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1. 1 AA 2 AA 1 AA 2 AA 1 AA 2 AA 1 AA 2 AA 1 AA 2 AA 8 9842012988325989E 02 0 7126564030053476 1 031628453483664 T8 9842001740361430E 02 0 7126564016386691 1 031628453483664 1 703606615756318 0 7960835641969248 0 2154630210843795 1 703606616107386 0 7960835644085253 0 2154630210843795 1 607104589746949 0 5686514782893214 2 104250946486463 71 607104586736009 0 5686514769482340 2 104250946486463 18 5 5 5 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 4 Examples 4 1 Polynomial Root Finder As an example of an application of OPTIMA we present a polynomial root finder The method we use is based on Bairstow s algorithm and is briefly described here Let Pir TR apr be a polynomial of degree n with real coefficients ag k 0 1 2 n The division with a quadratic polynomial P x x pit p2 Bn 2 a pi p2 R pi po x Q pr p2 1 can be made perfect by choosing properly p and po so as to make the remainder terms R and Q vanish The coefficients bg k 0 1 n 2 of B _2 as well as R and Q are obtained by synthetic division i e via a recursion relation If we determine such values for p and po then Pir has two roots that coincide with the
2. 2 10 X 1 4 H 2 1 1 H 2 2 48 X 2 2 8 END 3 2 PRICE Test Run To execute the test run issue the following command run merlin price d 6hump f The following output is produced on the standard output PRICE running with Number of parameters 2 Maximum function calls 10000 Termination criterion 1 0000E 04 Weighting factor 1 0000Et04 Sample size factor 20 Printout option 1 Output file PRICE OUT Bounds file POIMAR DAT Merlin input file in dat Merlin output file out dat Output format 1 Iter 30 Lower value 0 14170040690117 Calls 82 of 10000 Iter 35 Lower value 0 22434743776021 Calls 92 of 10000 Iter 48 Lower value 0 64718662239411 Calls 111 of 10000 Iter 115 Lower value 0 72788366844541 Calls 199 of 10000 Iter 140 Lower value 0 83495615401031 Calls 239 of 10000 Iter 141 Lower value 1 0275395980524 Calls 241 of 10000 12 1 0277942972752 Iter 239 Lower value 1 0284090174387 Calls 390 of 10000 Iter 257 Lower value 1 0313613560203 Calls 408 of 10000 Iter 312 Lower value 1 0313833436910 Calls 463 of 10000 Iter 335 Lower value 1 0316284204941 Calls 486 of 10000 The termination criterion has been satisfied Function value 1 0316284534837 Total function evaluations 661 PRICE Iterations 480 The minimum has been refined by Merlin GRMS Output parameters in file PRICE OUT 3 92365572601884E 10 The relevant input files are listed File
3. 299 Iterations 7 Coverage 0 99201 Est minima 6 6 Iterations 8 Coverage 0 99516 Est minima 6 5 Iterations 9 Coverage 0 99674 Est minima 6 4 Iterations 10 Coverage 0 99769 Est minima 6 3 TML run completed Number of local minimizers found The minimizers are disposed to file MINIMA Total number of Function calls 1914 Total number of Gradient calls 838 Total number of Jacobian calls U Total number of Hessian calls 0 14 Total number of local optimizations O51 Maximum number of starting points 13 Current state has been saved The relevant input files are listed File POIMAR DAT 0 0 5 0 5 0 1 0 0 5 0 5 0 1 File TML DAT 2 NOD 0 NOF 10 ITTHR 5 0000000000000 HEAL 2 NEARN 100 NSAMPL 4 0000000000000 SIGMA 0 99500000476837 COVTHR 0 10000000000000E 01 XDEPS 0 10000000000000E 03 FDEPS 0 10000000000000E 02 GDEPS 0 IREST 100 NDUMP in dat FINP out dat FOUT MINIMA FMIN POIMAR DAT POIFIL input INDIR File in dat ANAL BFGS NOC 2000 File MINIMA Contains the minimizers found 2 O 8 9842012988325989E 02 5 000 20 0 7126564030053476 5 000 Value 1 031628453483664 L 8 9842001740361430E 02 5 000 15 2 aaa 0 7126564016386691 5 Value 1 031628453483664 L 1 703606615756318 5 2 aaa 0 7960835641969248 5 Value 0 2154630210843795 L 1 TO3606616107 386 5 2 aaa 0 7960835644085253 5 Value 0 2154630210843795 L 1 6071045867 36
