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agomez |
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/* ABSTRACT: */
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/* Simulated annealing is a global optimization method that distinguishes */
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/* different local optima. Starting from an initial point, the algorithm */
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/* takes a step and the function is evaluated. When minimizing a function,*/
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/* any downhill step is accepted and the process repeats from this new */
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/* point. An uphill step may be accepted (thus, it can escape from local */
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/* optima). This uphill decision is made by the Metropolis criteria. As */
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/* the optimization process proceeds, the length of the steps decline and */
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/* the algorithm closes in on the global optimum. Since the algorithm */
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/* makes very few assumptions regarding the function to be optimized, it */
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/* is quite robust with respect to non-quadratic surfaces. The degree of */
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/* robustness can be adjusted by the user. In fact, simulated annealing */
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/* can be used as a local optimizer for difficult functions. */
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/* */
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/* The author can be contacted at h2zr1001@vm.cis.smu.edu */
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/* This file is a translation of a fortran code, which is an example of the*/
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/* Corana et al. simulated annealing algorithm for multimodal and robust */
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/* optimization as implemented and modified by by Goffe et al. */
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/* */
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/* Use the sample function from Judge with the following suggestions */
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/* to get a feel for how SA works. When you've done this, you should be */
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/* ready to use it on most any function with a fair amount of expertise. */
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/* 1. Run the program as is to make sure it runs okay. Take a look at */
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/* the intermediate output and see how it optimizes as temperature */
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/* (T) falls. Notice how the optimal point is reached and how */
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/* falling T reduces VM. */
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/* 2. Look through the documentation to SA so the following makes a */
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/* bit of sense. In line with the paper, it shouldn't be that hard */
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/* to figure out. The core of the algorithm is described on pp. 4-6 */
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/* and on pp. 28. Also see Corana et al. pp. 264-9. */
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/* 3. To see the importance of different temperatures, try starting */
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/* with a very low one (say T = 10E-5). You'll see (i) it never */
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/* escapes from the local optima (in annealing terminology, it */
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/* quenches) & (ii) the step length (VM) will be quite small. This */
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/* is a key part of the algorithm: as temperature (T) falls, step */
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/* length does too. In a minor point here, note how VM is quickly */
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/* reset from its initial value. Thus, the input VM is not very */
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/* important. This is all the more reason to examine VM once the */
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/* algorithm is underway. */
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/* 4. To see the effect of different parameters and their effect on */
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/* the speed of the algorithm, try RT = .95 & RT = .1. Notice the */
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/* vastly different speed for optimization. Also try NT = 20. Note */
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/* that this sample function is quite easy to optimize, so it will */
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/* tolerate big changes in these parameters. RT and NT are the */
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/* parameters one should adjust to modify the runtime of the */
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/* algorithm and its robustness. */
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/* 5. Try constraining the algorithm with either LB or UB. */
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/* Synopsis: */
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/* This routine implements the continuous simulated annealing global */
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/* optimization algorithm described in Corana et al.'s article */
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/* "Minimizing Multimodal Functions of Continuous Variables with the */
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/* "Simulated Annealing" Algorithm" in the September 1987 (vol. 13, */
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/* no. 3, pp. 262-280) issue of the ACM Transactions on Mathematical */
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/* Software. */
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/* A very quick (perhaps too quick) overview of SA: */
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/* SA tries to find the global optimum of an N dimensional function. */
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/* It moves both up and downhill and as the optimization process */
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/* proceeds, it focuses on the most promising area. */
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/* To start, it randomly chooses a trial point within the step length */
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/* VM (a vector of length N) of the user selected starting point. The */
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/* function is evaluated at this trial point and its value is compared */
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/* to its value at the initial point. */
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/* In a maximization problem, all uphill moves are accepted and the */
