COBYLA ~~~~~~ Here is a single-precision Fortran implementation of the algorithm for constrained optimization that is the subject of the report I have written on "A direct search optimization method that models the objective and constraint functions by linear interpolation". This report has the number DAMTP 1992/NA5, University of Cambridge, and it has been published in the proceedings of the conference on Numerical Analysis and Optimization that was held in Oaxaca, Mexico in January, 1992, which is the book "Advances in Optimization and Numerical Analysis" (eds. Susana Gomez and Jean-Pierre Hennart), Kluwer Academic Publishers (1994). The instructions for using the Fortran code are given in the comments of SUBROUTINE COBYLA, which is the interface between the user and the main calculation that is done by SUBROUTINE COBYLB. There is a need for a linear programming problem to be solved subject to a Euclidean norm trust region constraint. Therefore SUBROUTINE TRSTLP is provided too, but you may have some software that you prefer to use instead. These 3 subroutines are separated by lines of hyphens below. Further, there follows the main program, the CALCFC subroutine and the output that are appropriate to the numerical examples that are discussed in the last section of DAMTP 1992/NA5. Please note, however, that some cosmetic restructuring of the software has caused the given output to differ slightly from Table 1 of the report. There are no restrictions on the use of the software, nor do I offer any guarantees of success. Indeed, at the time of writing this note I had applied it only to test problems that have up to 10 variables. Mike Powell (May 7th, 1992). ------------------------------------------------------------------------------- SUBROUTINE COBYLA (N,M,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W,IACT) DIMENSION X(*),W(*),IACT(*) C C This subroutine minimizes an objective function F(X) subject to M C inequality constraints on X, where X is a vector of variables that has C N components. The algorithm employs linear approximations to the C objective and constraint functions, the approximations being formed by C linear interpolation at N+1 points in the space of the variables. C We regard these interpolation points as vertices of a simplex. The C parameter RHO controls the size of the simplex and it is reduced C automatically from RHOBEG to RHOEND. For each RHO the subroutine tries C to achieve a good vector of variables for the current size, and then C RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and C RHOEND should be set to reasonable initial changes to and the required C accuracy in the variables respectively, but this accuracy should be C viewed as a subject for experimentation because it is not guaranteed. C The subroutine has an advantage over many of its competitors, however, C which is that it treats each constraint individually when calculating C a change to the variables, instead of lumping the constraints together C into a single penalty function. The name of the subroutine is derived C from the phrase Constrained Optimization BY Linear Approximations. C C The user must set the values of N, M, RHOBEG and RHOEND, and must C provide an initial vector of variables in X. Further, the value of C IPRINT should be set to 0, 1, 2 or 3, which controls the amount of C printing during the calculation. Specifically, there is no output if C IPRINT=0 and there is output only at the end of the calculation if C IPRINT=1. Otherwise each new value of RHO and SIGMA is printed. C Further, the vector of variables and some function information are C given either when RHO is reduced or when each new value of F(X) is C