(* finding local minima for a function of two variables *)
(* To run on command line:  math -run "<<fmin2D.math" *)
(* The above line assumes that math kernel is aliased to "math" *)
(* For example, on my machine alias
math='/Applications/Mathematica.app/Contents/MacOS/MathKernel' *)
(* or simply use << fmin2D.math once math kernel is already running *)

<<JavaGraphics`

<<Optimization`UnconstrainedProblems`


(*FindMinimum[] is designed for local optimization.   *) 

(*  Value of option Method 
     is either "Automatic", "Gradient", "ConjugateGradient", "QuasiNewton",
     "Newton", or "LevenbergMarquardt".
 *) 

(* Define range [a, b] *) 

a=-2.0; 
b= 2.0; 

(* Iterations start at x0 *) 

x0 = 2.0; 



 
F[x_, y_] := x^2 + y^2; (* simplest possible, quadratic "wine bowl" *)


  
 
(* F(x,y) below designed to have many local minima, and
its global min. exactly at (0,0), value = -1 at min. *)

(*
F[x_, y_] := -Cos[30*x]*Cos[30*x]*Cos[30*y]*Cos[30*y]*Exp[-3*(x^2 + y^2)];  

*)  


(*

F[x_, y_] := 10000(y - x^2)^2 + (1-x)^2; (* Steepest descent (Gradient) fails *)

*)
    

					(* Need ConjugateGradient or Newton *)

ContourPlot[F[x, y], {x, a, b}, {y, a, b}, ContourShading->False]

Plot3D[F[x, y], {x, a, b}, {y, a, b}]

 

answer = FindMinimum[F[x, y], {x, x0}, {y, x0}, AccuracyGoal -> 2,
MaxIterations -> 50, Method->Gradient ] 

 

FindMinimumPlot[F[x, y], {{x, x0}, {y, x0}}, AccuracyGoal -> 2,
MaxIterations -> 50, Method->Gradient]
 
xmin =  x /. answer[ [2] ] 
ymin = y /. answer[ [2] ]
Fmin = answer[ [1] ]




Print["  "]
Print["Values of x,y that minimizes F, (xmin, ymin) =   ", xmin, "   ", ymin]
Print["  "]
Print["                          F(xmin, ymin)=     ", Fmin]
Print[" "]