444 Chapter 10. Minimization or Maximization of Functions Stoer,J.,and Bulirsch,R.1980,Introduction to Numerical Analysis(New York:Springer-Verlag). 84.10. Wilkinson,J.H.,and Reinsch,C.1971,Linear Algebra,vol.Il of Handbook for Automatic Com- putation (New York:Springer-Verlag).[5] 10.9 Simulated Annealing Methods 三 The method of simulated annealing [1,2]is a technique that has attracted signif- icant attention as suitable for optimization problems of large scale,especially ones where a desired global extremum is hidden among many,poorer,local extrema.For practical purposes,simulated annealing has effectively"solved"the famous traveling salesman problem of finding the shortest cyclical itinerary for a traveling salesman who must visit each of N cities in turn.(Other practical methods have also been found.)The method has also been used successfully for designing complex integrated circuits:The arrangement of several hundred thousand circuit elements on a tiny silicon substrate is optimized so as to minimize interference among their connecting 9 wires[3,4].Surprisingly,the implementation of the algorithm is relatively simple. Notice that the two applications cited are both examples of combinatorial minimization.There is an objective function to be minimized,as usual;but the space over which that function is defined is not simply the N-dimensional space of N continuously variable parameters.Rather,it is a discrete,but very large,configuration space,like the set of possible orders of cities,or the set of possible allocations of 兰∽6 9 silicon "real estate"blocks to circuit elements.The number of elements in the OF SCIENTIFIC configuration space is factorially large,so that they cannot be explored exhaustively. Furthermore,since the set is discrete,we are deprived of any notion of"continuing 6 downhill in a favorable direction."The concept of"direction"may not have any meaning in the configuration space. Below.we will also discuss how to use simulated annealing methods for spaces with continuous control parameters,like those of 810.4-10.7.This application is actually more complicated than the combinatorial one,since the familiar problem of Numerica 10621 "long,narrow valleys"again asserts itself.Simulated annealing,as we will see,tries "random"steps,but in a long,narrow valley,almost all random steps are uphill! 431 Some additional finesse is therefore required. Recipes At the heart of the method of simulated annealing is an analogy with thermody- namics,specifically with the way that liquids freeze and crystallize,or metals cool 腿 and anneal.At high temperatures,the molecules of a liquid move freely with respect North to one another.If the liquid is cooled slowly,thermal mobility is lost.The atoms are often able to line themselves up and form a pure crystal that is completely ordered over a distance up to billions of times the size of an individual atom in all directions. This crystal is the state of minimum energy for this system.The amazing fact is that, for slowly cooled systems,nature is able to find this minimum energy state.In fact,if a liquid metal is cooled quickly or "quenched,"it does not reach this state but rather ends up in a polycrystalline or amorphous state having somewhat higher energy. So the essence of the process is slow cooling,allowing ample time for redistribution of the atoms as they lose mobility.This is the technical definition of annealing,and it is essential for ensuring that a low energy state will be achieved
444 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag), §4.10. Wilkinson, J.H., and Reinsch, C. 1971, Linear Algebra, vol. II of Handbook for Automatic Computation (New York: Springer-Verlag). [5] 10.9 Simulated Annealing Methods The method of simulated annealing [1,2] is a technique that has attracted significant attention as suitable for optimization problems of large scale, especially ones where a desired global extremum is hidden among many, poorer, local extrema. For practical purposes, simulated annealing has effectively “solved” the famoustraveling salesman problem of finding the shortest cyclical itinerary for a traveling salesman who must visit each of N cities in turn. (Other practical methods have also been found.) The method has also been used successfully for designing complex integrated circuits: The arrangement of several hundred thousand circuit elements on a tiny silicon substrate is optimized so as to minimize interference among their connecting wires [3,4]. Surprisingly, the implementation of the algorithm is relatively simple. Notice that the two applications cited are both examples of combinatorial minimization. There is an objective function to be minimized, as usual; but the space over which that function is defined is not simply the N-dimensional space of N continuously variable parameters. Rather, it is a discrete, but very large, configuration space, like the set of possible orders of cities, or the set of possible allocations of silicon “real estate” blocks to circuit elements. The number of elements in the configuration space is factorially large, so that they cannot be explored exhaustively. Furthermore, since the set is discrete, we are deprived of any notion of “continuing downhill in a favorable direction.” The concept of “direction” may not have any meaning in the configuration space. Below, we will also discuss how to use simulated annealing methods for spaces with continuous control parameters, like those of §§10.4–10.7. This application is actually more complicated than the combinatorial one, since the familiar problem of “long, narrow valleys” again asserts itself. Simulated annealing, as we will see, tries “random” steps; but in a long, narrow valley, almost all random steps are uphill! Some additional finesse is therefore required. At the heart of the method of simulated annealing is an analogy with thermodynamics, specifically with the way that liquids freeze and crystallize, or metals cool and anneal. At high temperatures, the molecules of a liquid move freely with respect to one another. If the liquid is cooled slowly, thermal mobility is lost. The atoms are often able to line themselves up and form a pure crystal that is completely ordered over a distance up to billions of times the size of an individual atom in all directions. This crystal is the state of minimum energy for this system. The amazing fact is that, for slowly cooled systems, nature is able to find this minimum energy state. In fact, if a liquid metal is cooled quickly or “quenched,” it does not reach this state but rather ends up in a polycrystalline or amorphous state having somewhat higher energy. So the essence of the process is slow cooling, allowing ample time for redistribution of the atoms as they lose mobility. This is the technical definition of annealing, and it is essential for ensuring that a low energy state will be achieved