computed in the cases IPRINT=2 or IPRINT=3 respectively. Here SIGMA C is a penalty parameter, it being assumed that a change to X is an C improvement if it reduces the merit function C F(X)+SIGMA*MAX(0.0,-C1(X),-C2(X),...,-CM(X)), C where C1,C2,...,CM denote the constraint functions that should become C nonnegative eventually, at least to the precision of RHOEND. In the C printed output the displayed term that is multiplied by SIGMA is C called MAXCV, which stands for 'MAXimum Constraint Violation'. The C argument MAXFUN is an integer variable that must be set by the user to a C limit on the number of calls of CALCFC, the purpose of this routine being C given below. The value of MAXFUN will be altered to the number of calls C of CALCFC that are made. The arguments W and IACT provide real and C integer arrays that are used as working space. Their lengths must be at C least N*(3*N+2*M+11)+4*M+6 and M+1 respectively. C C In order to define the objective and constraint functions, we require C a subroutine that has the name and arguments C SUBROUTINE CALCFC (N,M,X,F,CON) C DIMENSION X(*),CON(*) . C The values of N and M are fixed and have been defined already, while C X is now the current vector of variables. The subroutine should return C the objective and constraint functions at X in F and CON(1),CON(2), C ...,CON(M). Note that we are trying to adjust X so that F(X) is as C small as possible subject to the constraint functions being nonnegative. C C Partition the working space array W to provide the storage that is needed C for the main calculation. C MPP=M+2 ICON=1 ISIM=ICON+MPP ISIMI=ISIM+N*N+N IDATM=ISIMI+N*N IA=IDATM+N*MPP+MPP IVSIG=IA+M*N+N IVETA=IVSIG+N ISIGB=IVETA+N IDX=ISIGB+N IWORK=IDX+N CALL COBYLB (N,M,MPP,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W(ICON), 1 W(ISIM),W(ISIMI),W(IDATM),W(IA),W(IVSIG),W(IVETA),W(ISIGB), 2 W(IDX),W(IWORK),IACT) RETURN END C------------------------------------------------------------------------------ SUBROUTINE COBYLB (N,M,MPP,X,RHOBEG,RHOEND,IPRINT,MAXFUN, 1 CON,SIM,SIMI,DATMAT,A,VSIG,VETA,SIGBAR,DX,W,IACT) DIMENSION X(*),CON(*),SIM(N,*),SIMI(N,*),DATMAT(MPP,*), 1 A(N,*),VSIG(*),VETA(*),SIGBAR(*),DX(*),W(*),IACT(*) C C Set the initial values of some parameters. The last column of SIM holds C the optimal vertex of the current simplex, and the preceding N columns C hold the displacements from the optimal vertex to the other vertices. C Further, SIMI holds the inverse of the matrix that is contained in the C first N columns of SIM. C IPTEM=MIN0(N,5) IPTEMP=IPTEM+1 NP=N+1 MP=M+1 ALPHA=0.25 BETA=2.1 GAMMA=0.5 DELTA=1.1 RHO=RHOBEG PARMU=0.0 IF (IPRINT .GE. 2) PRINT 10, RHO 10 FORMAT (/3X,'The initial value of RHO is',1PE13.6,2X, 1 'and PARMU is set to zero.') NFVALS=0 TEMP=1.0/RHO DO 30 I=1,N SIM(I,NP)=X(I) DO 20 J=1,N SIM(I,J)=0.0 20 SIMI(I,J)=0.0 SIM(I,I)=RHO 30 SIMI(I,I)=TEMP JDROP=NP IBRNCH=0 C C Make the next call of the user-supplied subroutine CALCFC. These C instructions are also used for calling CALCFC during the iterations of C the algorithm. C 40 IF (NFVALS .GE. MAXFUN .AND. NFVALS .GT. 0) THEN IF (IPRINT .GE. 1) PRINT 50 50 FORMAT (/3X,'Return from subroutine COBYLA because the ', 1 'MAXFUN limit has been reached.') GOTO 600 END IF NFVALS=NFVALS+1 CALL CALCFC (N,M,X,F,CON) RESMAX=0.0 IF (M .GT. 0) THEN DO 60 K=1,M 60 RESMAX=AMAX1(RESMAX,-CON(K)) END IF IF (NFVALS .EQ. IPRINT-1 .OR. IPRINT .EQ. 3) THEN PRINT 70, NFVALS,F,RESMAX,(X(I),I=1,IPTEM) 70 FORMAT (/3X,'NFVALS =',I5,3X,'F =',1PE13.6,4X,'MAXCV =', 1 1PE13.6/3X,'X =',1PE13.6,1P4E15.6) IF (IPTEM .LT. N) PRINT 80, (X(I),I=IPTEMP,N) 80 FORMAT (1PE19.6,1P4E15.6) END IF CON(MP)=F CON(MPP)=RESMAX IF (IBRNCH .EQ. 1) GOTO 440 C C Set