4. 30 1 MERLIN Set Up Before running the global optimization codes one should install properly the MERLIN opti mization package 1 2 1 1 MERLIN installation The following instructions are valid for installing MERLIN under a Unix system 1 Download the latest MERLIN package from http merlin cs uoi gr The corre sponding file will be of the form merlin x y tar gz where x y is the MERLIN version 2 Uncompress and untar the file tar xvfz merlin x y tar gz A directory named merlin x y will be created 3 Change into the newly created directory cd merlin x y 4 Edit Makefile inc and provide appropriate values for the variables DESTDIR and FOPTIONS found in the first few lines of Makefile inc DESTDIR is the destination directory where the MERLIN binaries and accompanying files will be installed You must specify DESTDIR using a full path Note that this directory must be different from the directory where you unpacked the MERLIN sources Also note that if installing MERLIN outside your home directory you must have super user privileges Example DESTDIR usr local merlin FOPTIONS is a string of options that must be passed to the Fortran compiler during the installation process This is a good place to pass appropriate optimization options to the compiler Example FOPTIONS 03 In addition the following options may be set LINKOPTIONS Options that must be passed to the linker during installation TYPE This specif
5. LEVE NOC 1000 TOLMIN NOC 1000 4 2 Sample code for multistart X X 4 PROGRAM MSTART Sample code for multistart Illustrates the use of subroutine optima IMPLICIT DOUBLE PRECISION A H 0 Z PARAMETER N 2 M 0 DIMENSION XP N XLL N XRL N IXAT N ICODE 4 CHARACTER 10 FINP FOUT The input Merlin instructions reside in file in dat Merlin s output will be disposed to dev null Trash file DATA FINP in dat DATA FOUT Zdevinull Instruct Merlin to initialize the parameters from array XP and to ignore the input for lower bounds upper bounds and fix status DATA ICODE 7 1 0 0 0 DO 2 J 1 10 Fill the XP array vith random numbers in the range 7 7 Routine RANM is provided by Merlin 24 DO 1 I 1 N XP I 7 2 RANM 1 1 CONTINUE Perform local optimization CALL OPTIMA N M XP XV XLL XRL IXAT ICODE amp FINP FOUT GRMS NF NG NH NJ IF J EQ 1 WRITE x 70 WRITE 71 J NF NG XV XP I I 1 N GRMS 2 CONTINUE 10 FORMAT 3X J 4x FE 4X GE 7X VAL 11X XP 2 amp 10x GRMS ti FORMAT 1X 13 1x 15 1X 15 2X D12 5 1X 10 2X D12 5 END The produced output is displayed below J FE GE VAL XP GRMS 1 47 13 0 10316E 01 0 89842E 01 0 71266E 00 0 89234E 08 2 28 19 0 21546E 00 0 17036E 01 0 79608E 00 0 49642E 08 3 13 10 0 10316E 01 0 89842E 01 0 71266E 00 0 81464E 09 4 22 17 0 10316E 01 0 89842E 01 0 71266E 00 0 34503E
6. 12 5 32 24 0 10316E 01 0 89842E 01 0 71266E 00 0 43907E 07 6 26 18 0 21546E 00 0 17036E 01 0 79608E 00 0 36890E 07 T 33 19 0 10316E 01 0 89842E 01 0 71266E 00 0 36883E OT 8 37 25 0 21043E 01 0 16071E 01 0 56865E 00 0 23094E 07 9 28 21 0 21043E 01 0 16071E 01 0 56865E 00 0 27245E 08 10 13 12 0 10316E 01 0 89842E 01 0 71266E 00 0 24530E 07 The file in dat that contains the Merlin instructions and specifies the method s used for the local search is displayed below ANAL DFP NOC 2000 The above is produced with the DFP method Davidon Fletcher Powell To change the minimization method to another say to the BFGS method edit the in dat file and change the second line to BFGS NOC 2000 4 3 Molecular conformation problem The geometrical structure is an important property for understanding and predicting the behavior of any molecular system It is the necessary starting point for the derivation of 25 structural features the estimation of steric requirements and the calculation of electronic properties For systems with many degrees of freedom the potential energy hypersurface may have a substantial number of local minima that correspond to stable molecular configurations Triglycerides are important biological compounds Among other functions they serve as structural components of the cell membranes and as a source of carbon atoms for biosynthetic reactions Glycerol triacetate also known as triacetin is a highly fl
7. 4 EF 546 G T5 Iterations 4 Coverage 0 9706831 Est minima 5 0 Number of minima found 5 Fi 123 G 130 Iterations 5 Coverage 0 9820669 Est minima 5 9 Number of minima found 6 F 1039 G 282 Iterations 6 Coverage 0 9870537 Est minima 6 8 Iterations 7 Coverage 0 9917658 Est minima 6 7 Iterations 8 Coverage 0 9953168 Est minima 6 5 Iterations 9 Coverage 0 9968641 Est minima 6 4 Iterations 10 Coverage 0 9978384 Fst minima 6 3 PTML run completed Number of local minimizers found 6 The minimizers are disposed to file MINIMA Total number of Function calls 2400 Total number of Gradient calls 986 Total number of Jacobian calls U Total number of Hessian calls U Total number of local optimizations 0 Maximum number of starting points 13 Current state has been saved Processor Utilization Proc Minim Graph Fcalls Gcalls U 0 U 1246 U 1 20 346 375 318 2 20 346 374 320 3 20 346 405 348 The relevant input files are listed File POIMAR DAT 0 0 5 0 5 0 1 0 0 5 0 5 0 1 File TML DAT 2 NOD 17 U 10 5 0000000000000 2 100 4 0000000000000 0 99500000476837 0 10000000000000E 01 0 10000000000000E 03 0 10000000000000E 02 U 100 in dat out dat MINIMA POIMAR DAT indir File in dat ANAL BFGS NOC 2000 NOF ITTHR HEAL NEARN NSAMPL SIGMA COVTHR XDEPS FDEPS GDEPS IREST NDUMP FINP FOUT FMIN POIFIL INDIR File MINIMA Contains the minimizers found 2 1 AA 2 AA
