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# FindMinValue

 FindMinValue[f, x]gives the value at a local minimum of f. FindMinValue[f, {x, x0}]gives the value at a local minimum of f, found by a search starting from the point x=x0. FindMinValue[f, {{x, x0}, {y, y0}, ...}]gives the value at a local minimum of a function of several variables. FindMinValue[{f, cons}, {{x, x0}, {y, y0}, ...}] gives the value at a local minimum subject to the constraints cons. FindMinValue[{f, cons}, {x, y, ...}]starts from a point within the region defined by the constraints.
• 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.
• FindMinValue first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
• FindMinValue[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.
• FindMinValue[f, {x, x0, xmin, xmax}] searches for a local minimum, 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 FindMinValue may correspond only to local, but not global, minima.
• By default, all variables are assumed to be real.
• For linear f and cons, can be used to specify that a variable can take on only integer values.
Find a minimum value of the univariate function:
Find a minimum value of a multivariate function::
Find the minimum value of a function subject to constraints:
Find a minimum value of the univariate function:
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Find a minimum value of a multivariate function::
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Find the minimum value of a function subject to constraints:
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 Scope   (6)
With different starting points, get the values of different local minima:
Value at a local minimum of a two-variable function starting from x=2, y=2:
Value at 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:
 Options   (8)
This enforces convergence criteria and :
This enforces convergence criteria and :
Setting a high WorkingPrecision makes the process convergent:
Plot convergence to the local minimum:
In this case the default derivative-based methods have difficulties:
Direct search methods which do not require derivatives can be helpful in these cases:
NMinimize also uses a range of direct search methods:
Steps taken by FindMinValue in finding the minimum of a function:
Set the working precision to ; by default AccuracyGoal and PrecisionGoal are set to :
FindMinimum gives both the value of the minimum and the minimizing argument:
FindArgMin gives the location of the minimum as a list:
FindMinValue gives the value at the minimum:
If the constraint region is empty, the algorithm will not converge:
If the minimum 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:
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