the recently calculated function values in a column of DATMAT. This C array has a column for each vertex of the current simplex, the entries of C each column being the values of the constraint functions (if any) C followed by the objective function and the greatest constraint violation C at the vertex. C DO 90 K=1,MPP 90 DATMAT(K,JDROP)=CON(K) IF (NFVALS .GT. NP) GOTO 130 C C Exchange the new vertex of the initial simplex with the optimal vertex if C necessary. Then, if the initial simplex is not complete, pick its next C vertex and calculate the function values there. C IF (JDROP .LE. N) THEN IF (DATMAT(MP,NP) .LE. F) THEN X(JDROP)=SIM(JDROP,NP) ELSE SIM(JDROP,NP)=X(JDROP) DO 100 K=1,MPP DATMAT(K,JDROP)=DATMAT(K,NP) 100 DATMAT(K,NP)=CON(K) DO 120 K=1,JDROP SIM(JDROP,K)=-RHO TEMP=0.0 DO 110 I=K,JDROP 110 TEMP=TEMP-SIMI(I,K) 120 SIMI(JDROP,K)=TEMP END IF END IF IF (NFVALS .LE. N) THEN JDROP=NFVALS X(JDROP)=X(JDROP)+RHO GOTO 40 END IF 130 IBRNCH=1 C C Identify the optimal vertex of the current simplex. C 140 PHIMIN=DATMAT(MP,NP)+PARMU*DATMAT(MPP,NP) NBEST=NP DO 150 J=1,N TEMP=DATMAT(MP,J)+PARMU*DATMAT(MPP,J) IF (TEMP .LT. PHIMIN) THEN NBEST=J PHIMIN=TEMP ELSE IF (TEMP .EQ. PHIMIN .AND. PARMU .EQ. 0.0) THEN IF (DATMAT(MPP,J) .LT. DATMAT(MPP,NBEST)) NBEST=J END IF 150 CONTINUE C C Switch the best vertex into pole position if it is not there already, C and also update SIM, SIMI and DATMAT. C IF (NBEST .LE. N) THEN DO 160 I=1,MPP TEMP=DATMAT(I,NP) DATMAT(I,NP)=DATMAT(I,NBEST) 160 DATMAT(I,NBEST)=TEMP DO 180 I=1,N TEMP=SIM(I,NBEST) SIM(I,NBEST)=0.0 SIM(I,NP)=SIM(I,NP)+TEMP TEMPA=0.0 DO 170 K=1,N SIM(I,K)=SIM(I,K)-TEMP 170 TEMPA=TEMPA-SIMI(K,I) 180 SIMI(NBEST,I)=TEMPA END IF C C Make an error return if SIGI is a poor approximation to the inverse of C the leading N by N submatrix of SIG. C ERROR=0.0 DO 200 I=1,N DO 200 J=1,N TEMP=0.0 IF (I .EQ. J) TEMP=TEMP-1.0 DO 190 K=1,N 190 TEMP=TEMP+SIMI(I,K)*SIM(K,J) 200 ERROR=AMAX1(ERROR,ABS(TEMP)) IF (ERROR .GT. 0.1) THEN IF (IPRINT .GE. 1) PRINT 210 210 FORMAT (/3X,'Return from subroutine COBYLA because ', 1 'rounding errors are becoming damaging.') GOTO 600 END IF C C Calculate the coefficients of the linear approximations to the objective C and constraint functions, placing minus the objective function gradient C after the constraint gradients in the array A. The vector W is used for C working space. C DO 240 K=1,MP CON(K)=-DATMAT(K,NP) DO 220 J=1,N 220 W(J)=DATMAT(K,J)+CON(K) DO 240 I=1,N TEMP=0.0 DO 230 J=1,N 230 TEMP=TEMP+W(J)*SIMI(J,I) IF (K .EQ. MP) TEMP=-TEMP 240 A(I,K)=TEMP C C Calculate the values of sigma and eta, and set IFLAG=0 if the current C simplex is not acceptable. C IFLAG=1 PARSIG=ALPHA*RHO PARETA=BETA*RHO DO 260 J=1,N WSIG=0.0 WETA=0.0 DO 250 I=1,N WSIG=WSIG+SIMI(J,I)**2 250 WETA=WETA+SIM(I,J)**2 VSIG(J)=1.0/SQRT(WSIG) VETA(J)=SQRT(WETA) IF (VSIG(J) .LT. PARSIG .OR. VETA(J) .GT. PARETA) IFLAG=0 260 CONTINUE C C If a new vertex is needed to improve acceptability, then decide which C vertex to drop from the simplex. C IF (IBRNCH .EQ. 1 .OR. IFLAG .EQ. 1) GOTO 370 JDROP=0 TEMP=PARETA DO 270 J=1,N IF (VETA(J) .GT. TEMP) THEN JDROP=J TEMP=VETA(J) END IF 270 CONTINUE IF (JDROP .EQ. 0) THEN DO 280 J=1,N IF (VSIG(J) .LT. TEMP) THEN JDROP=J TEMP=VSIG(J) END IF 280 CONTINUE END IF C C Calculate the step to the new vertex and its sign. C TEMP=GAMMA*RHO*VSIG(JDROP) DO 290 I=1,N 290 DX(I)=TEMP*SIMI(JDROP,I) CVMAXP=0.0 CVMAXM=0.0 DO 310 K=1,MP SUM=0.0 DO 300 I=1,N 300 SUM=SUM+A(I,K)*DX(I) IF (K .LT. MP) THEN TEMP=DATMAT(K,NP) CVMAXP=AMAX1(CVMAXP,-SUM-TEMP) CVMAXM=AMAX1(CVMAXM,SUM-TEMP) END IF 310 CONTINUE DXSIGN=1.0 IF (PARMU*(CVMAXP-CVMAXM) .GT. SUM+SUM) DXSIGN=-1.0 C C Update the elements