10.9 Simulated Annealing Methods 445 Although the analogy is not perfect.there is a sense in which all of the minimization algorithms thus far in this chapter correspond to rapid cooling or quenching.In all cases,we have gone greedily for the quick,nearby solution:From the starting point,go immediately downhill as far as you can go.This,as often remarked above,leads to a local,but not necessarily a global,minimum.Nature's own minimization algorithm is based on quite a different procedure.The so-called Boltzmann probability distribution, Prob (E)~exp(-E/kT) (10.9.1) 81 expresses the idea that a system in thermal equilibrium at temperature T has its energy probabilistically distributed among all different energy states E.Even at 县 low temperature,there is a chance,albeit very small,of a system being in a high energy state.Therefore,there is a corresponding chance for the system to get out of a local energy minimum in favor of finding a better,more global,one.The quantity (Boltzmann's constant)is a constant of nature that relates temperature to energy. In other words,the system sometimes goes uphill as well as downhill;but the lower the temperature,the less likely is any significant uphill excursion. 9 In 1953,Metropolis and coworkers [5]first incorporated these kinds of prin- ciples into numerical calculations.Offered a succession of options,a simulated thermodynamic system was assumed to change its configuration from energy E to energy E2 with probability p exp[-(E2-E1)/kT].Notice that if E2<E1,this probability is greater than unity;in such cases the change is arbitrarily assigned a probability p =1,i.e.,the system always took such an option.This general scheme, 、在总号分 of always taking a downhill step while sometimes taking an uphill step,has come to be known as the Metropolis algorithm. To make use ofthe Metropolis algorithm for other than thermodynamic systems, 61 one must provide the following elements: 1.A description of possible system configurations. 2.A generator of random changes in the configuration;these changes are the "options"presented to the system. 3.An objective function E(analog of energy)whose minimization is the Numerica 10621 goal of the procedure. 4.A control parameter T (analog of temperature)and an annealing schedule 营 431 which tells how it is lowered from high to low values,e.g.,after how many random changes in configuration is each downward step in T taken,and how large is that Recipes step.The meaning of"high"and "low"in this context,and the assignment of a schedule,may require physical insight and/or trial-and-error experiments. Combinatorial Minimization:The Traveling Salesman A concrete illustration is provided by the traveling salesman problem.The proverbial seller visits N cities with given positions(,yi),returning finally to his or her city of origin.Each city is to be visited only once,and the route is to be made as short as possible.This problem belongs to a class known as NP-complete problems, whose computation time for an exact solution increases with N as exp(const.x N), becoming rapidly prohibitive in cost as N increases.The traveling salesman problem also belongs to a class of minimization problems for which the objective function E
10.9 Simulated Annealing Methods 445 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Although the analogy is not perfect, there is a sense in which all of the minimization algorithms thus far in this chapter correspond to rapid cooling or quenching. In all cases, we have gone greedily for the quick, nearby solution: From the starting point, go immediately downhill as far as you can go. This, as often remarked above, leads to a local, but not necessarily a global, minimum. Nature’s own minimization algorithm is based on quite a different procedure. The so-called Boltzmann probability distribution, Prob (E) ∼ exp(−E/kT ) (10.9.1) expresses the idea that a system in thermal equilibrium at temperature T has its energy probabilistically distributed among all different energy states E. Even at low temperature, there is a chance, albeit very small, of a system being in a high energy state. Therefore, there is a corresponding chance for the system to get out of a local energy minimum in favor of finding a better, more global, one. The quantity k (Boltzmann’s constant) is a constant of nature that relates temperature to energy. In other words, the system sometimes goes uphill as well as downhill; but the lower the temperature, the less likely is any significant uphill excursion. In 1953, Metropolis and coworkers [5] first incorporated these kinds of principles into numerical calculations. Offered a succession of options, a simulated thermodynamic system was assumed to change its configuration from energy E 1 to energy E2 with probability p = exp[−(E2 − E1)/kT ]. Notice that if E2 < E1, this probability is greater than unity; in such cases the change is arbitrarily assigned a probability p = 1, i.e., the system always took such an option. This general scheme, of always taking a downhill step while sometimes taking an uphill step, has come to be known as the Metropolis algorithm. To make use of the Metropolis algorithm for other than thermodynamic systems, one must provide the following elements: 1. A description of possible system configurations. 2. A generator of random changes in the configuration; these changes are the “options” presented to the system. 3. An objective function E (analog of energy) whose minimization is the goal of the procedure. 4. A control parameter T (analog of temperature) and an annealing schedule which tells how it is lowered from high to low values, e.g., after how many random changes in configuration is each downward step in T taken, and how large is that step. The meaning of “high” and “low” in this context, and the assignment of a schedule, may require physical insight and/or trial-and-error experiments. Combinatorial Minimization: The Traveling Salesman A concrete illustration is provided by the traveling salesman problem. The proverbial seller visits N cities with given positions(xi, yi), returning finally to his or her city of origin. Each city is to be visited only once, and the route is to be made as short as possible. This problem belongs to a class known as NP-complete problems, whose computation time for an exact solution increases with N as exp(const. × N), becoming rapidly prohibitive in cost as N increases. The traveling salesman problem also belongs to a class of minimization problems for which the objective function E
