The following program calculates the minimum point of a multi-variable function using random search method. This method starts with an initial point and a search radius for each variable. The algorithm searches for a better optimum point within the search radius. This approach allows the search to drift towards the optimum point without being confined to absolute search ranges.

The program prompts you to enter:

1. The initial value for each variable

2. The search radius for each variable

3. The tolerance for the function value

The program displays the following results:

1. The coordinates of the minimum value.

2. The minimum function value.

3. The number of iterations

Here is a sample session to find the minimum of f(x) = 10 + (x1^2-1)^2 + (x2^2-2)^2 using the initial point of [0, 0], search radius of [1,1], and function tolerance of 1e-3. Remember your results are most likely to be different. The output show next is that of the EMU71 emulator for the HP-71B.

The first image shows the input:

The next image shows the output:

Here is the BASIC listing:

10 ! Random Drift Optimization 20 N = 2 30 I9 = 10000 40 RANDOMIZE 50 ! 60 ! This method implements a random walk algorithm. 70 ! The method starts with an initial point and a radius of search 80 ! Random numbers are generated within the radius of search allowing 90 ! for the current optimum to "drift" towards a minimum. 100 ! THIS METHOD HAS NO PREDEFINED RANGES, making it more flexible 110 ! to find an optimum, given enough iterations 120 ! 130 DIM X(N), X9(N), R(N), X0(N) 140 RANDOMIZE 150 For I = 1 To N 160 DISP "X(";I;")"; 170 INPUT X0(I) 180 DISP "Search Radius(";I;")"; 190 INPUT R(I) 200 X9(I) = X0(I) 210 Next I 220 INPUT "Enter function tolerance? "; E 230 ! calculate and display function value at initial point 240 CALL MyFx(X9, N, B) 250 L = SGN(B) * (100 + 100 * B) 260 I0 = 0 270 REM DO 280 I0 = I0 + 1 290 If I0 > I9 Then 470 300 For I = 1 To N 310 X(I) = X9(I) + (Rnd - 0.5) * R(I) 320 CALL MyFx(X, N, F) 330 If F >= B Then 440 340 X9(I) = X(I) 350 L = B 360 B = F 370 DISP "F = "; B;" "; 380 For J = 1 To N 390 DISP "X(";J;")=";X(J);" "; 400 Next J 410 DISP 420 ! test function value convergence 430 If Abs(B - L) < E Then 470 440 REM END_IF 450 Next I 460 GOTO 270 470 REM EXIT_DO 480 DISP "********FINAL RESULTS***********" 490 For I = 1 To N 500 DISP "X(";I;") = ";X9(I) @ PAUSE 510 Next I 520 DISP "Function value =";B @ PAUSE 530 DISP "Number of iterations = "; I0 540 END 550 SUB MyFx(X(), N, F0) 560 F0 = 10 570 For K = 1 To N 580 F0 = F0 + (X(I) ^ 2 - K) ^ 2 590 Next K 600 END SUBYou can customize the above listing by changing the definition of function in the subroutine

f(x1, x2) = 10 + (x1^2-1)^2 + (x2^2+4)^2

The above function has a minimum at x1=+/-1 and x2=+/-1.4142.

**Copyright (c) Namir Shammas. All rights reserved.**