of SIM and SIMI, and set the next X. C TEMP=0.0 DO 320 I=1,N DX(I)=DXSIGN*DX(I) SIM(I,JDROP)=DX(I) 320 TEMP=TEMP+SIMI(JDROP,I)*DX(I) DO 330 I=1,N 330 SIMI(JDROP,I)=SIMI(JDROP,I)/TEMP DO 360 J=1,N IF (J .NE. JDROP) THEN TEMP=0.0 DO 340 I=1,N 340 TEMP=TEMP+SIMI(J,I)*DX(I) DO 350 I=1,N 350 SIMI(J,I)=SIMI(J,I)-TEMP*SIMI(JDROP,I) END IF 360 X(J)=SIM(J,NP)+DX(J) GOTO 40 C C Calculate DX=x(*)-x(0). Branch if the length of DX is less than 0.5*RHO. C 370 IZ=1 IZDOTA=IZ+N*N IVMC=IZDOTA+N ISDIRN=IVMC+MP IDXNEW=ISDIRN+N IVMD=IDXNEW+N CALL TRSTLP (N,M,A,CON,RHO,DX,IFULL,IACT,W(IZ),W(IZDOTA), 1 W(IVMC),W(ISDIRN),W(IDXNEW),W(IVMD)) IF (IFULL .EQ. 0) THEN TEMP=0.0 DO 380 I=1,N 380 TEMP=TEMP+DX(I)**2 IF (TEMP .LT. 0.25*RHO*RHO) THEN IBRNCH=1 GOTO 550 END IF END IF C C Predict the change to F and the new maximum constraint violation if the C variables are altered from x(0) to x(0)+DX. C RESNEW=0.0 CON(MP)=0.0 DO 400 K=1,MP SUM=CON(K) DO 390 I=1,N 390 SUM=SUM-A(I,K)*DX(I) IF (K .LT. MP) RESNEW=AMAX1(RESNEW,SUM) 400 CONTINUE C C Increase PARMU if necessary and branch back if this change alters the C optimal vertex. Otherwise PREREM and PREREC will be set to the predicted C reductions in the merit function and the maximum constraint violation C respectively. C BARMU=0.0 PREREC=DATMAT(MPP,NP)-RESNEW IF (PREREC .GT. 0.0) BARMU=SUM/PREREC IF (PARMU .LT. 1.5*BARMU) THEN PARMU=2.0*BARMU IF (IPRINT .GE. 2) PRINT 410, PARMU 410 FORMAT (/3X,'Increase in PARMU to',1PE13.6) PHI=DATMAT(MP,NP)+PARMU*DATMAT(MPP,NP) DO 420 J=1,N TEMP=DATMAT(MP,J)+PARMU*DATMAT(MPP,J) IF (TEMP .LT. PHI) GOTO 140 IF (TEMP .EQ. PHI .AND. PARMU .EQ. 0.0) THEN IF (DATMAT(MPP,J) .LT. DATMAT(MPP,NP)) GOTO 140 END IF 420 CONTINUE END IF PREREM=PARMU*PREREC-SUM C C Calculate the constraint and objective functions at x(*). Then find the C actual reduction in the merit function. C DO 430 I=1,N 430 X(I)=SIM(I,NP)+DX(I) IBRNCH=1 GOTO 40 440 VMOLD=DATMAT(MP,NP)+PARMU*DATMAT(MPP,NP) VMNEW=F+PARMU*RESMAX TRURED=VMOLD-VMNEW IF (PARMU .EQ. 0.0 .AND. F .EQ. DATMAT(MP,NP)) THEN PREREM=PREREC TRURED=DATMAT(MPP,NP)-RESMAX END IF C C Begin the operations that decide whether x(*) should replace one of the C vertices of the current simplex, the change being mandatory if TRURED is C positive. Firstly, JDROP is set to the index of the vertex that is to be C replaced. C RATIO=0.0 IF (TRURED .LE. 0.0) RATIO=1.0 JDROP=0 DO 460 J=1,N TEMP=0.0 DO 450 I=1,N 450 TEMP=TEMP+SIMI(J,I)*DX(I) TEMP=ABS(TEMP) IF (TEMP .GT. RATIO) THEN JDROP=J RATIO=TEMP END IF 460 SIGBAR(J)=TEMP*VSIG(J) C C Calculate the value of ell. C EDGMAX=DELTA*RHO L=0 DO 480 J=1,N IF (SIGBAR(J) .GE. PARSIG .OR. SIGBAR(J) .GE. VSIG(J)) THEN TEMP=VETA(J) IF (TRURED .GT. 0.0) THEN TEMP=0.0 DO 470 I=1,N 470 TEMP=TEMP+(DX(I)-SIM(I,J))**2 TEMP=SQRT(TEMP) END IF IF (TEMP .GT. EDGMAX) THEN L=J EDGMAX=TEMP END IF END IF 480 CONTINUE IF (L .GT. 0) JDROP=L IF (JDROP .EQ. 0) GOTO 550 C C Revise the simplex by updating the elements of SIM, SIMI and DATMAT. C TEMP=0.0 DO 490 I=1,N SIM(I,JDROP)=DX(I) 490 TEMP=TEMP+SIMI(JDROP,I)*DX(I) DO 500 I=1,N 500 SIMI(JDROP,I)=SIMI(JDROP,I)/TEMP DO 530 J=1,N IF (J .NE. JDROP) THEN TEMP=0.0 DO 510 I=1,N 510 TEMP=TEMP+SIMI(J,I)*DX(I) DO 520 I=1,N 520 SIMI(J,I)=SIMI(J,I)-TEMP*SIMI(JDROP,I) END IF 530 CONTINUE DO 540 K=1,MPP 540 DATMAT(K,JDROP)=CON(K) C C Branch back for further iterations with the current RHO. C IF (TRURED .GT. 0.0 .AND. TRURED .GE. 0.1*PREREM) GOTO 140 550 IF (IFLAG .EQ. 0) THEN IBRNCH=0 GOTO 140 END IF C C Otherwise reduce RHO if it is not at its least value and reset PARMU. C IF (RHO .GT. RHOEND) THEN RHO=0.5*RHO IF (RHO .LE. 1.5*RHOEND) RHO=RHOEND IF (PARMU .GT. 0.0) THEN DENOM=0.0 DO 570 K=1,MP CMIN=DATMAT(K,NP) CMAX=CMIN