SECTION 16.3 GLOBAL OPTIMIZATION BY GENETIC ALGORITHM (GENALG)

Genetic algorithms start by generating a set of random feasible
solutions. The best of these solutions are permitted to "mate" and
produce offspring and mutations are occasionally introduced. The worst
solutions in every generation are discarded and the process continues
until either an iteration limit is reached or convergence takes place.
See R. L. Haupt and S. E. Haupt, *Practical Genetic Algorithms*, Wiley,
1998 and D. E. Goldberg, *Genetic Algorithms in Search, Optimization, and Machine Learning*, Addison-Wesley, 1989.

Suitable parameters for the algorithm should be determined by experimentation, since good values are problem dependent. This implementation is a continuous parameter (thus not a binary encoded) version of genetic algorithms.

In the MAIN program you must include

COMMON/GENALG2/NC,INITGEN,IPAIR,IMATE,KTERL,IMUTE, * ITEST,IHOWMAN COMMON/GENALG3/YMUTE,ALPH1,BET1,BACC,PM4,SHRINK1,EXPAND COMMON/BPRINT/IPT,NFILE,NDIG,NPUNCH,JPT,MFILE CALL DFLT OPEN (UNIT=nfile,file='filename')

where NFILE is initialized by DFLT to 8 and will be the output file.

The call to GENALG is

CALL GENALG(MX,NP,NIPOP,NPOP,NGOOD,XL,XH,CC,PP,CPP, * MATES,X,POP,U,FVAL,FUNC,ITERC,IER,IVAL,NCONT)

where

MX = 1 to maximize, = 2 to minimize NP = the number of parameters NIPOP = the initial population (a power of 2; 128 is often suitable) NPOP = the "working size" of the population (a power of 2, 64 is often suitable) NGOOD = the subpopulation that is permitted to mate (a power of 2; 32 is often suitable) XL = a DOUBLE PRECISION vector dimensioned XL(NP) containing the lower bounds of the search area XH = a DOUBLE PRECISION vector dimensioned XH(NP) containing the upper bounds of the search area CC,PP,CPP = DOUBLE PRECISION work arrays, each dimensioned NP, i.e., CC(NP), PP(NP), CPP(NP) MATES = a work array dimensioned MATES(2,NGOOD/2) X = a DOUBLE PRECISION work array dimensioned X(NP) POP = an INTEGER array dimensioned POP(NIPOP,NP) containing the initial and subsequent populations; after convergence POP(1,.) contains the parameter values for the best solution U = DOUBLE PRECISION work array dimensioned U((NIPOP,NP) FVAL = DOUBLE PRECISION array which will contain the function values corresponding to the population members; after convergence, FVAL(1) contains the best function value FUNC = the name of the function subroutine to be minimized or maximized; must be declared in an EXTERNAL statement in the MAIN program. See Section 1.1 for the conventions concerning FUNC subroutines. ITERC = the iteration (generation) count IVAL = the number of function evaluations IER = error return; =-1 iteration limit exceeded, =-3 input error, =-5 continual straying into a forbidden region NCONT = an integer value which causes the computations to be repeated if > 1. The repetition is as follows: after the first "pass" has converged, the best parameter values (in POP(1,.)) are retained, the feasible search area is shrunk (except in directions in which the optimum value is near the original boundary, in which case the search area is expanded in that direction), a new set of NIPOP-1 points are generated and the algorithm resumes. See also SHRINK1 and EXPAND below. A value of NCONT > 1 is useful to refine the location of a local optimum, but may hurt the chances of finding a global optimum. IHOWMAN = the number of consecutive generations (iterations) in which either the optimal function value or the mean function value is allowed to remain unchanged without terminationg the computations; must be <=9. (Default=5)

The following parameters, communicated to GENALG through labelled COMMON, are also important.

COMMON/GENALG2/NC,INITGEN,IPAIR,IMATE,KTERL,IMUTE, * ITEST,IHOWMAN

where

NC = 1 should not be altered; this is a provision for future modifications INITGEN = 2 initial population is generated randomly; 3 no initial population is generated; user is responsible for filling the NIPOP rows of POP; 4 one half of the population is generated randomly; from the ith row of POP (i=1,nipop/2) the remaining rows are generated by POP(nipop/2+i,j)=XH(j)-POP(i,j) (Default=2) IPAIR = 1 parents are selected for pairing as follows: the lowest-cost parent mates with the second-lowest cost parent, the third-lowest with the fourth-lowest, and so on; = 2 pairing by rank-weighting (See Haupt and Haupt, p. 39); = 3 pairing by cost weighting (See Haupt and Haupt, pp. 39-40) (default=3); = 4 random pairing IMATE = 1 each parent pair creates four potential offspring which differ in that for two the ic-th parameter of the offspring being a (BET1,1-BET1) and a (1-BET1,BET1) convex combination of the parents and for another two a (1+ALPH1,-ALPH1) and a (-ALPH1, ALPH1) combination, with the best two selected; = 2 each parent produces three potential offspring, the first of which is for the ic-th parameter a (u,1-u) combination of the parents (where u is a U(0,1) deviate), and the remaining two as immediately above, with the best two chosen; KTERL = iteration limit (default=99) IMUTE = 1 mutation rate remains constant; = 2 mutation rate decays 10% each generation; = 3 mutation is given by POP(I,.)*(1.5-u) where u is distributed as U(0,1); ITEST = 1 if testing for convergence should be based on the optimal value achieved in a population (default=1); = 2 if testing should be based on the mean value of the population (this latter is likely to give a better optimum, but may take a great deal more computation); IHOWMAN = the number of consecutive generations (iterations) in which either the optimal function value or the mean function value is allowed to remain unchanged without terminationg the computations; must be <=9. (default=5)

COMMON/GENALG3/YMUTE,ALPH1,BET1,BACC,PM4,SHRINK1,EXPAND

YMUTE = DOUBLE PRECISION constant defining the mutation rate; the number of mutations in a population will be (NPOP+NGOOD)*NP*YMUTE. The required number of individuals in the population will be chosen at random (except the best element in the population will not be subject to mutation) and then two randomly chosen city pairs will be exchanged (default=0.06) BACC = DOUBLE PRECISION required relative accuracy (default= 0.001) ALPH1 = extrapolation factor for mating (see IMATE above) (default =0.1) BET1 = convex combination factor for mating (see (IMATE above (default = 0.5) PM4 = used internally; user should not alter this SHRINK1 = Shrinkage factor for region of exploration in successive passes. XL and XH for each parameter are reset as XL = XL + SHRINK1*(POP(1,.)-XL) and XH = XH + SHRINK1*(POP(1,.)-XH) (default=0.95) EXPAND = Expansion factor for region of exploration in successive passes. XL and XH for each parameter are reset as XL = XL - EXPAND*(XH-XL) and XH = XH + EXPAND*(XH-XL) (default=0.05)