(* finding local minima for a function of two variables *)
(* To run on command line:  math -run "<<fmin2Dglobal.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 "RandomSearch",
 *) 

(* Define range [a, b] *) 

a=-2.0; 
b= 2.0; 


x0 = 2.0; (* Iterations start at x0 *)

Nstrt = 5; (* number of random start points for the random search method *)
 

(* F[x_, y_] := -Cos[5*x]*Cos[5*x]*Cos[5*y]*Cos[5*y]*Exp[-3*(x^2 + y^2)];
*)  

(* F(x,y) designed to have many local minima, and 
its global min. exactly at (0,0), value = -1 at min. *) 
   
F[x_, y_] := -(Cos[5*x]*Cos[5*y])^2*Exp[-3*(x^2 + y^2)];


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

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

 

answer = NMinimize[F[x, y], {x,y}, AccuracyGoal -> 2, Method -> {"RandomSearch",
Method -> "Gradient", "SearchPoints" -> Nstrt}]


xmin =  x /. answer[ [2] ]
ymin = y /. answer[ [2] ]
Fmin = answer[ [1] ]


pts = Reap[
    NMinimize[F[x, y], {x, y}, AccuracyGoal -> 2, Method ->
{"RandomSearch", Method -> "Gradient", "SearchPoints" -> Nstrt}, 
     StepMonitor :> Sow[{x, y}]]][[2, 1]];
pts = Join[{{-1.2, 1}}, pts];

ContourPlot[F[x, y], {x, -2, 2}, {y, -3, 3}, 
 Epilog -> {Arrow[pts], Point[pts]}, Contours -> 10, 
 ColorFunction -> (Hue[.6 (1 - #1)] &), ColorFunctionScaling -> False]



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