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/* algorithm continues from that trial point. Downhill moves may be */
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/* accepted; the decision is made by the Metropolis criteria. It uses T */
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/* (temperature) and the size of the downhill move in a probabilistic */
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/* manner. The smaller T and the size of the downhill move are, the more */
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/* likely that move will be accepted. If the trial is accepted, the */
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/* algorithm moves on from that point. If it is rejected, another point */
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/* is chosen instead for a trial evaluation. */
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/* Each element of VM periodically adjusted so that half of all */
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/* function evaluations in that direction are accepted. */
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/* A fall in T is imposed upon the system with the RT variable by */
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/* T(i+1) = RT*T(i) where i is the ith iteration. Thus, as T declines, */
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/* downhill moves are less likely to be accepted and the percentage of */
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/* rejections rise. Given the scheme for the selection for VM, VM falls. */
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/* Thus, as T declines, VM falls and SA focuses upon the most promising */
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/* area for optimization. */
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/* The importance of the parameter T: */
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/* The parameter T is crucial in using SA successfully. It influences */
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/* VM, the step length over which the algorithm searches for optima. For */
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/* a small intial T, the step length may be too small; thus not enough */
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/* of the function might be evaluated to find the global optima. The user */
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/* should carefully examine VM in the intermediate output (set IPRINT = */
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/* 1) to make sure that VM is appropriate. The relationship between the */
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/* initial temperature and the resulting step length is function */
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/* dependent. */
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/* To determine the starting temperature that is consistent with */
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/* optimizing a function, it is worthwhile to run a trial run first. Set */
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/* RT = 1.5 and T = 1.0. With RT > 1.0, the temperature increases and VM */
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/* rises as well. Then select the T that produces a large enough VM. */
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/* For modifications to the algorithm and many details on its use, */
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/* (particularly for econometric applications) see Goffe, Ferrier */
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/* and Rogers, "Global Optimization of Statistical Functions with */
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/* the Simulated Annealing," Journal of Econometrics (forthcoming) */
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/* For a pre-publication copy, contact */
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/* Bill Goffe */
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/* Department of Economics */
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/* Southern Methodist University */
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/* Dallas, TX 75275 */
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/* h2zr1001 @ smuvm1 (Bitnet) */
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/* h2zr1001 @ vm.cis.smu.edu (Internet) */
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/* As far as possible, the parameters here have the same name as in */
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/* the description of the algorithm on pp. 266-8 of Corana et al. */
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/* Input Parameters: */
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/* Note: The suggested values generally come from Corana et al. To */
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/* drastically reduce runtime, see Goffe et al., pp. 17-8 for */
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/* suggestions on choosing the appropriate RT and NT. */
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/* n - Number of variables in the function to be optimized. (INT) */
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/* x - The starting values for the variables of the function to be */
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/* optimized. (DP(N)) */
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/* max - Denotes whether the function should be maximized or */
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/* minimized. A true value denotes maximization while a false */
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/* value denotes minimization. */
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/* RT - The temperature reduction factor. The value suggested by */
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/* Corana et al. is .85. See Goffe et al. for more advice. (DP) */
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/* EPS - Error tolerance for termination. If the final function */
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/* values from the last neps temperatures differ from the */
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/* corresponding value at the current temperature by less than */
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/* EPS and the final function value at the current temperature */
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/* differs from the current optimal function value by less than */
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/* EPS, execution terminates and IER = 0 is returned. (EP) */
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/* NS - Number of cycles. After NS*N function evaluations, each element */
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/* of VM is adjusted so that approximately half of all function */
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/* evaluations are accepted. The suggested value is 20. (INT) */
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/* nt - Number of iterations before temperature reduction. After */
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/* NT*NS*N function evaluations, temperature (T) is changed */
