/* ABSTRACT: */
/* Simulated annealing is a global optimization method that distinguishes */
/* different local optima. Starting from an initial point, the algorithm */
/* takes a step and the function is evaluated. When minimizing a function,*/
/* any downhill step is accepted and the process repeats from this new */
/* point. An uphill step may be accepted (thus, it can escape from local */
/* optima). This uphill decision is made by the Metropolis criteria. As */
/* the optimization process proceeds, the length of the steps decline and */
/* the algorithm closes in on the global optimum. Since the algorithm */
/* makes very few assumptions regarding the function to be optimized, it */
/* is quite robust with respect to non-quadratic surfaces. The degree of */
/* robustness can be adjusted by the user. In fact, simulated annealing */
/* can be used as a local optimizer for difficult functions. */
/* */
/* The author can be contacted at h2zr1001@vm.cis.smu.edu */
/* This file is a translation of a fortran code, which is an example of the*/
/* Corana et al. simulated annealing algorithm for multimodal and robust */
/* optimization as implemented and modified by by Goffe et al. */
/* */
/* Use the sample function from Judge with the following suggestions */
/* to get a feel for how SA works. When you've done this, you should be */
/* ready to use it on most any function with a fair amount of expertise. */
/* 1. Run the program as is to make sure it runs okay. Take a look at */
/* the intermediate output and see how it optimizes as temperature */
/* (T) falls. Notice how the optimal point is reached and how */
/* falling T reduces VM. */
/* 2. Look through the documentation to SA so the following makes a */
/* bit of sense. In line with the paper, it shouldn't be that hard */
/* to figure out. The core of the algorithm is described on pp. 4-6 */
/* and on pp. 28. Also see Corana et al. pp. 264-9. */
/* 3. To see the importance of different temperatures, try starting */
/* with a very low one (say T = 10E-5). You'll see (i) it never */
/* escapes from the local optima (in annealing terminology, it */
/* quenches) & (ii) the step length (VM) will be quite small. This */
/* is a key part of the algorithm: as temperature (T) falls, step */
/* length does too. In a minor point here, note how VM is quickly */
/* reset from its initial value. Thus, the input VM is not very */
/* important. This is all the more reason to examine VM once the */
/* algorithm is underway. */
/* 4. To see the effect of different parameters and their effect on */
/* the speed of the algorithm, try RT = .95 & RT = .1. Notice the */
/* vastly different speed for optimization. Also try NT = 20. Note */
/* that this sample function is quite easy to optimize, so it will */
/* tolerate big changes in these parameters. RT and NT are the */
/* parameters one should adjust to modify the runtime of the */
/* algorithm and its robustness. */
/* 5. Try constraining the algorithm with either LB or UB. */
/* Synopsis: */
/* This routine implements the continuous simulated annealing global */
/* optimization algorithm described in Corana et al.'s article */
/* "Minimizing Multimodal Functions of Continuous Variables with the */
/* "Simulated Annealing" Algorithm" in the September 1987 (vol. 13, */
/* no. 3, pp. 262-280) issue of the ACM Transactions on Mathematical */
/* Software. */
/* A very quick (perhaps too quick) overview of SA: */
/* SA tries to find the global optimum of an N dimensional function. */
/* It moves both up and downhill and as the optimization process */
/* proceeds, it focuses on the most promising area. */
/* To start, it randomly chooses a trial point within the step length */
/* VM (a vector of length N) of the user selected starting point. The */
/* function is evaluated at this trial point and its value is compared */
/* to its value at the initial point. */
/* In a maximization problem, all uphill moves are accepted and the */
/* algorithm continues from that trial point. Downhill moves may be */
/* accepted; the decision is made by the Metropolis criteria. It uses T */
/* (temperature) and the size of the downhill move in a probabilistic */
/* manner. The smaller T and the size of the downhill move are, the more */
/* likely that move will be accepted. If the trial is accepted, the */
/* algorithm moves on from that point. If it is rejected, another point */
/* is chosen instead for a trial evaluation. */
/* Each element of VM periodically adjusted so that half of all */
/* function evaluations in that direction are accepted. */
/* A fall in T is imposed upon the system with the RT variable by */
/* T(i+1) = RT*T(i) where i is the ith iteration. Thus, as T declines, */
/* downhill moves are less likely to be accepted and the percentage of */
/* rejections rise. Given the scheme for the selection for VM, VM falls. */
/* Thus, as T declines, VM falls and SA focuses upon the most promising */