8. Merlin 3 0 A multidimensional optimization environment Comput Phys Commun 109 1998 227 249 12 D G Papageorgiou I N Demetropoulos I E Lagaris The Merlin Control Language for strategic optimization Comput Phys Commun 109 1998 250 275 Journal of Global Optimization 5 1994 349 358 3 N L Allinger Conformational analysis 180 MM2 A hydrocarbon force field utilizing V and Vo torsional terms J Am Chem Soc 99 1977 8127 8134 4 J W Ponder TINKER 3 7 of June 1999 Availability http dasher wustl edu tinker 5 D G Papageorgiou I N Demetropoulos I E Lagaris P T Papadimitriou How many conformers of the 1 2 3 Propanetriol triacetate are present in gas phase and in aqueous solution Tetrahedron 52 1996 677 686 30
9. POIMAR DAT 5 0 5 0 O O O O oo O O File PRICE DAT 2 NOD 10000 NOC 1 00000000000000E 04 EPS 10000 000000000 OMEGA 20 NF 1 IPRINT 1 NFORM PRICE QUT OUTFIL POIMAR DAT POIFIL in dat in dat out dat out dat File in dat ANAL BFGS NOC 2000 File PRICE OUT Contains the global minimizer 8 98420130599726E 02 0 71265640298440 1 0316284534837 13 3 3 TML Test Run To execute the test run issue the following command run merlin tml d 6hump f The following output is produced on the standard output TML running with Iteration threshold 10 Healing parameter D Number of nearest neighbors 2 Sample size 100 Sigma parameter 4 Coverage threshold X distance criterion 0 01 F distance criterion 0 0001 Gradient convergence criterion saving every 100 iterations Merlin input file in dat Merlin output file out dat 0 995000005 0 001 136 G 10 Est minima 3 9 Est minima 1 4 minimum is found F Coverage 0 79865 Coverage 0 96460 The first Iterations 1 Iterations 2 Number of minima found 2 F 361 G 32 Number of minima found 3 F 3 75 G 46 Iterations 3 Coverage 0 95274 Est minima 4 0 Number of minima found 4 F 489 G 60 Iterations 4 Coverage 0 97160 Est minima 5 0 Number of minima found 5 F 656 G 124 Iterations 5 Coverage 0 98355 Fst minima 5 8 Iterations 6 Coverage 0 99071 Est minima 5 6 Number of minima found 6 F 1042 G
10. roots of the quadratic The same procedure is then applied to the quotient polynomial Bn and so on so forth One way to search for proper values of p and po is by optimizing the quantity R Q It is here where OPTIMA can be used We list the source code in Fortran 77 PROGRAM ROOTF This is a polynomial root finder program the polynomials must have real coefficients The method applied is a modification of bairstow s i e division by a quadratic polynomial x 2 p1 x p2 determining pl amp p2 to be such that the remainder is vanishing Then a deflation is applied by synthetic division to reduce the polynomial degree by two and the procedure is repeated A check is always made if the degree is reduced to either one or two to use special finishing procedure x X KX KX XX XX KF KF KF KF IMPLICIT DOUBLE PRECISION A H 0 Z COMMON TOROOT A 0 100 B 0 100 KCO COMPLEX ROOT 100 DISC PARAMETER N 2 M 2 DIMENSION ICODE 4 XP 2 XLL 2 XRL 2 IXAT 2 CHARACTER LINE 78 DATA ICODE 1 0 0 0 DO 100 I 1 78 LINE I 1I 100 CONTINUE d Read in tolerance polynomial degree polynomial coefficients WRITE Enter a termination tolerance 19 READ x x EPS WRITE Enter the degree of the polynomial READ KCO WRITE Enter the polynomial coefficients ato a l READ x x ACI I 0 KCO Type the above input for reference WRITE A LI