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/* by the factor RT. Value suggested by Corana et al. is */
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/* MAX(100, 5*N). See Goffe et al. for further advice. (INT) */
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/* NEPS - Number of final function values used to decide upon termi- */
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/* nation. See EPS. Suggested value is 4. (INT) */
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/* maxevl - The maximum number of function evaluations. If it is */
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/* exceeded, IER = 1. (INT) */
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/* lb - The lower bound for the allowable solution variables. (DP(N)) */
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/* ub - The upper bound for the allowable solution variables. (DP(N)) */
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/* If the algorithm chooses X(I) .LT. LB(I) or X(I) .GT. UB(I), */
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/* I = 1, N, a point is from inside is randomly selected. This */
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/* This focuses the algorithm on the region inside UB and LB. */
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/* Unless the user wishes to concentrate the search to a par- */
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/* ticular region, UB and LB should be set to very large positive */
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/* and negative values, respectively. Note that the starting */
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/* vector X should be inside this region. Also note that LB and */
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/* UB are fixed in position, while VM is centered on the last */
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/* accepted trial set of variables that optimizes the function. */
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/* c - Vector that controls the step length adjustment. The suggested */
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/* value for all elements is 2.0. (DP(N)) */
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/* t - On input, the initial temperature. See Goffe et al. for advice. */
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/* On output, the final temperature. (DP) */
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/* vm - The step length vector. On input it should encompass the */
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/* region of interest given the starting value X. For point */
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/* X(I), the next trial point is selected is from X(I) - VM(I) */
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/* to X(I) + VM(I). Since VM is adjusted so that about half */
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/* of all points are accepted, the input value is not very */
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/* important (i.e. is the value is off, SA adjusts VM to the */
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/* correct value). (DP(N)) */
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/* Output Parameters: */
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/* xopt - The variables that optimize the function. (DP(N)) */
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/* fopt - The optimal value of the function. (DP) */
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/* JMB this has been modified to work with the gadget object structure */
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/* This means that the function has been replaced by a call to ecosystem */
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/* object, and we can use the vector objects that have been defined */
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#include "gadget.h" |
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#include "optinfo.h" |
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#include "mathfunc.h" |
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#include "doublevector.h" |
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#include "intvector.h" |
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#include "errorhandler.h" |
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#include "ecosystem.h" |
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#include "global.h" |
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extern Ecosystem* EcoSystem;
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void OptInfoSimann::OptimiseLikelihood() {
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//set initial values
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int nacc = 0; //The number of accepted function evaluations
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int nrej = 0; //The number of rejected function evaluations
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int naccmet = 0; //The number of metropolis accepted function evaluations
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double tmp, p, pp, ratio, nsdiv;
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double fopt, funcval, trialf;
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int a, i, j, k, l, offset, quit;
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int rchange, rcheck, rnumber; //Used to randomise the order of the parameters
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handle.logMessage(LOGINFO, "\nStarting Simulated Annealing optimisation algorithm\n");
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int nvars = EcoSystem->numOptVariables();
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DoubleVector x(nvars);
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DoubleVector init(nvars);
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DoubleVector trialx(nvars, 0.0);
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DoubleVector bestx(nvars);
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DoubleVector scalex(nvars);
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DoubleVector lowerb(nvars);
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DoubleVector upperb(nvars);
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DoubleVector fstar(tempcheck);
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DoubleVector vm(nvars, vminit);
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IntVector param(nvars, 0);
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IntVector nacp(nvars, 0);
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EcoSystem->resetVariables(); //JMB need to reset variables in case they have been scaled
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if (scale)
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EcoSystem->scaleVariables();
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EcoSystem->getOptScaledValues(x);
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EcoSystem->getOptLowerBounds(lowerb);
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EcoSystem->getOptUpperBounds(upperb);
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EcoSystem->getOptInitialValues(init);
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for (i = 0; i < nvars; i++) {
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bestx[i] = x[i];
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param[i] = i;
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}
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if (scale) {
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for (i = 0; i < nvars; i++) {
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scalex[i] = x[i];
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// Scaling the bounds, because the parameters are scaled
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lowerb[i] = lowerb[i] / init[i];
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upperb[i] = upperb[i] / init[i];
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if (lowerb[i] > upperb[i]) {
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tmp = lowerb[i];
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lowerb[i] = upperb[i];
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upperb[i] = tmp;
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}
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}
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}
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//funcval is the function value at x
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funcval = EcoSystem->SimulateAndUpdate(x);
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if (funcval != funcval) { //check for NaN
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handle.logMessage(LOGINFO, "Error starting Simulated Annealing optimisation with f(x) = infinity");
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converge = -1;
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iters = 1;
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return;
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}
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//the function is to be minimised so switch the sign of funcval (and trialf)
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funcval = -funcval;
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offset = EcoSystem->getFuncEval(); //number of function evaluations done before loop
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nacc++;
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cs /= lratio; //JMB save processing time
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nsdiv = 1.0 / ns;
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fopt = funcval;
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for (i = 0; i < tempcheck; i++)
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fstar[i] = funcval;
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//Start the main loop. Note that it terminates if
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//(i) the algorithm succesfully optimises the function or
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//(ii) there are too many function evaluations
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while (1) {
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for (a = 0; a < nt; a++) {
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//Randomize the order of the parameters once in a while, to avoid
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//the order having an influence on which changes are accepted
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rchange = 0;
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while (rchange < nvars) {
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rnumber = rand() % nvars;
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rcheck = 1;
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for (i = 0; i < rchange; i++)
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if (param[i] == rnumber)
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rcheck = 0;
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if (rcheck) {
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param[rchange] = rnumber;
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rchange++;
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}
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}
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for (j = 0; j < ns; j++) {
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for (l = 0; l < nvars; l++) {
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//Generate trialx, the trial value of x
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for (i = 0; i < nvars; i++) {
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if (i == param[l]) {
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trialx[i] = x[i] + ((randomNumber() * 2.0) - 1.0) * vm[i];
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//If trialx is out of bounds, try again until we find a point that is OK
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if ((trialx[i] < lowerb[i]) || (trialx[i] > upperb[i])) {
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//JMB - this used to just select a random point between the bounds
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k = 0;
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| 287 : |
|
|
while ((trialx[i] < lowerb[i]) || (trialx[i] > upperb[i])) {
|
| 288 : |
|
|
trialx[i] = x[i] + ((randomNumber() * 2.0) - 1.0) * vm[i];
|
| 289 : |
|
|
k++;
|
| 290 : |
|
|
if (k > 10) //we've had 10 tries to find a point neatly, so give up
|
| 291 : |
|
|
trialx[i] = lowerb[i] + (upperb[i] - lowerb[i]) * randomNumber();
|
| 292 : |
|
|
}
|
| 293 : |
|
|
}
|
| 294 : |
|
|
|
| 295 : |
|
|
} else
|
| 296 : |
|
|
trialx[i] = x[i];
|
| 297 : |
|
|
}
|
| 298 : |
|
|
|
| 299 : |
|
|
//Evaluate the function with the trial point trialx and return as -trialf
|
| 300 : |
|
|
trialf = EcoSystem->SimulateAndUpdate(trialx);
|
| 301 : |
|
|
trialf = -trialf;
|
| 302 : |
|
|
|
| 303 : |
|
|
//If too many function evaluations occur, terminate the algorithm
|
| 304 : |
|
|
iters = EcoSystem->getFuncEval() - offset;
|
| 305 : |
|
|
if (iters > simanniter) {
|
| 306 : |
|
|
handle.logMessage(LOGINFO, "\nStopping Simulated Annealing optimisation algorithm\n");
|
| 307 : |
|
|
handle.logMessage(LOGINFO, "The optimisation stopped after", iters, "function evaluations");
|
| 308 : |
|
|
handle.logMessage(LOGINFO, "The temperature was reduced to", t);
|
| 309 : |
|
|
handle.logMessage(LOGINFO, "The optimisation stopped because the maximum number of function evaluations");
|
| 310 : |
|
|
handle.logMessage(LOGINFO, "was reached and NOT because an optimum was found for this run");
|
| 311 : |
|
|
handle.logMessage(LOGINFO, "Number of directly accepted points", nacc);
|
| 312 : |
|
|
handle.logMessage(LOGINFO, "Number of metropolis accepted points", naccmet);
|
| 313 : |
|
|
handle.logMessage(LOGINFO, "Number of rejected points", nrej);
|
| 314 : |
|
|
|
| 315 : |
|
|
score = EcoSystem->SimulateAndUpdate(bestx);
|
| 316 : |
|
|
handle.logMessage(LOGINFO, "\nSimulated Annealing finished with a likelihood score of", score);
|
| 317 : |
|
|
return;
|
| 318 : |
|
|
}
|
| 319 : |
|
|
|
| 320 : |
|
|
//Accept the new point if the new function value better
|
| 321 : |
|
|
if ((trialf - funcval) > verysmall) {
|
| 322 : |
|
|
for (i = 0; i < nvars; i++)
|
| 323 : |
|
|
x[i] = trialx[i];
|
| 324 : |
|
|
funcval = trialf;
|
| 325 : |
|
|
nacc++;
|
| 326 : |
|
|
nacp[param[l]]++; //JMB - not sure about this ...
|
| 327 : |
|
|
|
| 328 : |
|
|
} else {
|
| 329 : |
|
|
//Accept according to metropolis condition
|
| 330 : |
|
|
p = expRep((trialf - funcval) / t);
|
| 331 : |
|
|
pp = randomNumber();
|
| 332 : |
|
|
if (pp < p) {
|
| 333 : |
|
|
//Accept point
|
| 334 : |
|
|
for (i = 0; i < nvars; i++)
|
| 335 : |
|
|
x[i] = trialx[i];
|
| 336 : |
|
|
funcval = trialf;
|
| 337 : |
|
|
naccmet++;
|
| 338 : |
|
|
nacp[param[l]]++;
|
| 339 : |
|
|
} else {
|
| 340 : |
|
|
//Reject point
|
| 341 : |
|
|
nrej++;
|
| 342 : |
|
|
}
|
| 343 : |
|
|
}
|
| 344 : |
|
|
|
| 345 : |
|
|
// JMB added check for really silly values
|
| 346 : |
|
|
if (isZero(trialf)) {
|
| 347 : |
|
|
handle.logMessage(LOGINFO, "Error in Simulated Annealing optimisation after", iters, "function evaluations, f(x) = 0");
|
| 348 : |
|
|
converge = -1;
|
| 349 : |
|
|
return;
|
| 350 : |
|
|
}
|
| 351 : |
|
|
|
| 352 : |
|
|
//If greater than any other point, record as new optimum
|
| 353 : |
|
|
if ((trialf > fopt) && (trialf == trialf)) {
|
| 354 : |
|
|
for (i = 0; i < nvars; i++)
|
| 355 : |
|
|
bestx[i] = trialx[i];
|
| 356 : |
|
|
fopt = trialf;
|
| 357 : |
|
|
|
| 358 : |
|
|
if (scale) {
|
| 359 : |
|
|
for (i = 0; i < nvars; i++)
|
| 360 : |
|
|
scalex[i] = bestx[i] * init[i];
|
| 361 : |
|
|
EcoSystem->storeVariables(-fopt, scalex);
|
| 362 : |
|
|
} else
|
| 363 : |
|
|
EcoSystem->storeVariables(-fopt, bestx);
|
| 364 : |
|
|
|
| 365 : |
|
|
handle.logMessage(LOGINFO, "\nNew optimum found after", iters, "function evaluations");
|
| 366 : |
|
|
handle.logMessage(LOGINFO, "The likelihood score is", -fopt, "at the point");
|
| 367 : |
|
|
EcoSystem->writeBestValues();
|
| 368 : |
|
|
}
|
| 369 : |
|
|
}
|
| 370 : |
|
|
}
|
| 371 : |
|
|
|
| 372 : |
|
|
//Adjust vm so that approximately half of all evaluations are accepted
|
| 373 : |
|
|
for (i = 0; i < nvars; i++) {
|
| 374 : |
|
|
ratio = nsdiv * nacp[i];
|
| 375 : |
|
|
nacp[i] = 0;
|
| 376 : |
|
|
if (ratio > uratio) {
|
| 377 : |
|
|
vm[i] = vm[i] * (1.0 + cs * (ratio - uratio));
|
| 378 : |
|
|
} else if (ratio < lratio) {
|
| 379 : |
|
|
vm[i] = vm[i] / (1.0 + cs * (lratio - ratio));
|
| 380 : |
|
|
}
|
| 381 : |
|
|
|
| 382 : |
|
|
if (vm[i] < rathersmall)
|
| 383 : |
|
|
vm[i] = rathersmall;
|
| 384 : |
|
|