DO 560 I=1,N CMIN=AMIN1(CMIN,DATMAT(K,I)) 560 CMAX=AMAX1(CMAX,DATMAT(K,I)) IF (K .LE. M .AND. CMIN .LT. 0.5*CMAX) THEN TEMP=AMAX1(CMAX,0.0)-CMIN IF (DENOM .LE. 0.0) THEN DENOM=TEMP ELSE DENOM=AMIN1(DENOM,TEMP) END IF END IF 570 CONTINUE IF (DENOM .EQ. 0.0) THEN PARMU=0.0 ELSE IF (CMAX-CMIN .LT. PARMU*DENOM) THEN PARMU=(CMAX-CMIN)/DENOM END IF END IF IF (IPRINT .GE. 2) PRINT 580, RHO,PARMU 580 FORMAT (/3X,'Reduction in RHO to',1PE13.6,' and PARMU =', 1 1PE13.6) IF (IPRINT .EQ. 2) THEN PRINT 70, NFVALS,DATMAT(MP,NP),DATMAT(MPP,NP), 1 (SIM(I,NP),I=1,IPTEM) IF (IPTEM .LT. N) PRINT 80, (X(I),I=IPTEMP,N) END IF GOTO 140 END IF C C Return the best calculated values of the variables. C IF (IPRINT .GE. 1) PRINT 590 590 FORMAT (/3X,'Normal return from subroutine COBYLA') IF (IFULL .EQ. 1) GOTO 620 600 DO 610 I=1,N 610 X(I)=SIM(I,NP) F=DATMAT(MP,NP) RESMAX=DATMAT(MPP,NP) 620 IF (IPRINT .GE. 1) THEN PRINT 70, NFVALS,F,RESMAX,(X(I),I=1,IPTEM) IF (IPTEM .LT. N) PRINT 80, (X(I),I=IPTEMP,N) END IF MAXFUN=NFVALS RETURN END C------------------------------------------------------------------------------ SUBROUTINE TRSTLP (N,M,A,B,RHO,DX,IFULL,IACT,Z,ZDOTA,VMULTC, 1 SDIRN,DXNEW,VMULTD) DIMENSION A(N,*),B(*),DX(*),IACT(*),Z(N,*),ZDOTA(*), 1 VMULTC(*),SDIRN(*),DXNEW(*),VMULTD(*) C C This subroutine calculates an N-component vector DX by applying the C following two stages. In the first stage, DX is set to the shortest C vector that minimizes the greatest violation of the constraints C A(1,K)*DX(1)+A(2,K)*DX(2)+...+A(N,K)*DX(N) .GE. B(K), K=2,3,...,M, C subject to the Euclidean length of DX being at most RHO. If its length is C strictly less than RHO, then we use the resultant freedom in DX to C minimize the objective function C -A(1,M+1)*DX(1)-A(2,M+1)*DX(2)-...-A(N,M+1)*DX(N) C subject to no increase in any greatest constraint violation. This C notation allows the gradient of the objective function to be regarded as C the gradient of a constraint. Therefore the two stages are distinguished C by MCON .EQ. M and MCON .GT. M respectively. It is possible that a C degeneracy may prevent DX from attaining the target length RHO. Then the C value IFULL=0 would be set, but usually IFULL=1 on return. C C In general NACT is the number of constraints in the active set and C IACT(1),...,IACT(NACT) are their indices, while the remainder of IACT C contains a permutation of the remaining constraint indices. Further, Z is C an orthogonal matrix whose first NACT columns can be regarded as the C result of Gram-Schmidt applied to the active constraint gradients. For C J=1,2,...,NACT, the number ZDOTA(J) is the scalar product of the J-th C column of Z with the gradient of the J-th active constraint. DX is the C current vector of variables and here the residuals of the active C constraints should be zero. Further, the active constraints have C nonnegative Lagrange multipliers that are held at the beginning of C VMULTC. The remainder of this vector holds the residuals of the inactive C constraints at DX, the ordering of the components of VMULTC being in C agreement with the permutation of the indices of the constraints that is C in IACT. All these residuals are nonnegative, which is achieved by the C shift RESMAX that makes the least residual zero. C C Initialize Z and some other variables. The value of RESMAX will be C appropriate to DX=0, while ICON will be the index of a most violated C constraint if RESMAX is positive. Usually during the first stage the C vector SDIRN gives a search direction that reduces all the active C constraint violations by one simultaneously. C IFULL=1 MCON=M NACT=0 RESMAX=0.0 DO 20 I=1,N DO 10 J=1,N 10 Z(I,J)=0.0 Z(I,I)=1.0 20 DX(I)=0.0 