446 Chapter 10.Minimization or Maximization of Functions has many local minima.In practical cases,it is often enough to be able to choose from these a minimum which,even if not absolute,cannot be significantly improved upon.The annealing method manages to achieve this,while limiting its calculations to scale as a small power of N. As a problem in simulated annealing,the traveling salesman problem is handled as follows: 1.Configuration.The cities are numberedi=1...N and each has coordinates (i,y).A configuration is a permutation of the number 1...N,interpreted as the order in which the cities are visited. 2.Rearrangements.An efficient set of moves has been suggested by Lin [61. The moves consist of two types:(a)A section of path is removed and then replaced with the same cities running in the opposite order,or(b)a section of path is removed and then replaced in between two cities on another,randomly chosen,part of the path. 3.Objective Function.In the simplest form of the problem,E is taken just as the total length of journey, N E=L=∑V-x+1P+(-+1P (10.9.2) s 9 with the convention that point N+1 is identified with point 1.To illustrate the flexibility of the method,however,we can add the following additional wrinkle: Suppose that the salesman has an irrational fear of flying over the Mississippi River. In that case,we would assign each city a parameter ui,equal to+1 if it is east of the 026 Mississippi,-1 if it is west,and take the objective function to be (工-工+12+(-+1P+A(山-4：+1)2 (10.9.3) 6 A penalty 4A is thereby assigned to any river crossing.The algorithm now finds the shortest path that avoids crossings.The relative importance that it assigns to length of path versus river crossings is determined by our choice of A.Figure 10.9.1 shows the results obtained.Clearly,this technique can be generalized to include Numerica 10621 many conflicting goals in the minimization. 4.Annealing schedule.This requires experimentation.We first generate some 431 random rearrangements,and use them to determine the range of values of AE that Recipes will be encountered from move to move.Choosing a starting value for the parameter (outside 腿 T which is considerably larger than the largest AE normally encountered,we proceed downward in multiplicative steps each amounting to a 10 percent decrease North in T.We hold each new value of T constant for,say,100N reconfigurations,or for 10N successful reconfigurations,whichever comes first.When efforts to reduce E further become sufficiently discouraging,we stop. The following traveling salesman program,using the Metropolis algorithm, illustrates the main aspects of the simulated annealing technique for combinatorial problems
446 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). has many local minima. In practical cases, it is often enough to be able to choose from these a minimum which, even if not absolute, cannot be significantly improved upon. The annealing method manages to achieve this, while limiting its calculations to scale as a small power of N. As a problem in simulated annealing, the traveling salesman problem is handled as follows: 1. Configuration. The cities are numbered i = 1 ...N and each has coordinates (xi, yi). A configuration is a permutation of the number 1 ...N, interpreted as the order in which the cities are visited. 2. Rearrangements. An efficient set of moves has been suggested by Lin [6]. The moves consist of two types: (a) A section of path is removed and then replaced with the same cities running in the opposite order; or (b) a section of path is removed and then replaced in between two cities on another, randomly chosen, part of the path. 3. Objective Function. In the simplest form of the problem, E is taken just as the total length of journey, E = L ≡ N i=1 (xi − xi+1)2 + (yi − yi+1)2 (10.9.2) with the convention that point N + 1 is identified with point 1. To illustrate the flexibility of the method, however, we can add the following additional wrinkle: Suppose that the salesman has an irrational fear of flying over the Mississippi River. In that case, we would assign each city a parameter µi, equal to +1 if it is east of the Mississippi, −1 if it is west, and take the objective function to be E = N i=1 (xi − xi+1)2 + (yi − yi+1)2 + λ(µi − µi+1) 2 (10.9.3) A penalty 4λ is thereby assigned to any river crossing. The algorithm now finds the shortest path that avoids crossings. The relative importance that it assigns to length of path versus river crossings is determined by our choice of λ. Figure 10.9.1 shows the results obtained. Clearly, this technique can be generalized to include many conflicting goals in the minimization. 4. Annealing schedule. This requires experimentation. We first generate some random rearrangements, and use them to determine the range of values of ∆E that will be encountered from move to move. Choosing a starting value for the parameter T which is considerably larger than the largest ∆E normally encountered, we proceed downward in multiplicative steps each amounting to a 10 percent decrease in T . We hold each new value of T constant for, say, 100N reconfigurations, or for 10N successful reconfigurations, whichever comes first. When efforts to reduce E further become sufficiently discouraging, we stop. The following traveling salesman program, using the Metropolis algorithm, illustrates the main aspects of the simulated annealing technique for combinatorial problems.
10.9 Simulated Annealing Methods 447 Permission is Sample page a b http://w.nr.com or call 1-800-872-7423(North America only),orsend email to directcustserv@cambridge.org(outside North America). granted for internet users to make one paper copy for thei Copyright (C)1988-1992 by Cambridge University Press.Programs Copyright(C)1988-1992 by Numerical Recipes from NUMERICAL RECIPES IN C:THE ART OF SCIENTIFIC COMPUTING(ISBN 0-521-43108-5) 1 Software. (c) 0 .5 Figure 10.9.1.Traveling salesman problem solved by simulated annealing.The (nearly)shortest path among 100 randomly positioned cities is shown in(a).The dotted line is a river,but there is no penalty in crossing.In (b)the river-crossing penalty is made large,and the solution restricts itself to the minimum number of crossings,two.In(c)the penalty has been made negative:the salesman is actually a smuggler who crosses the river on the flimsiest excuse!
10.9 Simulated Annealing Methods 447 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 0 .5 1 0 .5 1 0 .5 1 0 .5 1 0 .5 1 0 .5 1 (a) (b) (c) Figure 10.9.1. Traveling salesman problem solved by simulated annealing. The (nearly) shortest path among 100 randomly positioned cities is shown in (a). The dotted line is a river, but there is no penalty in crossing. In (b) the river-crossing penalty is made large, and the solution restricts itself to the minimum number of crossings, two. In (c) the penalty has been made negative: the salesman is actually a smuggler who crosses the river on the flimsiest excuse!