11. 009 5 2 aaa 0 5686514769482340 5 Value 2 104250946486463 L 1 607104589746949 5 2 aaa 0 5686514782893214 5 Value 2 104250946486463 3 4 PIML Test Run 000 000 000 000 000 000 000 000 000 The test run assumes that LAM MPI is installed and operational processors To execute the test run issue the following commands src create ptml dirs 4 indir setenv MERLIN_F77 mpif 7 compile merlin ptml d 6hump f lamboot mpirun np 4 merlin executable wipe The following output is produced on the standard output Parallel TML running with Iteration threshold 10 Healing parameter 5 0000000000000 Number of nearest neighbors 2 Sample size 100 Sigma parameter 4 0000000000000 Coverage threshold 0 99500000476837 X distance criterion 1 0000000000000D 02 F distance criterion 1 0000000000000D 04 Gradient convergence criterion 1 0000000000000D 03 saving every 100 iterations Merlin input file in dat Merlin output file out dat Directory containing input files indir 16 5 000 5 000 5 000 5 000 5 000 5 000 5 000 5 000 5 000 The test run uses 4 The first minimum is found F 136 G 9 Iterations 1 Coverage 0 6350206 Est minima 2 0 Iterations 2 Coverage 0 9556767 Est minima 1 5 Number of minima found 2 F 379 G 31 Number of minima found 3 F 407 G 47 Iterations 3 Coverage 0 94 76247 Est minima 4 1 Number of minima found
12. FM COMMON TOROOT A 0 100 B 0 100 KCO Pi X 1 P2 X 2 22 CALL SYNDIV A KCO P1 P2 B R Q F 1 R F 2 END II D IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X N GRAD N COMMON TOROOT A 0 100 B 0 100 KCO DIMENSION B1 0 100 B2 0 100 Pi X 1 P2 X 2 CALL SYNDER A KCO P1 P2 B R Q B1 B2 R1 R2 Q1 Q2 GRAD 1 2 R R1 Q Q1 GRAD 2 2 R R2 Q Q2 END IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X N FJ LD N COMMON TOROOT A 0 100 B 0 100 KCO DIMENSION B1 0 100 B2 0 100 Pi X 1 P2 X 2 CALL SYNDER A KCO P1 P2 B R Q B1 B2 R1 R2 Q1 Q2 FJ 1 1 R1 FJ 1 2 R2 FJ 2 1 Q1 FJ 2 2 Q2 END We list the Input Output of a run for obtaining the roots of the polynomial x dx 2x1 82 7x 4x 4 which is an expansion of x 1 x 1 x 2 Enter a termination tolerance 1 e 12 Enter the degree of the polynomial 6 Enter the polynomial coefficients ao a l 4 4 7 8 2 4 1 Running with the following input 23 Tolerance 1 E 12 Degre The roots of the polynomial are root 1 1 000000 0 000000 root 2 1 000000 0 000000 root 3 1 000000 0 000000 root 4 1 000000 0 000000 root 5 2 000000 0 000000 root 6 2 000000 0 000000 The input instructions to OPTIMA that reside in the file in dat are as ANAL JANAL e 6 Coefficients ali i 0 1 4 4 7 8 2 4 1
13. ND IF GO TO 30 END IMPLICIT DOUBLE PRECISION A H 0 Z Given a polynomial a n x n a 1 x a 0 and a quadratic x 2 plxx p2 calculate the coefficients of the quotient polynomial b k k 0 1 n 2 and the remainder terms r and q DIMENSION A O B O x B N 2 A N B N 3 A N 1 P1 B N 2 DO 1 K N 2 2 1 B K 2 A K P1 B K 1 P2 B K CONTINUe R A 1 PixB 0 P2xB 1 Q A O P2 B 0 SUBROUTINE SYNDER A N P1 P2 B R Q B1 B2 R1 R2 Q1 Q2 21 X KX KX KXK F amp F KF lt IMPLICIT DOUBLE PRECISION A H 0 Z Given a polynomial a n x n ta l xx a 0 and a quadratic x 2 pixx p2 calculate the coefficients of the quotient polynomial b k k 0 1 n 2 and the remainder terms r and q Calculate the derivatives bi k r1 qi of b k r and q with respect to pl and the derivatives b2 k r2 q2 of b k r and q with respect to p2 DIMENSION A 0 x B O B1 0 B2 0 B N 2 A N B N 3 A N 1 P1i B N 2 DO 1 K N 2 2 1 B K 2 ACK P1 B K 1 PaxB K CONTINUE R A 1 PixB 0 P2xB 1 Q A O P2 B 0 Bi N 2 U B1 N 3 A N DO 2 K N 2 2 1 B1 K 2 B K 1 P1 B1 K 1 P2 B1 K CONTINUE Ri B O P1 B1 0 P2xB1 1 QI P2xB1 0 B2 N 2 U B2 N 3 U DO 3 K N 2 2 1 B2 K 2 PIxB2 K 1 B K P2 B2 K CONTINUE R2 PixB2 0 B 1 P2 B2 1 Q2 B O P2 B2 0 END IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X N
14. NE WRITE 0 Running with the following input WRITE Tolerance eps WRITE Degree gt kco WRITE Coefficients a i i 0 1 WRITE A T I 0 KCO WRITE A LINE NOR 0 30 CONTINUE Check if the degree is one and if so solve a linear equation IF KCO EQ 1 THEN WRITE The roots of the polynomial are NOR NOR 1 ROOT NOR CMPLX A 0 A 1 0 DO 1 I 1 NOR WRITE 19 I ROOT I 1 CONTINUE 19 FORMAT 2X Root i4 t17 g14 7 g14 7 STOP END IF Check if the degree is two and if so solve a quadratic equation IF KCO EQ 2 THEN WRITE The roots of the polynomial are NOR NOR 1 DISC CMPLX A 1 2 AxA 2 xA 0 0 DO ROUT NORD A 1 CSQRT DISC 2 A 2 NOR NOR i ROUT NORD A 1 CSQRT DISC 2 A 2 DO 2 I 1 NOR WRITE 19 I ROOT I 2 CONTINUE STOP END IF d Initialize randomly in 1 1 Pi 2 RANM 1 P2 2 RANM 1 20 X X X XxX P1 P2 XP 1 XP 2 Minimize the remainder down to zero CALL OPTIMA N M XP XV XLL XRL IXAT ICODE amp in dat Idevinull GRMS NF NG NHE NJA IF SQRT XV LE EPS THEN Pi XP 1 P2 XP 2 DISC CMPLX P1i 2 4 P2 0 DO NOR NOR 1 ROOT NOR P1 CSQRT DISC 2 NOR NOR 1 ROOT NOR P1 CSQRT DISC 2 CALL SYNDIV A KCO P1 P2 B R Q KCO KCO 2 DO 3 I 0 KCO A I BCT CONTINUE E