IF (M .GE. 1) THEN DO 30 K=1,M IF (B(K) .GT. RESMAX) THEN RESMAX=B(K) ICON=K END IF 30 CONTINUE DO 40 K=1,M IACT(K)=K 40 VMULTC(K)=RESMAX-B(K) END IF IF (RESMAX .EQ. 0.0) GOTO 480 DO 50 I=1,N 50 SDIRN(I)=0.0 C C End the current stage of the calculation if 3 consecutive iterations C have either failed to reduce the best calculated value of the objective C function or to increase the number of active constraints since the best C value was calculated. This strategy prevents cycling, but there is a C remote possibility that it will cause premature termination. C 60 OPTOLD=0.0 ICOUNT=0 70 IF (MCON .EQ. M) THEN OPTNEW=RESMAX ELSE OPTNEW=0.0 DO 80 I=1,N 80 OPTNEW=OPTNEW-DX(I)*A(I,MCON) END IF IF (ICOUNT .EQ. 0 .OR. OPTNEW .LT. OPTOLD) THEN OPTOLD=OPTNEW NACTX=NACT ICOUNT=3 ELSE IF (NACT .GT. NACTX) THEN NACTX=NACT ICOUNT=3 ELSE ICOUNT=ICOUNT-1 IF (ICOUNT .EQ. 0) GOTO 490 END IF C C If ICON exceeds NACT, then we add the constraint with index IACT(ICON) to C the active set. Apply Givens rotations so that the last N-NACT-1 columns C of Z are orthogonal to the gradient of the new constraint, a scalar C product being set to zero if its nonzero value could be due to computer C rounding errors. The array DXNEW is used for working space. C IF (ICON .LE. NACT) GOTO 260 KK=IACT(ICON) DO 90 I=1,N 90 DXNEW(I)=A(I,KK) TOT=0.0 K=N 100 IF (K .GT. NACT) THEN SP=0.0 SPABS=0.0 DO 110 I=1,N TEMP=Z(I,K)*DXNEW(I) SP=SP+TEMP 110 SPABS=SPABS+ABS(TEMP) ACCA=SPABS+0.1*ABS(SP) ACCB=SPABS+0.2*ABS(SP) IF (SPABS .GE. ACCA .OR. ACCA .GE. ACCB) SP=0.0 IF (TOT .EQ. 0.0) THEN TOT=SP ELSE KP=K+1 TEMP=SQRT(SP*SP+TOT*TOT) ALPHA=SP/TEMP BETA=TOT/TEMP TOT=TEMP DO 120 I=1,N TEMP=ALPHA*Z(I,K)+BETA*Z(I,KP) Z(I,KP)=ALPHA*Z(I,KP)-BETA*Z(I,K) 120 Z(I,K)=TEMP END IF K=K-1 GOTO 100 END IF C C Add the new constraint if this can be done without a deletion from the C active set. C IF (TOT .NE. 0.0) THEN NACT=NACT+1 ZDOTA(NACT)=TOT VMULTC(ICON)=VMULTC(NACT) VMULTC(NACT)=0.0 GOTO 210 END IF C C The next instruction is reached if a deletion has to be made from the C active set in order to make room for the new active constraint, because C the new constraint gradient is a linear combination of the gradients of C the old active constraints. Set the elements of VMULTD to the multipliers C of the linear combination. Further, set IOUT to the index of the C constraint to be deleted, but branch if no suitable index can be found. C RATIO=-1.0 K=NACT 130 ZDOTV=0.0 ZDVABS=0.0 DO 140 I=1,N TEMP=Z(I,K)*DXNEW(I) ZDOTV=ZDOTV+TEMP 140 ZDVABS=ZDVABS+ABS(TEMP) ACCA=ZDVABS+0.1*ABS(ZDOTV) ACCB=ZDVABS+0.2*ABS(ZDOTV) IF (ZDVABS .LT. ACCA .AND. ACCA .LT. ACCB) THEN TEMP=ZDOTV/ZDOTA(K) IF (TEMP .GT. 0.0 .AND. IACT(K) .LE. M) THEN TEMPA=VMULTC(K)/TEMP IF (RATIO .LT. 0.0 .OR. TEMPA .LT. RATIO) THEN RATIO=TEMPA IOUT=K END IF END IF IF (K .GE. 2) THEN KW=IACT(K) DO 150 I=1,N 150 DXNEW(I)=DXNEW(I)-TEMP*A(I,KW) END IF VMULTD(K)=TEMP ELSE VMULTD(K)=0.0 END IF K=K-1 IF (K .GT. 0) GOTO 130 IF (RATIO .LT. 0.0) GOTO 490 C C Revise the Lagrange multipliers and reorder the active constraints so C that the one to be replaced is at the end of the list. Also calculate the C new value of ZDOTA(NACT) and branch if it is not acceptable. C DO 160 K=1,NACT 160 VMULTC(K)=AMAX1(0.0,VMULTC(K)-RATIO*VMULTD(K)) IF (ICON .LT. NACT) THEN ISAVE=IACT(ICON) VSAVE=VMULTC(ICON) K=ICON 170 KP=K+1 KW=IACT(KP) SP=0.0 DO 180 I=1,N 180 SP=SP+Z(I,K)*A(I,KW) TEMP=SQRT(SP*SP+ZDOTA(KP)**2) ALPHA=ZDOTA(KP)/TEMP BETA=SP/TEMP ZDOTA(KP)=ALPHA*ZDOTA(K) ZDOTA(K)=TEMP DO 190 I=1,N TEMP=ALPHA*Z(I,KP)+BETA*Z(I,K) Z(I,KP)=ALPHA*Z(I,K)-BETA*Z(I,KP) 190 Z(I,K)=TEMP IACT(K)=KW VMULTC(K)=VMULTC(KP) K=KP IF (K .LT. NACT) GOTO 170 IACT(K)=ISAVE VMULTC(K)=VSAVE END