FindArgMin

FindArgMin[f,x]

gives the position x_min of a local minimum of f.

FindArgMin[f,{x,x₀}]

gives the position x_min of a local minimum of f, found by a search starting from the point x=x₀.

FindArgMin[f,{{x,x₀},{y,y₀},…}]

gives the position {x_min,y_min,…} of a local minimum of a function of several variables.

FindArgMin[{f,cons},{{x,x₀},{y,y₀},…}]

gives the position of a local minimum subject to the constraints cons.

FindArgMin[{f,cons},{x,y,…}]

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

Details and Options

FindArgMin[…,{x,y,…}] is effectively equivalent to {x,y,…}/.Last[FindMinimum[…,{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.
The constraints cons can be any logical combination of:
lhs==rhs equations

lhs>rhs or lhs>=rhs inequalities

{x,y,…}∈reg region specification
FindArgMin first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
FindArgMin has attribute HoldAll, and effectively uses Block to localize variables.
FindArgMin[f,{x,x₀,x₁}] searches for a local minimum in f using x₀ and x₁ as the first two values of x, avoiding the use of derivatives.
FindArgMin[f,{x,x₀,x_min,x_max}] searches for a local minimum, 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 FindArgMin may correspond only to local, but not global, minima.
By default, all variables are assumed to be real.
For linear f and cons, x∈Integers can be used to specify that a variable can take on only integer values.
FindArgMin takes the same options as FindMinimum.

List of all options

AccuracyGoal	Automatic	the accuracy sought
EvaluationMonitor	None	expression to evaluate whenever f is evaluated
Gradient	Automatic	the list of gradient components for f
MaxIterations	Automatic	maximum number of iterations to use
Method	Automatic	method to use
PrecisionGoal	Automatic	the precision sought
StepMonitor	None	expression to evaluate whenever a step is taken
WorkingPrecision	MachinePrecision	the precision used in internal computations

Examples

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Basic Examples (4)

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

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

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

Find a minimizer point in a geometric region:

Plot it:

Scope (12)

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

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

Location of a local minimum 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 minimum in a region:

Plot it:

Find the minimum distance between two regions:

Plot it:

Find the minimum such that the triangle and ellipse still intersect:

Plot it:

Find the disk of minimum radius that contains the given three points:

Plot it:

Using Circumsphere gives the same result directly:

Use to specify that is a vector in :

Find the minimum distance between two regions:

Plot it:

Options (7)

AccuracyGoal & PrecisionGoal (2)

This enforces convergence criteria $TemplateBox[{{{x, _, k}, -, {x, ^, *}}}, Norm]<=max(10^(-9),10^(-8)TemplateBox[{{x, _, k}}, Norm])$ and $TemplateBox[{{del , {f, (, {x, _, k}, )}}}, Norm]<=10^(-9)$ :

This enforces convergence criteria $TemplateBox[{{{x, _, k}, -, {x, ^, *}}}, Norm]<=max(10^(-20),10^(-18)TemplateBox[{{x, _, k}}, Norm])$ and $TemplateBox[{{del , {f, (, {x, _, k}, )}}}, Norm]<=10^(-20)$ :