15. USER MANUAL PANMIN Sequential and Parallel Global Optimization Procedures with a variety of options for the Local Search Strategy F V Theos I E Lagaris Department of Computer Science D G Papageorgiou Department of Materials Science and Engineering UNIVERSITY OF IOANNINA P O Box 1186 Ioannina 45110 GREECE October 16 2003 http merlin cs uoi gr panmin Contents 1 MERLIN Set Up 11 MERLIN installation 202 00 0 0 ee ee 1 2 Running MERLIN 2 2 2 Running and customizing the global optimization codes 2 1 Some general remarks 2 1 2 2 Running PRICE 2 2 20 00 00 00000 2 2 3 Customizing PRICE 2 0 00 20 000002 2 ee 2 44 Running TME 2 5 Customizing TL 2 6 Running Parallel TML PTM 2 7 Customizing PML 2 8 Files common to all methods 0 0 0 0 00000058 2 8 1 The MERLIN input fle 0 0 0 0 020484 2 8 2 Specifying parameter bounds 000040 3 Test Runs 3 1 Six Hump Camel Back code 00 00 0002 eee ee 3 2 PRICE Test Run 2 0 00 00 00000 02 3 3 TML Test Run 2 2 0 00 0000 a 3 4 PTML Test Run 2 2 ee 4 Examples 4 1 Polynomial Root Finder 0 2 0 0 0 00002 eee ee 4 2 Sample code for multistart 2 a a 0000 2 ee ee 4 3 Molecular conformation problem 2 0 0 000 eee 5 Using MCL with the OPTIMA interface References ds Oo N N Ot Or Oo 11 11 12 14 16 19 19 24 25 26
16. alculation of the critical dis tance COVTHR specifies the required relative domain coverage Default value 0 995 XDEPS specifies the minimum distance between two minima to be considered as different FDEPS specifies the minimum relative difference in the values of two minima in order to be considered as different GDEPS specifies the maximum value for the RMS gradient in order that a point is accepted as a local minimum IREST If set to a non zero value starts the procedure from a previously saved dump NDUMP Specifies the iteration period between dumps FINP is a character string that specifies the name of the Merlin instructions file as described in section 2 8 1 FOUT is a character string that specifies the name of the Merlin output file FMIN is a character string that specifies the name of the file containing the discovered local minima POIFIL is a character string that specifies the name of the file containing an initial point parameter bounds and the fix status as described in section 2 8 2 INDIR is a character string that specifies the name of a Unix directory where auxiliary files may reside as described in section 2 6 This applies only to the parallel version of the software The sequential version ignores it 2 Prepare a file containing the parameter bounds as described in section 2 8 2 3 Prepare the MERLIN input file as described in section 2 8 1 4 Compile the program and the objective funct
17. cepts one or more file names as arguments that can be of the following types e Files ending in d These are files that must be processed by the MERLIN preprocessor before they can be compiled The MERLIN preprocessor will use the definitions file DEFS in the current directory or if no such file exists the one used for the installation of the MERLIN package found in DESTDIR files After preprocessing the files the script will compile and link them with the rest of the MERLIN package e Files ending in f These are files containing Fortran 77 code that must be compiled e Files ending in o These are files containing object code already compiled In addition the run merlin script recognizes the following environment variables 4 e MERLIN F77 This is the name of the compiler that will be used to compile the user written code e MERLIN FOPTIONS Options that will be used when compiling the user written code The default is to use c compile only do not link and any other flags that were specified when editing Makefile inc Note that if you set this environment variable you must include the c compiler flag e MERLIN_LDOPTIONS Options that will be passed to the linker when building the final MERLIN executable Here you can specify any libraries required by the user written subprograms 2 Running and customizing the global optimization codes 2 1 Some general remarks Before running any of the global optimization codes
