Minimizer

Function Minimizer

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creator: Peter Knipp

Background:

This API finds a local minimum of a multidimensional function if given the function's string representation. It uses Nelder-Mead method , as presented on page 292 of Chapter 10 of the 1987 edition of Numerical Recipes, by William Press, Brian Flannery, Saul Teukolsky, and william Vetterling. In addition to solving this optimization problem, this API also performs two more basic tasks that require the parsing of a string: expression evaluation and function evaluation .

Params:

:exprStr is is a string representation of a valid arithmetic expression. As such it contains numbers (including PI and E), parentheses, binary operations (addition, subtraction, multiplication, division, and exponentiation), and unary operations (sin, sqrt, log, etc). Addition and subtraction are simply represented by + and -, respectively. Multiplication is represented by * or nothing (for the case of implied multiplication). Division is represented by DIV or D, owing to the fact that the usual symbol for division (/) has a special meaning in a url. Exponentiation may be represented by either ** or ^, but the former is preferable owing to the fact that the latter will be replaced by %5E in the url. The unary may be any of the static Math methods listed in the javascript docs. Spaces are allowed but discouraged.
:vars is is a bracket-delimited list of strings, each of which is the name for a variable which appears in a function (described below). Each string satisfies the same rules as javascript variables. The first character must be either a letter (upper-case or lower-case) or underscore. Any subsequent character may be that or a numeral. No variable name can be a substring of the name of a unary (for which the previous bullet includes a link to a list).
:fnStr is a string that defines a function. It includes expressions and variable names, as defined above.
:coords is a bracket-delimited list of the coordinates of a point (ie, independent variables) where a function is to be evaluated.
:simplex is the list of :coords that constitute a simplex, a data structure used for function minimization. A simplex contains one more point than the dimensionality of the space: a segment in one dimension, a triangle in two dimensions, or a tetrahedron in three. This param is optional, in that you can replace it by random if you are willing to allow the API to use a random simplex.
:maxIter (optional) allows the user to increase above 500 the maximum number of iterations of the Nelder-Mead method, as is sometimes necessary in order to converge to an answer.

Endpoints:

task	url fragment(s)	example(s)
evaluates an expression	`/evaluate-expression/:exprStr`	[6 - 5sin(4)] / (3² + 1)
evaluates a function	`/evaluate-function/:fnStr/:vars/:coords`	[x - 5sin(y)] / (z² + 1) evaluated at (x, y, z) = (6, 4, 3)
minimizes a function	`/minimize/:fnStr/:vars/random`	4x⁴ + 2xy + 3y² + 4x + 5y + 6 minimized with a random simplex
	`/minimize/:fnStr/:vars/:simplex`	same as above, but specifying a simplex
	`/minimize/:fnStr/:vars/random/40`	same as first example, but reducing iteration-limit from 500 to 40

Notes:

If you want the response as json rather than as html, simply insert api after ...heroku.com in the address.
If you want to maximize a function instead of minimize it, of course you simply need to wrap your function string with -(...).