448 Chapter 10. Minimization or Maximization of Functions #include #include #define TFACTR 0.9 Annealing schedule:reduce t by this factor on each step. #define ALEN(a,b,c,d) sqrt(((b)-(a))*((b)-(a)+(d)-(c))*((d)-(c)) void anneal(float x[],float y[],int iorder[],int ncity) This algorithm finds the shortest round-trip path to ncity cities whose coordinates are in the arrays x[1..ncity],y[1..ncity].The array iorder [1..ncity]specifies the order in which the cities are visited.On input,the elements of iorder may be set to any permutation of the numbers 1 to ncity.This routine will return the best alternative path it can find. int irbiti(unsigned long *iseed); 常 int metrop(float de,float t); float ran3(long *idum); float revcst(float x[],float y,int iorder[],int ncity,int n[]); void reverse(int iorder[],int ncity,int n[]); granted for 19881992 float trncst(float x[],float y[],int iorder[],int ncity,int n[]); void trnspt(int iorder[],int ncity,int n[]); 1800 int ans,nover,nlimit,i1,12; int i,1,k,nsucc,nn,idec; static int n[7]; to any Cambridge from NUMERICAL RECIPES I long idum; unsigned long iseed; float path,de,ti (Nor to make nover=100*ncity; Maximum number of paths tried at any temperature. THE nlimit=10*ncity; Maximum number of successful path changes before con- America server computer, e University Press. path=0.0; tinuing. one paper ART t=0.5; for(i=1;1=n[1])+n[2]; 1988-1992 by Numerical Recipes -43108 nn=1+((n[1]-n[2]+ncity-1)%ncity); nn is the number of cities while (nn<3); not on the segment. idec=irbit1(&iseed); (outside Decide whether to do a segment reversal or transport. if (idec==0){ Do a transport. North Software. n[3]=n[2]+(int)(abs(nn-2)*ran3(&idum))+1; n[3]=1+(n[3]-1)%ncity); Transport to a location not on the path de=trncst(x,y,iorder,ncity,n); Calculate cost. ans=metrop(de,t); Consult the oracle if (ans){ ++nsucc; path +=de; trnspt(iorder,ncity,n); Carry out the transport. else Do a path reversal de=revcst(x,y,iorder,ncity,n); Calculate cost. ans=metrop(de,t); Consult the oracle
448 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). #include #include #define TFACTR 0.9 Annealing schedule: reduce t by this factor on each step. #define ALEN(a,b,c,d) sqrt(((b)-(a))*((b)-(a))+((d)-(c))*((d)-(c))) void anneal(float x[], float y[], int iorder[], int ncity) This algorithm finds the shortest round-trip path to ncity cities whose coordinates are in the arrays x[1..ncity],y[1..ncity]. The array iorder[1..ncity] specifies the order in which the cities are visited. On input, the elements of iorder may be set to any permutation of the numbers 1 to ncity. This routine will return the best alternative path it can find. { int irbit1(unsigned long *iseed); int metrop(float de, float t); float ran3(long *idum); float revcst(float x[], float y[], int iorder[], int ncity, int n[]); void reverse(int iorder[], int ncity, int n[]); float trncst(float x[], float y[], int iorder[], int ncity, int n[]); void trnspt(int iorder[], int ncity, int n[]); int ans,nover,nlimit,i1,i2; int i,j,k,nsucc,nn,idec; static int n[7]; long idum; unsigned long iseed; float path,de,t; nover=100*ncity; Maximum number of paths tried at any temperature. nlimit=10*ncity; Maximum number of successful path changes before conpath=0.0; tinuing. t=0.5; for (i=1;i= n[1]) ++n[2]; nn=1+((n[1]-n[2]+ncity-1) % ncity); nn is the number of cities } while (nn<3); not on the segment. idec=irbit1(&iseed); Decide whether to do a segment reversal or transport. if (idec == 0) { Do a transport. n[3]=n[2]+(int) (abs(nn-2)*ran3(&idum))+1; n[3]=1+((n[3]-1) % ncity); Transport to a location not on the path. de=trncst(x,y,iorder,ncity,n); Calculate cost. ans=metrop(de,t); Consult the oracle. if (ans) { ++nsucc; path += de; trnspt(iorder,ncity,n); Carry out the transport. } } else { Do a path reversal. de=revcst(x,y,iorder,ncity,n); Calculate cost. ans=metrop(de,t); Consult the oracle
10.9 Simulated Annealing Methods 449 if (ans){ ++nsucc; path de; reverse(iorder,ncity,n); Carry out the reversal ] if (nsucc >nlimit)break; Finish early if we have enough suc- 2 cessful changes printf("\n %s %10.6f %s %12.6f In","T =",t, Path Length ="path); printf("Successful Moves:%6d\n",nsucc); t *=TFACTR; Annealing schedule. if (nsucc ==0)return; If no success,we are done. (includi #include #define ALEN(a,b,c,d)sqrt(((b)-(a))*((b)-(a))+((d)-(c))*((d)-(c))) NUMERICAL float revcst(float x[],float y[],int iorder[],int ncity,int n[) This function returns the value of the cost function for a proposed path reversal.ncity is the RECIPES I number of cities,and arrays x[1..ncity],y[1..ncity]give the coordinates of these cities. iorder [1..ncity]holds the present itinerary.The first two values n[1]and n[2]of array n give the starting and ending cities along the path segment which is to be reversed.On output, de is the cost of making the reversal.The actual reversal is not performed by this routine. Press. float xx[5],yy[5],de; int j,ii; n[3]=1+((n[1]+ncity-2)%ncity); Find the city before n[1].. n[4]=1+(n[2]%ncity); .and the city after n[2] for(j=1:j<=4;j+){ ii=iorder[n[i]]; Find coordinates for the four cities in- IENTIFIC xx[j]=x[11]; volved. yy[j]=y[ii]; de =-ALEN(xx[1],xx[3],yy[1],yy [3]); Calculate cost of disconnecting the seg- de -ALEN(xx[2],xx[4],yy[2],yy [4]); ment at both ends and reconnecting de +ALEN(xx[1],xx[4],yy[1],yy[4]); in the opposite order. 1920 COMPUTING(ISBN de+=ALE(xx[2],xx[3],yy[2],yy[3]); return de; Numerical 1021 Recipes 43108 void reverse(int iorder[],int ncity,int n[]) This routine performs a path segment reversal.iorder [1..ncity]is an input array giving the present itinerary.The vector n has as its first four elements the first and last cities n[1],n[2] (outside of the path segment to be reversed,and the two cities n[3]and n[4]that immediately precede and follow this segment.n[3]and n[4]are found by function revcst.On output, 首 Software. iorder [1..ncity]contains the segment from n[1]to n[2]in reversed order. f int nn,j,k,1,itmp; nn=(1+((n[2]-n[1]+ncity)%ncity))/2; This many cities must be swapped to for (j=1;j<=nn;j++) effect the reversal. k=1+(n[1]+j-2)%ncity): Start at the ends of the segment and 1=1 ((n[2]-j+ncity)ncity); swap pairs of cities,moving toward itmp=iorder[k]; the center. iorder[k]=iorder[1]; iorder[l]=itmp;