18. exible molecule hence its properties do not depend only on the globally optimal conformational state Triacetin consists of 29 atoms which form three branches named a 0 and y as shown in fig 1 The 15 heavy atoms form a backbone structure with 8 rotational degrees of freedom while the rotation of the terminal methyl groups add 3 more For the potential function we used the MM2 3 force field with 1991 parameters as implemented in the Tinker JA molecular modeling package Triacetin was modelled using internal coordinates Bond lenghts and angles were fixed to their equilibrium values while the 11 dihedral angles were allowed to vary We applied the parallel TML algorithm using 4 processors on a SUN Enterprise 450 with the following default parameter values ITTHR 10 HEAL 5 NEARN 2 NSAMPL 100 SIGMA 4 COVTHR 0 995 XDEPS 1 e 2 FDEPS 1 e 4 GDEPS 1 e 5 For the local optimization we used the TOLMIN method with a maximum of 5000 function calls The TML algorithm performed 29780 iterations 3982011 function and 993272 gradient calls A total of 2181 local minima were found After TML completed all minima were further refined using the full set of internal coordinates Using this procedure we recovered all 109 low energy conformers described in 5 The global minimum is shown in fig 2 while the next lowest is shown in fig 3 5 Using MCL with the OPTIMA interface We present a general strategy coded in MCL and then show how it can be
19. ies whether MERLIN will be compiled using REAL or DOUBLE PRECISION arithmetic MXV The maximum number of optimization parameters MERLIN will handle MXT The maximum number of terms in a sum of squares objective function MCLBUF The maximum size in bytes of a compiled MCL program MCLMEM The maximum size in words of memory available to MCL programs The default values provided for DESTDIR and FOPTIONS are adequate for most systems Note for Cygwin users You must set F77 g77 and uncomment EXESUFFIX exe in Makefile inc 5 Build the package make 6 Install the package make install 1 2 Running MERLIN In order to compile the user written subprograms with the rest of the MERLIN package one must use the run merlin script After installation the run merlin script as well as other binaries are located in DESTDIR bin Hence one needs to add this directory to its Unix PATH To add DESTDIR bin to your Unix path and assuming that DESTDIR is set to usr local merlin e Users of csh and tcsh add the following line in the file cshre path path usr local merlin bin e Users of sh add the following lines in the file profile PATH PATH usr local merlin bin EXPORT PATH e Users of bash add the following lines in the file bashrc PATH PATH usr local merlin bin EXPORT PATH Alternatively you can invoke the run merlin script using a full path For example usr local merlin bin run merlin funmin f The run merlin script ac
20. ion link with the rest of the MERLIN package and run assuming the objective function is in file funmin f run merlin tml d funmin f 5 2 5 After the run completes the results are disposed in the output file Its file name is specified in the input file TML DAT the default being MINIMA The output file contains the values of the minimization parameters along with the corresponding value of the objective function and is written as a series of MERLIN records see the MERLIN Users Manual for a description of MERLIN records Customizing TML The following parameter statements are used to customize the code in accord with the problem at hand e PARAMETER MXNM 10000 2 6 Sets the maximum number of local minima that will be stored The program stops when the number of discovered minima becomes equal to MXNM PARAMETER MXNS 500 The maximum number of points to sample at each iteration Maximum allowed value for the NSAMPL input parameter PARAMETER MXNNN 8 The maximum number of nearest neighbors considered Maximum allowed value for the NEARN input parameter Running Parallel TML PTML Prepare the input file The file name and contents are the same as for the plain TML code section 2 4 Prepare a file containing the parameter bounds as described in section 2 8 2 Prepare the MERLIN input file as described in section 2 8 1 In parallel TML it is necessary for every processor to operate on a
