(* 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`



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

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

  
(* 

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, -1, 1}, {y, -1, 1}, ContourShading->False]

Plot3D[F[x, y], {x, -1, 1}, {y, -1, 1}]


answer = FindMinimum [F[x, y], {x, 2}, {y, 2}, 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[" "]