# FindMaxValue

FindMaxValue[f,x]

gives the value at a local maximum of f.

FindMaxValue[f,{x,x0}]

gives the value at a local maximum of f, found by a search starting from the point x=x0.

FindMaxValue[f,{{x,x0},{y,y0},}]

gives the value at a local maximum of a function of several variables.

FindMaxValue[{f,cons},{{x,x0},{y,y0},}]

gives the value at a local maximum subject to the constraints cons.

FindMaxValue[{f,cons},{x,y,}]

starts from a point within the region defined by the constraints.

# Details and Options • FindMaxValue[] is effectively equivalent to First[FindMaximum[]].
• 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.
• The constraints cons can be any logical combination of:
•  lhs==rhs equations lhs>rhs or lhs>=rhs inequalities {x,y,…}∈reg region specification
• FindMaxValue first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
• FindMaxValue has attribute HoldAll, and effectively uses Block to localize variables.
• FindMaxValue[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.
• FindMaxValue[f,{x,x0,xmin,xmax}] searches for a local maximum, stopping the search if x ever gets outside the range xmin to xmax.
• Except when f and cons are both linear, the results found by FindMaxValue 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.
• FindMaxValue takes the same options as FindMaximum.

# Examples

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## Basic Examples(4)

Find a maximum value of a univariate function:

Find a maximum value of a multivariate function:

Find a maximum value of a function subject to constraints:

Find a maximum value of a function in a geometric region:

## Scope(12)

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

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

Value at 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:

Find a maximum value of a function in a geometric region:

Plot it:

Find the maximum distance between points in two regions:

Find the maximum such that the rectangle and ellipse still intersect:

Plot it:

Find the maximum for which contains the given three points:

Use to specify that is a vector in :

Find the maximum distance between points in two regions:

## Options(7)

### AccuracyGoal & PrecisionGoal(2)

This enforces convergence criteria and :

This enforces convergence criteria and : Setting a high WorkingPrecision makes the process convergent:

### EvaluationMonitor(1)

Plot convergence to the local minimum:

Use a given gradient; the Hessian is computed automatically:

### Method(1)

In this case the default derivative-based methods have difficulties: Direct search methods that do not require derivatives can be helpful in these cases:

NMaximize also uses a range of direct search methods:

### StepMonitor(1)

Steps taken by FindMaxValue in finding the minimum of a function:

### WorkingPrecision(1)

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 minimizing argument:

FindArgMax gives the location of the maximum as a list:

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: