀.

FindArgMax[f, {{x, x₀}, {y, y₀}, ...}]
gives the position {x_max, y_max, ...} of a local maximum of a function of several variables.

FindArgMax[{f, cons}, {{x, x₀}, {y, y₀}, ...}]
gives the position of a local maximum subject to the constraints cons.

FindArgMax[{f, cons}, {x, y, ...}]
starts from a point within the region defined by the constraints.

MORE INFORMATION

FindArgMax[..., {x, y, ...}] is effectively equivalent to {x, y, ...}/.Last[FindMaximum[..., {x, y, ...}, ...].

If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.

cons can contain equations, inequalities or logical combinations of these.

FindArgMax first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.

FindArgMax has attribute HoldAll, and effectively uses Block to localize variables.

FindArgMax[f, {x, x₀, x₁}] searches for a local maximum in f using x₀ and x₁ as the first two values of x, avoiding the use of derivatives.

FindArgMax[f, {x, x₀, x_min, x_max}] searches for a local maximum, stopping the search if x ever gets outside the range x_min to x_max.

Except when f and cons are both linear, the results found by FindArgMax may correspond only to local, but not global, maxima.

By default, all variables are assumed to be real.

For linear f and cons, xIntegers can be used to specify that a variable can take on only integer values.

FindArgMax takes the same options as FindMaximum.

EXAMPLES

CLOSE ALL

Basic Examples (3)

Find a point {x} at which the univariate function -2x^2-3x+5 has a maximum:

Find a point {x, y} at which the function Sin[x]Sin[2y] has a maximum:

Find a point at which a function is a maximum subject to constraints:

Find a point {x} at which the univariate function -2x^2-3x+5 has a maximum:

In[1]:=

Out[1]=

Find a point {x, y} at which the function Sin[x]Sin[2y] has a maximum:

In[1]:=

Out[1]=

Find a point at which a function is a maximum subject to constraints:

In[1]:=

Out[1]=

Scope (6)

With different starting points, get the locations of different local maxima:

Location of a local maximum of a two-variable function starting from x=2, y=2:

Location of a local maximum constrained within a disk:

Starting point does not have to be provided:

For linear objective and constraints, integer constraints can be imposed:

Or constraints can be specified:

Options (7)

This enforces convergence criteria

and

:

This enforces convergence criteria

and

:

Setting a high WorkingPrecision makes the process convergent:

Plot convergence to the local maximum:

Use a given gradient; the Hessian is computed automatically:

Supply both gradient and Hessian:

In this case the default derivative-based methods have difficulties:

Direct search methods which do not require derivatives can be helpful in these cases:

NMaximize also uses a range of direct search methods:

Steps taken by FindArgMax in finding the maximum of a function:

Set the working precision to

; by default AccuracyGoal and PrecisionGoal are set to

:

Properties & Relations (1)

FindMaximum gives both the value of the maximum and the maximizer point:

FindArgMax gives the location of the maximum:

FindMaxValue gives the value at the maximum:

Possible Issues (4)

If the constraint region is empty, the algorithm will not converge:

If the maximum value is not finite, the algorithm will not converge:

Integer linear programming algorithm is only available for machine-number problems:

Sometimes providing a suitable starting point can help the algorithm to converge: