In the previous article, the basic principle of simulation annealing is described. The following will be described below with an actual example, where all source code has been posted, and you can learn a lot of details.
Use analog annealing method F (X, Y) = 5sin (xy) x2 y2 minimum
Solution: According to the question, we design the cooling table schedule:
That is, the initial temperature is 100
Attenuation parameters are 0.95
The length of the Marco chain is 10000
Metropolis' s steps are 0.02
The end condition is based on the difference between the latest optimal solution and the latest optimal solution less than a certain tolerance.
Use the Metropolis acceptance criteria to simulate, the program is as follows
/ *
* Simulated annealing method function f (x, y) = 5sin (xy) x ^ 2 y ^ 2 minimum
* Date: 2004-4-16
* Author: ARMYLAU
* Email: armylulau2@163.com
* The end condition is the difference between the two optimal solutions less than a small amount.
* /
Using system;
Namespace Simulateanneland
{
Class class1
{
/ / Target function to require an optimal value
Static Double ObjectFunction (Double X, Double Y)
{
Double Z = 0.0;
Z = 5.0 * math.sin (x * y) x * x y * y;
Return Z;
}
[Stathread]
Static void
Main
(String [] ARGS)
{
/ / Searching the maximum interval
Const Double XMAX = 4;
Const Double Ymax = 4;
// Cooling table parameters
INT MARKOVLENGTH = 10000; // Marco chain length
Double decayscale = 0.95; // Attenuation parameters
Double stepFactor = 0.02; // Step length factor
Double Temperature = 100; // Initial temperature
Double tolerance = 1e-8; // tolerance
Double prex, Next;;; x x
Double prey, nexty; // prior and next value of y
Double prebestx, prebesty; // Previous optimal solution
Double Bestx, Besty; // Final Solution
Double Acceptpoints = 0.0; // General Accepted Point in the metropolis
Random rnd = new random ();
// Random selection
Prex = -Xmax * rnd.nextdouble ();
PREY = -ymax * rnd.nextdouble ();
Prebestx = bestx = prex;
Prebesty = BESTY = PREY;
/ / Each iteration is annealed once (cool down) until it is satisfied
DO
{
Temperature * = decayscale;
Acceptpoints = 0.0;
// Detert LOOP (ie Markov Chain Length) at the current temperature t
For (int i = 0; i { // 1) Randomly select a little near this point DO { Nextx = prex stepfactor * xmax * (rnd.nextdouble () - 0.5); Next = prey stepfactor * ymax * (rnd.nextdouble () - 0.5); } While (! (NEXTX> = -Xmax && nextx <= xmax && nexty> = -ymax && nexty <= ymax); // 2) Whether or not the global optimal solution IF (objectFunction> ObjectFunction (nextx, nexty)) { / / Keep the previous best solution Prebestx = BESTX; Prebesty = BESTY; // This is the new best solution BESTX = NEXTX; Besty = nexty; } // 3) Metropolis process IF (ObjectFunction - ObjectFunction (Nextx, Next)> 0) { // Accepted, here LastPoint is the next iteration point at the beginning of the newly accepted point Prex = nextx; Prey = nexty; AcceptPoints ; } Else { Double change = -1 * (objectFunction (nextx, nexty) - ObjectFunction (prex, prey)) / temperature IF (Math.exp (Change)> rnd.nextdouble ()) { Prex = nextx; Prey = nexty; AcceptPoints ; } // Do not accept, save the original solution } } Console.writeline ("{0}, {1}, {2}, {3}", prex, pregra, ibjectfunction (prex, prey), temperature; WHIL (Math.Abs (objectFunction (Bestx, Besty) - ObjectFunction (Prebestx, Prebesty)> Tolerance); Console.writeline ("minimum at point: {0}, {1}", bestx, besty; Console.Writeline ("Minimum value: {0}", ObjectFunction (Bestx, Best)); } } } l Results: Minimum value at the point: -1.07678129318956, 1.07669421564618 The minimum is: -2.26401670947686 l Postscript: It turns out that a series of articles, introduces how to use C # decodes, because there are very few numerical calculations on 9CBS, so I hope I can add. Attack in the online search to simulate annealing data and want to be an example of C # value calculation, no ready-to-range source code. Later, I have been experimenting for a long time, I finally wrote this procedure, I didn't dare to hide, and took out an entry example of an annealing or use C # decoding value algorithm. This article tries to avoid too academicization, such as mathematics and physical names and formulas, rushing the pen, there are many places that may not be very clear, I hope you understand. Any questions or criticism, Email and me: armylau2@163.com In addition, analog annealing can also be applied to other more complex issues, such as combined optimization issues such as Salesman issues. This example is just a minimum problem with a two-dimensional function, and its choice of cooling table parameters is also too simple, only a preliminary entry profile, please pay attention. l Reference: 1. Http://www.computer-dictionary-online.org/index.asp?q=simulated annealing computer dictionary 2. Numeric Recipes in C 3. Calculation Method Book Non-Numerical Parallel Algaith (Volume 1) Simulation Annealing Algorithm