/* area for optimization. */
/* The importance of the parameter T: */
/* The parameter T is crucial in using SA successfully. It influences */
/* VM, the step length over which the algorithm searches for optima. For */
/* a small intial T, the step length may be too small; thus not enough */
/* of the function might be evaluated to find the global optima. The user */
/* should carefully examine VM in the intermediate output (set IPRINT = */
/* 1) to make sure that VM is appropriate. The relationship between the */
/* initial temperature and the resulting step length is function */
/* dependent. */
/* To determine the starting temperature that is consistent with */
/* optimizing a function, it is worthwhile to run a trial run first. Set */
/* RT = 1.5 and T = 1.0. With RT > 1.0, the temperature increases and VM */
/* rises as well. Then select the T that produces a large enough VM. */
/* For modifications to the algorithm and many details on its use, */
/* (particularly for econometric applications) see Goffe, Ferrier */
/* and Rogers, "Global Optimization of Statistical Functions with */
/* the Simulated Annealing," Journal of Econometrics (forthcoming) */
/* For a pre-publication copy, contact */
/* Bill Goffe */
/* Department of Economics */
/* Southern Methodist University */
/* Dallas, TX 75275 */
/* h2zr1001 @ smuvm1 (Bitnet) */
/* h2zr1001 @ vm.cis.smu.edu (Internet) */
/* As far as possible, the parameters here have the same name as in */
/* the description of the algorithm on pp. 266-8 of Corana et al. */
/* Input Parameters: */
/* Note: The suggested values generally come from Corana et al. To */
/* drastically reduce runtime, see Goffe et al., pp. 17-8 for */
/* suggestions on choosing the appropriate RT and NT. */
/* n - Number of variables in the function to be optimized. (INT) */
/* x - The starting values for the variables of the function to be */
/* optimized. (DP(N)) */
/* max - Denotes whether the function should be maximized or */
/* minimized. A true value denotes maximization while a false */
/* value denotes minimization. */
/* RT - The temperature reduction factor. The value suggested by */
/* Corana et al. is .85. See Goffe et al. for more advice. (DP) */
/* EPS - Error tolerance for termination. If the final function */
/* values from the last neps temperatures differ from the */
/* corresponding value at the current temperature by less than */
/* EPS and the final function value at the current temperature */
/* differs from the current optimal function value by less than */
/* EPS, execution terminates and IER = 0 is returned. (EP) */
/* NS - Number of cycles. After NS*N function evaluations, each element */
/* of VM is adjusted so that approximately half of all function */
/* evaluations are accepted. The suggested value is 20. (INT) */
/* nt - Number of iterations before temperature reduction. After */
/* NT*NS*N function evaluations, temperature (T) is changed */
/* by the factor RT. Value suggested by Corana et al. is */
/* MAX(100, 5*N). See Goffe et al. for further advice. (INT) */
/* NEPS - Number of final function values used to decide upon termi- */
/* nation. See EPS. Suggested value is 4. (INT) */
/* maxevl - The maximum number of function evaluations. If it is */
/* exceeded, IER = 1. (INT) */
/* lb - The lower bound for the allowable solution variables. (DP(N)) */
/* ub - The upper bound for the allowable solution variables. (DP(N)) */
/* If the algorithm chooses X(I) .LT. LB(I) or X(I) .GT. UB(I), */
/* I = 1, N, a point is from inside is randomly selected. This */
/* This focuses the algorithm on the region inside UB and LB. */
/* Unless the user wishes to concentrate the search to a par- */
/* ticular region, UB and LB should be set to very large positive */
/* and negative values, respectively. Note that the starting */
/* vector X should be inside this region. Also note that LB and */
/* UB are fixed in position, while VM is centered on the last */
/* accepted trial set of variables that optimizes the function. */
/* c - Vector that controls the step length adjustment. The suggested */
/* value for all elements is 2.0. (DP(N)) */
/* t - On input, the initial temperature. See Goffe et al. for advice. */
/* On output, the final temperature. (DP) */
/* vm - The step length vector. On input it should encompass the */
/* region of interest given the starting value X. For point */
/* X(I), the next trial point is selected is from X(I) - VM(I) */
/* to X(I) + VM(I). Since VM is adjusted so that about half */
/* of all points are accepted, the input value is not very */
/* important (i.e. is the value is off, SA adjusts VM to the */
/* correct value). (DP(N)) */
/* Output Parameters: */
/* xopt - The variables that optimize the function. (DP(N)) */
/* fopt - The optimal value of the function. (DP) */
/* JMB this has been modified to work with the gadget object structure */
/* This means that the function has been replaced by a call to ecosystem */