21. m allowed value for the NSAMPL input parameter e PARAMETER MXNNN 8 The maximum number of nearest neighbors considered Maximum allowed value for the NEARN input parameter 2 8 Files common to all methods 2 8 1 The MERLIN input file The MERLIN input file contains commands that are to be executed by MERLIN in order to perform a local minimization The name of this file is specified in files PRICE DAT or TML DAT according to the method the default being in dat A sample is presented below anal bfgs noc 5000 Note that it not allowed to change the current point using commands such as point init etc Note also that this may be the object file of an MCL program 10 2 8 2 Specifying parameter bounds The name of this file is specified in files PRICE DAT or TML DAT according to the method the default being POIMAR DAT Each line in the file corresponds to one of the minimization parameters and must contain 4 values An initial value for the parameter the lower and upper bound and an integer specifying whether the parameter is fixed 0 or not 1 Example for a two parameter objective function om ol 5 0 5 0 8 0 17 0 NO e Note that if you attempt to run any of the global optimization codes without preparing this file the program will create a sample file with default values and stop You must then edit the file specify the correct values for your problem and rerun the program 3 Test Runs We employ for the tes
22. make sure you have installed the latest version of the MERLIN optimization package as described in section 1 1 In addition you should prepare the objective function following the instructions given in the MERLIN Users Manual You may prepare either FUNCTION FUNMIN or SUBROUTINE SUBSUM Optionally the gradient vector Hessian and Jacobian matrices may be programmed as well 2 2 Running PRICE 1 Prepare the input file This file must be named PRICE DAT A sample file is displayed below 2 NOD U NOF 10000 NOC 1 00000000000000E 04 EPS 10000 000000000 OMEGA 20 NF 1 IPRINT 1 NFORM PRICE OUT OUTFIL POIMAR DAT POIFIL in dat FINP out dat FOUT Only the first column is required by the software The second column contains only code names that serve as a reminder to the user to ease the data entry 1 NOD is the dimensionality of the problem NOF is the number of terms when the SUBROUTINE SUBSUM has been prepared If NOF is set to zero FUNCTION FUNMIN is used NOC is an upper bound for the number of function evaluations EPS is a small tolerance used to terminate the search OMEGA is the value of the w parameter NF is an integer used to set the sample size M via the relation M NF x NOD IPRINT is a flag If it is set to zero no intermediate printout is produced If it is set to one informative printout is issued at every iteration NFORM controls the format of the final output Takes on the values 0 1 2 and the correspo
23. nding formats are described at the end of this section OUTFIL is a character string that specifies the name of the output file POIFIL is a character string that specifies the name of the file containing an initial point parameter bounds and the fix status as described in section 2 8 2 FINP is a character string that specifies the name of the Merlin instructions file as described in section 2 8 1 If FINP is left blank then the final polishing is not performed FOUT is a character string that specifies the name of the Merlin output file Prepare the MERLIN input file as described in section 2 8 1 Prepare a file containing the parameter bounds as described in section 2 8 2 Compile the program and the objective function link with the rest of the MERLIN package and run assuming the objective function is in file funmin f run merlin price d funmin f After the run completes the results are disposed in the output file Its file name is specified in the input file PRICE DAT The default being PRICE OUT The output file contains the values of the minimization parameters along with the corresponding value of the objective function Three output formats are supported and may be specified in the input file NFORM e NFORM 0 Raw unformatted output Output is written using unformatted write statements as in the following program segment DO 10 Iz1 N WRITE 1 X I 10 CONTINUE WRITE 1 VALUE e NFORM 1 Raw formatted outp
24. separate directory of the file system so that intermediate files created by MERLIN and the objective function do not get mixed up The user must create a directory and copy all input files required by MERLIN and the objective function in this directory Note that the MERLIN panel description file PDESC must be copied as well mkdir input cp PDESC input cp an_input_file input Create temporary directories for all processors and copy the necessary input files as suming 4 processors create ptml dirs 4 input 6 Compile the program and the objective function and link with MERLIN and the MPI environment assuming the objective function is in file funmin f setenv MERLIN F77 mpif 7 compile merlin ptml d funmin f 7 Run the program assuming 4 processors mpirun np 4 merlin executable 8 The results are disposed in the output file its file name is specified in the input file TML DAT the default being MINIMA 2 7 Customizing PTML The following parameter statements are used to customize the code in accord with the problem at hand These statements reside in the include file ptml h e PARAMETER MXPROC 100 The maximum number of processors that can be utilized e PARAMETER MXNM 10000 Sets the maximum number of local minima that will be stored The program stops when the number of discovered minima becomes equal to MXNM e PARAMETER MXNS 500 The maximum number of points to sample at each iteration Maximu