10.9 Simulated Annealing Methods 449 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). if (ans) { ++nsucc; path += de; reverse(iorder,ncity,n); Carry out the reversal. } } if (nsucc >= nlimit) break; Finish early if we have enough suc- } cessful changes. printf("\n %s %10.6f %s %12.6f \n","T =",t, " Path Length =",path); printf("Successful Moves: %6d\n",nsucc); t *= TFACTR; Annealing schedule. if (nsucc == 0) return; If no success, we are done. } } #include #define ALEN(a,b,c,d) sqrt(((b)-(a))*((b)-(a))+((d)-(c))*((d)-(c))) float revcst(float x[], float y[], int iorder[], int ncity, int n[]) This function returns the value of the cost function for a proposed path reversal. ncity is the number of cities, and arrays x[1..ncity],y[1..ncity] give the coordinates of these cities. iorder[1..ncity] holds the present itinerary. The first two values n[1] and n[2] of array n give the starting and ending cities along the path segment which is to be reversed. On output, de is the cost of making the reversal. The actual reversal is not performed by this routine. { float xx[5],yy[5],de; int j,ii; n[3]=1 + ((n[1]+ncity-2) % ncity); Find the city before n[1] .. n[4]=1 + (n[2] % ncity); .. and the city after n[2]. for (j=1;j<=4;j++) { ii=iorder[n[j]]; Find coordinates for the four cities inxx[j]=x[ii]; volved. yy[j]=y[ii]; } de = -ALEN(xx[1],xx[3],yy[1],yy[3]); Calculate cost of disconnecting the segment at both ends and reconnecting in the opposite order. de -= ALEN(xx[2],xx[4],yy[2],yy[4]); de += ALEN(xx[1],xx[4],yy[1],yy[4]); de += ALEN(xx[2],xx[3],yy[2],yy[3]); return de; } void reverse(int iorder[], int ncity, int n[]) This routine performs a path segment reversal. iorder[1..ncity] is an input array giving the present itinerary. The vector n has as its first four elements the first and last cities n[1],n[2] of the path segment to be reversed, and the two cities n[3] and n[4] that immediately precede and follow this segment. n[3] and n[4] are found by function revcst. On output, iorder[1..ncity] contains the segment from n[1] to n[2] in reversed order. { int nn,j,k,l,itmp; nn=(1+((n[2]-n[1]+ncity) % ncity))/2; This many cities must be swapped to for (j=1;j<=nn;j++) { effect the reversal. k=1 + ((n[1]+j-2) % ncity); Start at the ends of the segment and swap pairs of cities, moving toward the center. l=1 + ((n[2]-j+ncity) % ncity); itmp=iorder[k]; iorder[k]=iorder[l]; iorder[l]=itmp; } }
450 Chapter 10.Minimization or Maximization of Functions #include #define ALEN(a,b,c,d)sqrt(((b)-(a))*((b)-(a))+((d)-(c))*((d)-(c))) float trncst(float x[],float y[],int iorder[],int ncity,int n[]) This routine returns the value of the cost function for a proposed path segment transport.ncity is the number of cities,and arrays x[1..ncity]and y[1..ncity]give the city coordinates. iorder [1..ncity]is an array giving the present itinerary.The first three elements of array n give the starting and ending cities of the path to be transported,and the point among the remaining cities after which it is to be inserted.On output,de is the cost of the change.The actual transport is not performed by this routine. f float xx[7],yy[7],de; int j,ii; n[4]=1+(n[3]%ncity); Find the city following n[3].. (includi n[5]=1 +((n[1]+ncity-2)%ncity); .and the one preceding n[1].. n[6]=1+(n[2]%ncity); .and the one following n[2]. for(j=1;j<=6;j++){ ii=iorder[n[j]]; Determine coordinates for the six cities xx[j]=x[11]; involved. yy[j]=y[ii]; NUMERICAL RECIPES I de =-ALEN(xx[2],xx[6],yy[2],yy [6]) Calculate the cost of disconnecting the de -ALEN(xx[1],xx[5],yy[1],yy [5]); path segment from n[1]to n[2]. (Nort server 2 de -ALEN(xx [3],xx[4],yy[3],yy [4]); opening a space between n[3]and de +ALEN(xx[1],xx[3],yy[1],yy [3]); n [4],connecting the segment in the de+=ALE(xx[2],xx[4],yy[2],y[4]) space,and connecting n[5]to n[6] Americ computer e University Press. THE de +ALEN(xx[5],xx[6],yy [5],yy[6]); ART return de; Programs #include "nrutil.h" void trnspt(int iorder[],int ncity,int n) to dir This routine does the actual path transport,once metrop has approved.iorder [1..ncity] is an input array giving the present itinerary.The array n has as its six elements the beginning n[1]and end n[2]of the path to be transported,the adjacent cities n[3]and n[4]between SCIENTIFIC COMPUTING(ISBN which the path is to be placed,and the citiesn[]and n[6]that precede and follow the path. 198918920 n[4],n[5],and n[6]are calculated by function trncst.On output,iorder is modified to reflect the movement of the path segment. 10-621 int m1,m2,m3,nn,j,jj,*jorderi jorder=ivector(1,ncity); Numerical Recipes 43108 mi=1+((n[2]-n[1]+ncity)ncity); Find number of cities from n[1]to n[2] m2=1 ((n[5]-n[4]+ncity)ncity); ..and the number from n[4]to n[5] m3=1 ((n[3]-n[6]+ncity)%ncity); ...and the number from n[6]to n[3]. (outside nn=1; North Software. for(j=1;j<=m1;j++) jj=1+((j+n[1]-2)%ncity); Copy the chosen segment. jorder[nn++]=iorder[jj]; for(j=1;j<=m2;j++){ Then copy the segment from n[4]to jj=1+(j+n[4]-2)%ncity); n[5]. jorder [nn++]=iorder[jj]; for(j=1:j<=m3;j++) Finally,the segment from n[6]to n[3] jj=1+((j+n[6]-2)%ncity); jorder[nn++]=iorder[jj]; for (j=1;j<=ncity;j++) Copy jorder back into iorder. iorder[j]=jorder[j];