IF TEMP=0.0 DO 200 I=1,N 200 TEMP=TEMP+Z(I,NACT)*A(I,KK) IF (TEMP .EQ. 0.0) GOTO 490 ZDOTA(NACT)=TEMP VMULTC(ICON)=0.0 VMULTC(NACT)=RATIO C C Update IACT and ensure that the objective function continues to be C treated as the last active constraint when MCON>M. C 210 IACT(ICON)=IACT(NACT) IACT(NACT)=KK IF (MCON .GT. M .AND. KK .NE. MCON) THEN K=NACT-1 SP=0.0 DO 220 I=1,N 220 SP=SP+Z(I,K)*A(I,KK) TEMP=SQRT(SP*SP+ZDOTA(NACT)**2) ALPHA=ZDOTA(NACT)/TEMP BETA=SP/TEMP ZDOTA(NACT)=ALPHA*ZDOTA(K) ZDOTA(K)=TEMP DO 230 I=1,N TEMP=ALPHA*Z(I,NACT)+BETA*Z(I,K) Z(I,NACT)=ALPHA*Z(I,K)-BETA*Z(I,NACT) 230 Z(I,K)=TEMP IACT(NACT)=IACT(K) IACT(K)=KK TEMP=VMULTC(K) VMULTC(K)=VMULTC(NACT) VMULTC(NACT)=TEMP END IF C C If stage one is in progress, then set SDIRN to the direction of the next C change to the current vector of variables. C IF (MCON .GT. M) GOTO 320 KK=IACT(NACT) TEMP=0.0 DO 240 I=1,N 240 TEMP=TEMP+SDIRN(I)*A(I,KK) TEMP=TEMP-1.0 TEMP=TEMP/ZDOTA(NACT) DO 250 I=1,N 250 SDIRN(I)=SDIRN(I)-TEMP*Z(I,NACT) GOTO 340 C C Delete the constraint that has the index IACT(ICON) from the active set. C 260 IF (ICON .LT. NACT) THEN ISAVE=IACT(ICON) VSAVE=VMULTC(ICON) K=ICON 270 KP=K+1 KK=IACT(KP) SP=0.0 DO 280 I=1,N 280 SP=SP+Z(I,K)*A(I,KK) TEMP=SQRT(SP*SP+ZDOTA(KP)**2) ALPHA=ZDOTA(KP)/TEMP BETA=SP/TEMP ZDOTA(KP)=ALPHA*ZDOTA(K) ZDOTA(K)=TEMP DO 290 I=1,N TEMP=ALPHA*Z(I,KP)+BETA*Z(I,K) Z(I,KP)=ALPHA*Z(I,K)-BETA*Z(I,KP) 290 Z(I,K)=TEMP IACT(K)=KK VMULTC(K)=VMULTC(KP) K=KP IF (K .LT. NACT) GOTO 270 IACT(K)=ISAVE VMULTC(K)=VSAVE END IF NACT=NACT-1 C C If stage one is in progress, then set SDIRN to the direction of the next C change to the current vector of variables. C IF (MCON .GT. M) GOTO 320 TEMP=0.0 DO 300 I=1,N 300 TEMP=TEMP+SDIRN(I)*Z(I,NACT+1) DO 310 I=1,N 310 SDIRN(I)=SDIRN(I)-TEMP*Z(I,NACT+1) GO TO 340 C C Pick the next search direction of stage two. C 320 TEMP=1.0/ZDOTA(NACT) DO 330 I=1,N 330 SDIRN(I)=TEMP*Z(I,NACT) C C Calculate the step to the boundary of the trust region or take the step C that reduces RESMAX to zero. The two statements below that include the C factor 1.0E-6 prevent some harmless underflows that occurred in a test C calculation. Further, we skip the step if it could be zero within a C reasonable tolerance for computer rounding errors. C 340 DD=RHO*RHO SD=0.0 SS=0.0 DO 350 I=1,N IF (ABS(DX(I)) .GE. 1.0E-6*RHO) DD=DD-DX(I)**2 SD=SD+DX(I)*SDIRN(I) 350 SS=SS+SDIRN(I)**2 IF (DD .LE. 0.0) GOTO 490 TEMP=SQRT(SS*DD) IF (ABS(SD) .GE. 1.0E-6*TEMP) TEMP=SQRT(SS*DD+SD*SD) STPFUL=DD/(TEMP+SD) STEP=STPFUL IF (MCON .EQ. M) THEN ACCA=STEP+0.1*RESMAX ACCB=STEP+0.2*RESMAX IF (STEP .GE. ACCA .OR. ACCA .GE. ACCB) GOTO 480 STEP=AMIN1(STEP,RESMAX) END IF C C Set DXNEW to the new variables if STEP is the steplength, and reduce C RESMAX to the corresponding maximum residual if stage one is being done. C Because DXNEW will be changed during the calculation of some Lagrange C multipliers, it will be restored to the following value later. C DO 360 I=1,N 360 DXNEW(I)=DX(I)+STEP*SDIRN(I) IF (MCON .EQ. M) THEN RESOLD=RESMAX RESMAX=0.0 DO 380 K=1,NACT KK=IACT(K) TEMP=B(KK) DO 370 I=1,N 370 TEMP=TEMP-A(I,KK)*DXNEW(I) RESMAX=AMAX1(RESMAX,TEMP) 380 CONTINUE END IF C C Set VMULTD to the VMULTC vector that would occur if DX became DXNEW. A C device is included to force VMULTD(K)=0.0 if deviations from this value C can be attributed to computer rounding errors. First calculate the new C Lagrange multipliers. C K=NACT 390 ZDOTW=0.0 ZDWABS=0.0 DO 400 I=1,N TEMP=Z(I,K)*DXNEW(I) ZDOTW=ZDOTW+TEMP 400 ZDWABS=ZDWABS+ABS(TEMP) ACCA=ZDWABS+0.1*ABS(ZDOTW) ACCB=ZDWABS+0.2*ABS(ZDOTW) IF (ZDWABS .GE. ACCA .OR. ACCA .GE. ACCB) ZDOTW=0.0 VMULTD(K)=ZDOTW/ZDOTA(K) IF (K .GE. 2) THEN