if (vm[i] > (upperb[i] - lowerb[i]))
|
| 385 : |
|
|
vm[i] = upperb[i] - lowerb[i];
|
| 386 : |
|
|
}
|
| 387 : |
|
|
}
|
| 388 : |
|
|
|
| 389 : |
|
|
//Check termination criteria
|
| 390 : |
|
|
for (i = tempcheck - 1; i > 0; i--)
|
| 391 : |
|
|
fstar[i] = fstar[i - 1];
|
| 392 : |
|
|
fstar[0] = funcval;
|
| 393 : |
|
|
|
| 394 : |
|
|
quit = 0;
|
| 395 : |
|
|
if (fabs(fopt - funcval) < simanneps) {
|
| 396 : |
|
|
quit = 1;
|
| 397 : |
|
|
for (i = 0; i < tempcheck - 1; i++)
|
| 398 : |
|
|
if (fabs(fstar[i + 1] - fstar[i]) > simanneps)
|
| 399 : |
|
|
quit = 0;
|
| 400 : |
|
|
}
|
| 401 : |
|
|
|
| 402 : |
|
|
handle.logMessage(LOGINFO, "Checking convergence criteria after", iters, "function evaluations ...");
|
| 403 : |
|
|
|
| 404 : |
|
|
//Terminate SA if appropriate
|
| 405 : |
|
|
if (quit) {
|
| 406 : |
|
|
handle.logMessage(LOGINFO, "\nStopping Simulated Annealing optimisation algorithm\n");
|
| 407 : |
|
|
handle.logMessage(LOGINFO, "The optimisation stopped after", iters, "function evaluations");
|
| 408 : |
|
|
handle.logMessage(LOGINFO, "The temperature was reduced to", t);
|
| 409 : |
|
|
handle.logMessage(LOGINFO, "The optimisation stopped because an optimum was found for this run");
|
| 410 : |
|
|
handle.logMessage(LOGINFO, "Number of directly accepted points", nacc);
|
| 411 : |
|
|
handle.logMessage(LOGINFO, "Number of metropolis accepted points", naccmet);
|
| 412 : |
|
|
handle.logMessage(LOGINFO, "Number of rejected points", nrej);
|
| 413 : |
|
|
|
| 414 : |
|
|
converge = 1;
|
| 415 : |
|
|
score = EcoSystem->SimulateAndUpdate(bestx);
|
| 416 : |
|
|
handle.logMessage(LOGINFO, "\nSimulated Annealing finished with a likelihood score of", score);
|
| 417 : |
|
|
return;
|
| 418 : |
|
|
}
|
| 419 : |
|
|
|
| 420 : |
|
|
//If termination criteria is not met, prepare for another loop.
|
| 421 : |
|
|
t *= rt;
|
| 422 : |
|
|
if (t < rathersmall)
|
| 423 : |
|
|
t = rathersmall; //JMB make sure temperature doesnt get too small
|
| 424 : |
|
|
|
| 425 : |
|
|
handle.logMessage(LOGINFO, "Reducing the temperature to", t);
|
| 426 : |
|
|
funcval = fopt;
|
| 427 : |
|
|
for (i = 0; i < nvars; i++)
|
| 428 : |
|
|
x[i] = bestx[i];
|
| 429 : |
|
|
}
|
| 430 : |
|
|
}
|
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