/* object, and we can use the vector objects that have been defined */
#include "gadget.h" //All the required standard header files are in here
#include "optinfo.h"
#include "mathfunc.h"
#include "doublevector.h"
#include "intvector.h"
#include "errorhandler.h"
#include "ecosystem.h"
#include "global.h"
extern Ecosystem* EcoSystem;
void OptInfoSimann::OptimiseLikelihood() {
//set initial values
int nacc = 0; //The number of accepted function evaluations
int nrej = 0; //The number of rejected function evaluations
int naccmet = 0; //The number of metropolis accepted function evaluations
double tmp, p, pp, ratio, nsdiv;
double fopt, funcval, trialf;
int a, i, j, k, l, offset, quit;
int rchange, rcheck, rnumber; //Used to randomise the order of the parameters
handle.logMessage(LOGINFO, "\nStarting Simulated Annealing optimisation algorithm\n");
int nvars = EcoSystem->numOptVariables();
DoubleVector x(nvars);
DoubleVector init(nvars);
DoubleVector trialx(nvars, 0.0);
DoubleVector bestx(nvars);
DoubleVector scalex(nvars);
DoubleVector lowerb(nvars);
DoubleVector upperb(nvars);
DoubleVector fstar(tempcheck);
DoubleVector vm(nvars, vminit);
IntVector param(nvars, 0);
IntVector nacp(nvars, 0);
EcoSystem->resetVariables(); //JMB need to reset variables in case they have been scaled
if (scale)
EcoSystem->scaleVariables();
EcoSystem->getOptScaledValues(x);
EcoSystem->getOptLowerBounds(lowerb);
EcoSystem->getOptUpperBounds(upperb);
EcoSystem->getOptInitialValues(init);
for (i = 0; i < nvars; i++) {
bestx[i] = x[i];
param[i] = i;
}
if (scale) {
for (i = 0; i < nvars; i++) {
scalex[i] = x[i];
// Scaling the bounds, because the parameters are scaled
lowerb[i] = lowerb[i] / init[i];
upperb[i] = upperb[i] / init[i];
if (lowerb[i] > upperb[i]) {
tmp = lowerb[i];
lowerb[i] = upperb[i];
upperb[i] = tmp;
}
}
}
//funcval is the function value at x
funcval = EcoSystem->SimulateAndUpdate(x);
if (funcval != funcval) { //check for NaN
handle.logMessage(LOGINFO, "Error starting Simulated Annealing optimisation with f(x) = infinity");
converge = -1;
iters = 1;
return;
}
//the function is to be minimised so switch the sign of funcval (and trialf)
funcval = -funcval;
offset = EcoSystem->getFuncEval(); //number of function evaluations done before loop
nacc++;
cs /= lratio; //JMB save processing time
nsdiv = 1.0 / ns;
fopt = funcval;
for (i = 0; i < tempcheck; i++)
fstar[i] = funcval;
//Start the main loop. Note that it terminates if
//(i) the algorithm succesfully optimises the function or
//(ii) there are too many function evaluations
while (1) {
for (a = 0; a < nt; a++) {
//Randomize the order of the parameters once in a while, to avoid
//the order having an influence on which changes are accepted
rchange = 0;
while (rchange < nvars) {
rnumber = rand() % nvars;
rcheck = 1;
for (i = 0; i < rchange; i++)
if (param[i] == rnumber)
rcheck = 0;
if (rcheck) {
param[rchange] = rnumber;
rchange++;
}
}
for (j = 0; j < ns; j++) {
for (l = 0; l < nvars; l++) {
//Generate trialx, the trial value of x
for (i = 0; i < nvars; i++) {
if (i == param[l]) {
trialx[i] = x[i] + ((randomNumber() * 2.0) - 1.0) * vm[i];
//If trialx is out of bounds, try again until we find a point that is OK
if ((trialx[i] < lowerb[i]) || (trialx[i] > upperb[i])) {
//JMB - this used to just select a random point between the bounds
k = 0;
while ((trialx[i] < lowerb[i]) || (trialx[i] > upperb[i])) {
trialx[i] = x[i] + ((randomNumber() * 2.0) - 1.0) * vm[i];
k++;
if (k > 10) //we've had 10 tries to find a point neatly, so give up
trialx[i] = lowerb[i] + (upperb[i] - lowerb[i]) * randomNumber();
}
}
} else
trialx[i] = x[i];
}
//Evaluate the function with the trial point trialx and return as -trialf
trialf = EcoSystem->SimulateAndUpdate(trialx);
trialf = -trialf;
//If too many function evaluations occur, terminate the algorithm
iters = EcoSystem->getFuncEval() - offset;
if (iters > simanniter) {
handle.logMessage(LOGINFO, "\nStopping Simulated Annealing optimisation algorithm\n");
handle.logMessage(LOGINFO, "The optimisation stopped after", iters, "function evaluations");
handle.logMessage(LOGINFO, "The temperature was reduced to", t);
handle.logMessage(LOGINFO, "The optimisation stopped because the maximum number of function evaluations");
handle.logMessage(LOGINFO, "was reached and NOT because an optimum was found for this run");
handle.logMessage(LOGINFO, "Number of directly accepted points", nacc);
handle.logMessage(LOGINFO, "Number of metropolis accepted points", naccmet);
handle.logMessage(LOGINFO, "Number of rejected points", nrej);
score = EcoSystem->SimulateAndUpdate(bestx);
handle.logMessage(LOGINFO, "\nSimulated Annealing finished with a likelihood score of", score);
return;
}
//Accept the new point if the new function value better
if ((trialf - funcval) > verysmall) {
for (i = 0; i < nvars; i++)
x[i] = trialx[i];
funcval = trialf;
nacc++;
nacp[param[l]]++; //JMB - not sure about this ...