450 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). #include #define ALEN(a,b,c,d) sqrt(((b)-(a))*((b)-(a))+((d)-(c))*((d)-(c))) float trncst(float x[], float y[], int iorder[], int ncity, int n[]) This routine returns the value of the cost function for a proposed path segment transport. ncity is the number of cities, and arrays x[1..ncity] and y[1..ncity] give the city coordinates. iorder[1..ncity] is an array giving the present itinerary. The first three elements of array n give the starting and ending cities of the path to be transported, and the point among the remaining cities after which it is to be inserted. On output, de is the cost of the change. The actual transport is not performed by this routine. { float xx[7],yy[7],de; int j,ii; n[4]=1 + (n[3] % ncity); Find the city following n[3].. n[5]=1 + ((n[1]+ncity-2) % ncity); ..and the one preceding n[1].. n[6]=1 + (n[2] % ncity); ..and the one following n[2]. for (j=1;j<=6;j++) { ii=iorder[n[j]]; Determine coordinates for the six cities xx[j]=x[ii]; involved. yy[j]=y[ii]; } de = -ALEN(xx[2],xx[6],yy[2],yy[6]); Calculate the cost of disconnecting the path segment from n[1] to n[2], opening a space between n[3] and n[4], connecting the segment in the space, and connecting n[5] to n[6]. de -= ALEN(xx[1],xx[5],yy[1],yy[5]); de -= ALEN(xx[3],xx[4],yy[3],yy[4]); de += ALEN(xx[1],xx[3],yy[1],yy[3]); de += ALEN(xx[2],xx[4],yy[2],yy[4]); de += ALEN(xx[5],xx[6],yy[5],yy[6]); return de; } #include "nrutil.h" void trnspt(int iorder[], int ncity, int n[]) This routine does the actual path transport, once metrop has approved. iorder[1..ncity] is an input array giving the present itinerary. The array n has as its six elements the beginning n[1] and end n[2] of the path to be transported, the adjacent cities n[3] and n[4] between which the path is to be placed, and the cities n[5] and n[6] that precede and follow the path. n[4], n[5], and n[6] are calculated by function trncst. On output, iorder is modified to reflect the movement of the path segment. { int m1,m2,m3,nn,j,jj,*jorder; jorder=ivector(1,ncity); m1=1 + ((n[2]-n[1]+ncity) % ncity); Find number of cities from n[1] to n[2] m2=1 + ((n[5]-n[4]+ncity) % ncity); ...and the number from n[4] to n[5] m3=1 + ((n[3]-n[6]+ncity) % ncity); ...and the number from n[6] to n[3]. nn=1; for (j=1;j<=m1;j++) { jj=1 + ((j+n[1]-2) % ncity); Copy the chosen segment. jorder[nn++]=iorder[jj]; } for (j=1;j<=m2;j++) { Then copy the segment from n[4] to jj=1+((j+n[4]-2) % ncity); n[5]. jorder[nn++]=iorder[jj]; } for (j=1;j<=m3;j++) { Finally, the segment from n[6] to n[3]. jj=1 + ((j+n[6]-2) % ncity); jorder[nn++]=iorder[jj]; } for (j=1;j<=ncity;j++) Copy jorder back into iorder. iorder[j]=jorder[j];
10.9 Simulated Annealing Methods 451 free_ivector(jorder,1,ncity); 2 #include int metrop(float de,float t) Metropolis algorithm.metrop returns a boolean variable that issues a verdict on whether to accept a reconfiguration that leads to a change de in the objective function e.If de0,metrop is only true with probability exp(-de/t),where t is a temperature determined by the annealing schedule. float ran3(long *idum); 83g static long gljdum=1; return de 0.0 II ran3(&gljdum)<exp(-de/t); 1-800 NUMERICAL RECIPES I Continuous Minimization by Simulated Annealing computer, The basic ideas of simulated annealing are also applicable to optimization problems with continuous N-dimensional control spaces,e.g.,finding the (ideally. 9 Programs global)minimum of some function f(x).in the presence of many local minima. where x is an N-dimensional vector.The four elements required by the Metropolis 萱 OF SCIENTIFIC procedure are now as follows:The value of f is the objective function.The system state is the point x.The control parameter T is,as before,something like a temperature,with an annealing schedule by which it is gradually reduced.And there must be a generator of random changes in the configuration,that is,a procedure for taking a random step from x to x+Ax. The last of these elements is the most problematical.The literature to date [7-10] describes several different schemes for choosing Ax.none of which,in our view. 10621 inspire complete confidence.The problem is one of efficiency:A generator of Recipes Numerica random changes is inefficient if,when local downhill moves exist,it nevertheless 43106 almost always proposes an uphill move.A good generator,we think,should not Recipes become inefficient in narrow valleys:nor should it become more and more inefficient (outside as convergence to a minimum is approached.Except possibly for [7],all of the schemes that we have seen are inefficient in one or both of these situations. Software. 首 Our own way of doing simulated annealing minimization on continuous control spaces is to use a modification of the downhill simplex method (10.4).This amounts to replacing the single point x as a description of the system state by a simplex of N+1 points.The "moves"are the same as described in $10.4,namely reflections, expansions,and contractions of the simplex.The implementation of the Metropolis procedure is slightly subtle:We add a positive,logarithmically distributed random variable,proportional to the temperature T,to the stored function value associated with every vertex of the simplex,and we subtract a similar random variable from the function value of every new point that is tried as a replacement point.Like the ordinary Metropolis procedure,this method always accepts a true downhill step,but