25. t runs the Six hump Camel Back function 1 f a y 4 217 sa a ACL yy zy This function has six local minima two of which are global The code for the function and its gradient is given in section 3 1 The PRICE and the TML results are presented in sections 3 2 and 3 3 correspondingly In section 4 2 we list a code that makes use of the subroutine OPTIMA to search for the global minimum of the same test function The simplistic multistart approach is being used i e we generate a number of points at random and from each one we start a local search 3 1 Six Hump Camel Back code The code for the Six hump Camel Back its gradient and its Hessian is given below IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X N X1 X 1 X2 X 2 FUNMIN 4 2 1 X1 2 X1 4 3 X1 2 amp X1 X2 4 AxX2xx2 X2 2 END 11 C IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X N GRAD N GRAD 1 8 X 1 8 4 X 1 3 2 X 1 5 X 2 GRAD 2 X 1 8 X 2 16 X 2 3 END C SUBROUTINE HANAL H LD N X C IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION H LD N X N H 1 1 8 25 2 X 1
26. used with the OPTIMA interface program var geps nocalls geps 1 e 5 nocalls 1000 call local geps nocalls end h It is a local minimization strategy 26 Figure 1 The triacetin molecule Figure 2 Lowest energy conformer of triacetin 27 Figure 3 The second lovvest energy triacetin conformer h GEPS is input and is a tolerance for the rms gradient A NOCALLS is input and adjusts the number of calls to the objective h function h Care is taken to use proper methods depending on the existance h of analytic derivatives jacobian and the functional form var ic z ic 0 REDO if funmode 0 then h General form FUNCTION FUNMIN if deriva 1 then h Analytic gradient exists tolmin NOC nocalls when grms z gt geps just simplex NOC nocalls else h Analytic gradient does not exist simplex NOC nocalls when grms z gt geps just bfgs NOC nocalls endif else h Sum of Squares form SUBSUM if jacomo 1 then h Analytic Jacobian exists 28 leve NOC nocalls when grms z gt geps just tolmin NOC nocalls else A Numerically estimated Jacobian leve NOC nocalls when grmslz gt geps just simplex NOC nocalls endif endif ic ic 1 if grms z gt geps then when ic lt 10 just move to REDO endif end Compile the above MCL program and let the object code be the input file FINP to subroutine optima 29 References 1 D G Papageorgiou I N Demetropoulos I E Lagaris
27. ut Output is written using formatted write statements as in the following program segment DO 10 1 1 N WRITE 1 20 X I 10 CONTINUE WRITE 1 20 VALUE 20 FORMAT 1PG21 14 e NFORM 2 Output is written as a MERLIN record This is the default 2 3 Customizing PRICE The following parameter statement is used to customize the code in accord with the problem at hand e PARAMETER MAXFAC 100 This sets the maximum value for the NF factor 2 4 Running TML 1 Prepare the input file This file must be named TML DAT A sample file is displayed below 2 0 10 5 0000000000000 2 100 4 0000000000000 0 99500000000000 0 10000000000000E 01 0 10000000000000E 03 0 10000000000000E 02 0 100 in dat out dat MINIMA PU TL MAR DAT input NOD NOF ITTHR HEAL NEARN NSAMPL SIGMA COVTHR XDEPS FDEPS GDEPS IREST NDUMP FINP FOUT FMIN POIFIL INDIR Only the first column is required by the software The second column contains only code names that serve as a reminder to the user to ease the data entry NOD is the dimensionality of the problem NOF is the number of terms in case where the SUBROUTINE SUBSUM is being prepared If NOF is set to zero FUNCTION FUNMIN is used ITTHR is the enforced minimum number of iterations of the algorithm HEAL is the healing parameter NEARN specifies the number of nearest neighbors NSAMPL specifies the sample size SIGMA specifies the o parameter involved in the c

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