} else {
//Accept according to metropolis condition
p = expRep((trialf - funcval) / t);
pp = randomNumber();
if (pp < p) {
//Accept point
for (i = 0; i < nvars; i++)
x[i] = trialx[i];
funcval = trialf;
naccmet++;
nacp[param[l]]++;
} else {
//Reject point
nrej++;
}
}
// JMB added check for really silly values
if (isZero(trialf)) {
handle.logMessage(LOGINFO, "Error in Simulated Annealing optimisation after", iters, "function evaluations, f(x) = 0");
converge = -1;
return;
}
//If greater than any other point, record as new optimum
if ((trialf > fopt) && (trialf == trialf)) {
for (i = 0; i < nvars; i++)
bestx[i] = trialx[i];
fopt = trialf;
if (scale) {
for (i = 0; i < nvars; i++)
scalex[i] = bestx[i] * init[i];
EcoSystem->storeVariables(-fopt, scalex);
} else
EcoSystem->storeVariables(-fopt, bestx);
handle.logMessage(LOGINFO, "\nNew optimum found after", iters, "function evaluations");
handle.logMessage(LOGINFO, "The likelihood score is", -fopt, "at the point");
EcoSystem->writeBestValues();
}
}
}
//Adjust vm so that approximately half of all evaluations are accepted
for (i = 0; i < nvars; i++) {
ratio = nsdiv * nacp[i];
nacp[i] = 0;
if (ratio > uratio) {
vm[i] = vm[i] * (1.0 + cs * (ratio - uratio));
} else if (ratio < lratio) {
vm[i] = vm[i] / (1.0 + cs * (lratio - ratio));
}
if (vm[i] < rathersmall)
vm[i] = rathersmall;
if (vm[i] > (upperb[i] - lowerb[i]))
vm[i] = upperb[i] - lowerb[i];
}
}
//Check termination criteria
for (i = tempcheck - 1; i > 0; i--)
fstar[i] = fstar[i - 1];
fstar[0] = funcval;
quit = 0;
if (fabs(fopt - funcval) < simanneps) {
quit = 1;
for (i = 0; i < tempcheck - 1; i++)
if (fabs(fstar[i + 1] - fstar[i]) > simanneps)
quit = 0;
}
handle.logMessage(LOGINFO, "Checking convergence criteria after", iters, "function evaluations ...");
//Terminate SA if appropriate
if (quit) {
handle.logMessage(LOGINFO, "\nStopping Simulated Annealing optimisation algorithm\n");
handle.logMessage(LOGINFO, "The optimisation stopped after", iters, "function evaluations");
handle.logMessage(LOGINFO, "The temperature was reduced to", t);
handle.logMessage(LOGINFO, "The optimisation stopped because an optimum was found for this run");
handle.logMessage(LOGINFO, "Number of directly accepted points", nacc);
handle.logMessage(LOGINFO, "Number of metropolis accepted points", naccmet);
handle.logMessage(LOGINFO, "Number of rejected points", nrej);
converge = 1;
score = EcoSystem->SimulateAndUpdate(bestx);
handle.logMessage(LOGINFO, "\nSimulated Annealing finished with a likelihood score of", score);
return;
}
//If termination criteria is not met, prepare for another loop.
t *= rt;
if (t < rathersmall)
t = rathersmall; //JMB make sure temperature doesnt get too small
handle.logMessage(LOGINFO, "Reducing the temperature to", t);
funcval = fopt;
for (i = 0; i < nvars; i++)
x[i] = bestx[i];
}
}
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