10.9 Simulated Annealing Methods 451 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). free_ivector(jorder,1,ncity); } #include int metrop(float de, float t) Metropolis algorithm. metrop returns a boolean variable that issues a verdict on whether to accept a reconfiguration that leads to a change de in the objective function e. If de0, metrop is only true with probability exp(-de/t), where t is a temperature determined by the annealing schedule. { float ran3(long *idum); static long gljdum=1; return de < 0.0 || ran3(&gljdum) < exp(-de/t); } Continuous Minimization by Simulated Annealing The basic ideas of simulated annealing are also applicable to optimization problems with continuous N-dimensional control spaces, e.g., finding the (ideally, global) minimum of some function f(x), in the presence of many local minima, where x is an N-dimensional vector. The four elements required by the Metropolis procedure are now as follows: The value of f is the objective function. The system state is the point x. The control parameter T is, as before, something like a temperature, with an annealing schedule by which it is gradually reduced. And there must be a generator of random changes in the configuration, that is, a procedure for taking a random step from x to x + ∆x. The last of these elements is the most problematical. The literature to date [7-10] describes several different schemes for choosing ∆x, none of which, in our view, inspire complete confidence. The problem is one of efficiency: A generator of random changes is inefficient if, when local downhill moves exist, it nevertheless almost always proposes an uphill move. A good generator, we think, should not become inefficient in narrow valleys; nor should it become more and more inefficient as convergence to a minimum is approached. Except possibly for [7], all of the schemes that we have seen are inefficient in one or both of these situations. Our own way of doing simulated annealing minimization on continuous control spaces is to use a modification of the downhill simplex method (§10.4). This amounts to replacing the single point x as a description of the system state by a simplex of N + 1 points. The “moves” are the same as described in §10.4, namely reflections, expansions, and contractions of the simplex. The implementation of the Metropolis procedure is slightly subtle: We add a positive, logarithmically distributed random variable, proportional to the temperature T , to the stored function value associated with every vertex of the simplex, and we subtract a similar random variable from the function value of every new point that is tried as a replacement point. Like the ordinary Metropolis procedure, this method always accepts a true downhill step, but
452 Chapter 10.Minimization or Maximization of Functions sometimes accepts an uphill one.In the limit T-0.this algorithm reduces exactly to the downhill simplex method and converges to a local minimum. At a finite value of T,the simplex expands to a scale that approximates the size of the region that can be reached at this temperature,and then executes a stochastic, tumbling Brownian motion within that region,sampling new,approximately random, points as it does so.The efficiency with which a region is explored is independent of its narrowness (for an ellipsoidal valley,the ratio of its principal axes)and orientation.If the temperature is reduced sufficiently slowly,it becomes highly likely that the simplex will shrink into that region containing the lowest relative minimum encountered 81 As in all applications of simulated annealing,there can be quite a lot of problem-dependent subtlety in the phrase "sufficiently slowly";success or failure is quite often determined by the choice of annealing schedule.Here are some possibilities worth trying: Reduce T to (1-e)T after every m moves,where e/m is determined by experiment. Budget a total of K moves,and reduce T after every m moves to a value T=To(1-k/K),where k is the cumulative number of moves thus far, 令 and a is a constant,say 1,2,or 4.The optimal value for a depends on the statistical distribution of relative minima of various depths.Larger values of a spend more iterations at lower temperature. After every m moves,set T to B times fi-fo,where B is an experimentally 、兰a2。d determined constant of order 1,f is the smallest function value currently represented in the simplex,and fo is the best function ever encountered. However,never reduce T by more than some fraction y at a time. 人了 OF SCIENTIFIC Another strategic question is whether to do an occasional restart,where a vertex of the simplex is discarded in favor of the "best-ever"point.(You must be sure that to dir the best-ever point is not currently in the simplex when you do this!)We have found problems for which restarts-every time the temperature has decreased by a factor of 3,say-are highly beneficial;we have found other problems for which restarts have no positive,or a somewhat negative,effect. You should compare the following routine,amebsa,with its counterpart amoeba in 810.4.Note that the argument iter is used in a somewhat different manner. #include Numerical Recipes 10621 43106 #include "nrutil.h" #define GET_PSUM for (n=1;n<-ndimin++){\ (outside for (sum=0.0,m=1;m<=mpts;m++)sum +p[m][n]; Software. psum [n]=sum;} extern long idum; Defined and initialized in main float tt; Communicates with amotsa. void amebsa(float **p,float y[],int ndim,float pb[],float *yb,float ftol, float (*funk)(float [])int *iter,float temptr) Multidimensional minimization of the function funk(x)where x[1..ndim]is a vector in ndim dimensions,by simulated annealing combined with the downhill simplex method of Nelder and Mead.The input matrix p[1..ndim+1][1..ndim]has ndim+1 rows,each an ndim- dimensional vector which is a vertex of the starting simplex.Also input are the following:the vector y[1..ndim+1].whose components must be pre-initialized to the values of funk eval- uated at the ndim+1 vertices (rows)of p;ftol,the fractional convergence tolerance to be achieved in the function value for an early return;iter,and temptr.The routine makes iter function evaluations at an annealing temperature temptr,then returns.You should then de-
