1. First, let's look at the language description of SA to summarize the core idea of simulated annealing (SA:
Appropriate acceptance of poor solutions. Why"
Appropriate"Accept bad solutions? In the process of iteration, a new solution is generated for each iteration, and a new fitness value can be obtained for the new solution, if the new fitness value is smaller than the original fitness value (taking the minimum value as an example), a good solution will be accepted at this time. However, there are also a lot of bad solutions. If a bad solution is fully accepted every time, it is "completely random", and its convergence speed is slow; so it is not good to completely accept the bad solution. The result is that it is easy to fall into a local minimization point, so sa is not a "global optimization algorithm", so we need"
Appropriate"Accept bad solutions to balance the two extremes. What kind of receiving rules are "appropriate"? Take the initial version as an example. P = exp (-delta F/kt)> R, R is a random number between 0 and 1, and K is the Boltzmann constant; delta F is the difference between the fitness value of the first and second generations. This random reception looks good. It should be noted that here t is exponential to the number of iterations: T-> α T, α (0, 1 ). Here are some constants (K, α) and initial values (t) that need to be determined, but these are all related to specific problems. Let's take a look at how the following examples are selected, the initial values of these parameters have a great impact on the optimization problem. 2. after reading the description, let's take a look at the description in the Code Format: after reading the above pseudo code: if we want to improve SA, we can start from the following aspects: 1, how to move randomly to new locations2, P generation strategy, that is, how to change the "appropriate" Policy 3, temperature T drop strategy, that is, Alpha change 3. after reading the description in the form of code, let's look at the implementation of the real code: sa_simpledemo.m
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% Find the minimum of a function by Simulated Annealing
% Programmed by x s Yang (Cambridge University)
% Usage: sa_simpledemo
% Show lab information % %
Disp ('simulating... it will take a minute or so! ');
% Fitness function and its graphics % %
% Rosenbrock test function with f * = 0 at (1, 1)
Fstr = '(3o) ^ 2 + 100 * (Y-x ^ 2) ^ 2 ';
% Convert into an inline function
F = vectorize (Inline (fstr ));
% Show the topography of the objective function
Range = [-2 2-2 2];
Xgrid = range (1): 0.1: range (2 );
Ygrid = range (3): 0.1: range (4 );
[X, y] = meshgrid (xgrid, ygrid );
Surfc (X, Y, f (x, y ));% Initialize SA parameter % %
% Initializing parameters and settings
T_init = 1.0; % initial temperature
T_min = 1e-10; % finial stopping Temperature
% (Eg., t_min = 1e-10)
F_min =-1E + 100; % min value of the Function
Max_rej = 500; % maximum number of rejections
Max_execute = 100; % maximum number of runs
Max_accept = 15; % maximum number of accept
K = 1; % Boltzmann constant
Alpha = 0.9; % cooling factor
Enorm = 1e-5; % energy norm (eg, enorm = 1e-8)
Guess = [2 2]; % initial guess
% Initializing the counters I, j etc
I = 0; j = 0;
Accept = 0; totaleval = 0;
% Initializing Various values
T = t_init;
E_init = f (guess (1), guess (2 ));
E_old = e_init; e_new = e_old;
Best = guess; % initially guessed values
% SA iteration process % %
% Starting the simulated annealling
While (T> t_min) & (j <= max_rej) & e_new> f_min)
I = I + 1;
% Check if Max numbers of run/accept are met
If (I> = max_run) | (accept> = max_accept)
% Cooling according to a cooling schedule
T = Alpha * t;
Totaleval = totaleval + I;
% Reset the counters
I = 1; accept = 1;
End
% Function evaluations at new locations
NS = guess + rand (1, 2) * randn;
E_new = f (NS (1), NS (2 ));
% Decide to accept the new solution
DeltaE = E_new-E_old;
% Accept if improved
If (-deltaE> enorm)
Best = ns; e_old = e_new;
Accept = accept + 1; j = 0;
End
% Accept with a small probability if not improved
If (deltaE <= enorm & exp (-deltaE/(K * t)> rand );
Best = ns; e_old = e_new;
Accept = accept + 1;
Else
J = J + 1;
End
% Update the estimated Optimal Solution
F_opt = e_old;
End
% Show experiment result % %
% Display the final results
Disp (strcat ('obj function: ', fstr ));
Disp (strcat ('evaluations: ', num2str (totaleval )));
Disp (strcat ('best location: ', num2str (BEST )));
Disp (strcat ('best estimate: ', num2str (f_opt ))); |
Experiment results: Reference: Introduction to mathematical optimization (Xin-She Yang)