452 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). sometimes accepts an uphill one. In the limit T → 0, this algorithm reduces exactly to the downhill simplex method and converges to a local minimum. At a finite value of T , the simplex expands to a scale that approximates the size of the region that can be reached at this temperature, and then executes a stochastic, tumbling Brownian motion within that region, sampling new, approximately random, points as it does so. The efficiency with which a region is explored is independent of its narrowness (for an ellipsoidal valley, the ratio of its principal axes) and orientation. If the temperature is reduced sufficiently slowly, it becomes highly likely that the simplex will shrink into that region containing the lowest relative minimum encountered. As in all applications of simulated annealing, there can be quite a lot of problem-dependent subtlety in the phrase “sufficiently slowly”; success or failure is quite often determined by the choice of annealing schedule. Here are some possibilities worth trying: • Reduce T to (1 − )T after every m moves, where /m is determined by experiment. • Budget a total of K moves, and reduce T after every m moves to a value T = T0(1 − k/K)α, where k is the cumulative number of moves thus far, and α is a constant, say 1, 2, or 4. The optimal value for α depends on the statistical distribution of relative minima of various depths. Larger values of α spend more iterations at lower temperature. • After every m moves, set T to β times f1−fb, where β is an experimentally determined constant of order 1, f1 is the smallest function value currently represented in the simplex, and fb is the best function ever encountered. However, never reduce T by more than some fraction γ at a time. Another strategic question is whether to do an occasionalrestart, where a vertex of the simplex is discarded in favor of the “best-ever” point. (You must be sure that the best-ever point is not currently in the simplex when you do this!) We have found problems for which restarts — every time the temperature has decreased by a factor of 3, say — are highly beneficial; we have found other problems for which restarts have no positive, or a somewhat negative, effect. You should compare the following routine, amebsa, with its counterpart amoeba in §10.4. Note that the argument iter is used in a somewhat different manner. #include #include "nrutil.h" #define GET_PSUM \ for (n=1;n<=ndim;n++) {\ for (sum=0.0,m=1;m<=mpts;m++) sum += p[m][n];\ psum[n]=sum;} extern long idum; Defined and initialized in main. float tt; Communicates with amotsa. void amebsa(float **p, float y[], int ndim, float pb[], float *yb, float ftol, float (*funk)(float []), int *iter, float temptr) Multidimensional minimization of the function funk(x) where x[1..ndim] is a vector in ndim dimensions, by simulated annealing combined with the downhill simplex method of Nelder and Mead. The input matrix p[1..ndim+1][1..ndim] has ndim+1 rows, each an ndimdimensional vector which is a vertex of the starting simplex. Also input are the following: the vector y[1..ndim+1], whose components must be pre-initialized to the values of funk evaluated at the ndim+1 vertices (rows) of p; ftol, the fractional convergence tolerance to be achieved in the function value for an early return; iter, and temptr. The routine makes iter function evaluations at an annealing temperature temptr, then returns. You should then de-
10.9 Simulated Annealing Methods 453 crease temptr according to your annealing schedule,reset iter,and call the routine again (leaving other arguments unaltered between calls).If iter is returned with a positive value, then early convergence and return occurred.If you initialize yb to a very large value on the first call,then yb and pb[1..ndim]will subsequently return the best function value and point ever encountered (even if it is no longer a point in the simplex). float amotsa(float **p,float y[],float psum[],int ndim,float pb[], float *yb,float (*funk)(float [])int ihi,float *yhi,float fac); float ran1(long *idum); int i,ihi,ilo,j,m,n,mpts=ndim+1; float rtol,sum,swap,yhi,ylo,ynhi,ysave,yt,ytry,*psum; 常 psum=vector(1,ndim); tt =-temptri GET_PSUM 83 for(;;)[ 18881892 110=1: Determine which point is the highest(worst), ihi=2; next-highest,and lowest (best). g ynhi=ylo=y [1]+tt*log(ran1(&idum)); Whenever we "look at"a vertex,it gets yhi=y[2]+tt*log(ran1(&idum)); a random thermal fluctuation. if (ylo yhi){ from NUMERICAL RECIPES I 110=2: ynhisyhi; yhi=ylo; ylo=ynhi; America computer, for (i-3;iynhi){ ynhi-yt; ectcustser 18881920 SCIENTIFIC COMPUTING(ISBN rtol=2.0*fabs(yhi-ylo)/(fabs(yhi)+fabs(ylo)); v@cam Compute the fractional range from highest to lowest and return if satisfactory. 10-621 if (rtol ftol ll *iter 0){ If returning,put best point and value in swap=y[1]; slot 1. y[1]=y[i1o]; Numerical Recipes -43108 y[ilo]-swap; for (n=1;nynhi){ The reflected point is worse than the second-highest,so look for an intermediate
10.9 Simulated Annealing Methods 453 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). crease temptr according to your annealing schedule, reset iter, and call the routine again (leaving other arguments unaltered between calls). If iter is returned with a positive value, then early convergence and return occurred. If you initialize yb to a very large value on the first call, then yb and pb[1..ndim] will subsequently return the best function value and point ever encountered (even if it is no longer a point in the simplex). { float amotsa(float **p, float y[], float psum[], int ndim, float pb[], float *yb, float (*funk)(float []), int ihi, float *yhi, float fac); float ran1(long *idum); int i,ihi,ilo,j,m,n,mpts=ndim+1; float rtol,sum,swap,yhi,ylo,ynhi,ysave,yt,ytry,*psum; psum=vector(1,ndim); tt = -temptr; GET_PSUM for (;;) { ilo=1; Determine which point is the highest (worst), ihi=2; next-highest, and lowest (best). ynhi=ylo=y[1]+tt*log(ran1(&idum)); Whenever we “look at” a vertex, it gets yhi=y[2]+tt*log(ran1(&idum)); a random thermal fluctuation. if (ylo > yhi) { ihi=1; ilo=2; ynhi=yhi; yhi=ylo; ylo=ynhi; } for (i=3;i yhi) { ynhi=yhi; ihi=i; yhi=yt; } else if (yt > ynhi) { ynhi=yt; } } rtol=2.0*fabs(yhi-ylo)/(fabs(yhi)+fabs(ylo)); Compute the fractional range from highest to lowest and return if satisfactory. if (rtol = ynhi) { The reflected point is worse than the second-highest, so look for an intermediate