KK=IACT(K) DO 410 I=1,N 410 DXNEW(I)=DXNEW(I)-VMULTD(K)*A(I,KK) K=K-1 GOTO 390 END IF IF (MCON .GT. M) VMULTD(NACT)=AMAX1(0.0,VMULTD(NACT)) C C Complete VMULTC by finding the new constraint residuals. C DO 420 I=1,N 420 DXNEW(I)=DX(I)+STEP*SDIRN(I) IF (MCON .GT. NACT) THEN KL=NACT+1 DO 440 K=KL,MCON KK=IACT(K) SUM=RESMAX-B(KK) SUMABS=RESMAX+ABS(B(KK)) DO 430 I=1,N TEMP=A(I,KK)*DXNEW(I) SUM=SUM+TEMP 430 SUMABS=SUMABS+ABS(TEMP) ACCA=SUMABS+0.1*ABS(SUM) ACCB=SUMABS+0.2*ABS(SUM) IF (SUMABS .GE. ACCA .OR. ACCA .GE. ACCB) SUM=0.0 440 VMULTD(K)=SUM END IF C C Calculate the fraction of the step from DX to DXNEW that will be taken. C RATIO=1.0 ICON=0 DO 450 K=1,MCON IF (VMULTD(K) .LT. 0.0) THEN TEMP=VMULTC(K)/(VMULTC(K)-VMULTD(K)) IF (TEMP .LT. RATIO) THEN RATIO=TEMP ICON=K END IF END IF 450 CONTINUE C C Update DX, VMULTC and RESMAX. C TEMP=1.0-RATIO DO 460 I=1,N 460 DX(I)=TEMP*DX(I)+RATIO*DXNEW(I) DO 470 K=1,MCON 470 VMULTC(K)=AMAX1(0.0,TEMP*VMULTC(K)+RATIO*VMULTD(K)) IF (MCON .EQ. M) RESMAX=RESOLD+RATIO*(RESMAX-RESOLD) C C If the full step is not acceptable then begin another iteration. C Otherwise switch to stage two or end the calculation. C IF (ICON .GT. 0) GOTO 70 IF (STEP .EQ. STPFUL) GOTO 500 480 MCON=M+1 ICON=MCON IACT(MCON)=MCON VMULTC(MCON)=0.0 GOTO 60 C C We employ any freedom that may be available to reduce the objective C function before returning a DX whose length is less than RHO. C 490 IF (MCON .EQ. M) GOTO 480 IFULL=0 500 RETURN END C------------------------------------------------------------------------------ C Main program of test problems in Report DAMTP 1992/NA5. C------------------------------------------------------------------------------ COMMON NPROB DIMENSION X(10),XOPT(10),W(3000),IACT(51) DO 180 NPROB=1,10 IF (NPROB .EQ. 1) THEN C C Minimization of a simple quadratic function of two variables. C PRINT 10 10 FORMAT (/7X,'Output from test problem 1 (Simple quadratic)') N=2 M=0 XOPT(1)=-1.0 XOPT(2)=0.0 ELSE IF (NPROB .EQ. 2) THEN C C Easy two dimensional minimization in unit circle. C PRINT 20 20 FORMAT (/7X,'Output from test problem 2 (2D unit circle ', 1 'calculation)') N=2 M=1 XOPT(1)=SQRT(0.5) XOPT(2)=-XOPT(1) ELSE IF (NPROB .EQ. 3) THEN C C Easy three dimensional minimization in ellipsoid. C PRINT 30 30 FORMAT (/7X,'Output from test problem 3 (3D ellipsoid ', 1 'calculation)') N=3 M=1 XOPT(1)=1.0/SQRT(3.0) XOPT(2)=1.0/SQRT(6.0) XOPT(3)=-1.0/3.0 ELSE IF (NPROB .EQ. 4) THEN C C Weak version of Rosenbrock's problem. C PRINT 40 40 FORMAT (/7X,'Output from test problem 4 (Weak Rosenbrock)') N=2 M=0 XOPT(1)=-1.0 XOPT(2)=1.0 ELSE IF (NPROB .EQ. 5) THEN C C Intermediate version of Rosenbrock's problem. C PRINT 50 50 FORMAT (/7X,'Output from test problem 5 (Intermediate ', 1 'Rosenbrock)') N=2 M=0 XOPT(1)=-1.0 XOPT(2)=1.0 ELSE IF (NPROB .EQ. 6) THEN C C This problem is taken from Fletcher's book Practical Methods of C Optimization and has the equation number (9.1.15). C PRINT 60 60 FORMAT (/7X,'Output from test problem 6 (Equation ', 1 '(9.1.15) in Fletcher)') N=2 M=2 XOPT(1)=SQRT(0.5) XOPT(2)=XOPT(1) ELSE IF (NPROB .EQ. 7) THEN C C This problem is taken from Fletcher's book Practical Methods of C Optimization and has the equation number (14.4.2). C PRINT 70 70 FORMAT (/7X,'Output from test problem 7 (Equation ', 1 '(14.4.2) in Fletcher)') N=3 M=3 XOPT(1)=0.0 XOPT(2)=-3.0 XOPT(3)=-3.0 ELSE IF (NPROB .EQ. 8) THEN C C This problem is taken from page 66 of Hock and Schittkowski's book Test C Examples for Nonlinear Programming Codes. It is their test problem Number C 43, and has the name Rosen-Suzuki. C PRINT 80 80 FORMAT (/7X,'Output from test problem 8 (Rosen-Suzuki)') N=4 M=3 XOPT(1)=0.0 XOPT(2)=1.0 XOPT(3)=2.0 XOPT(4)=-1.0 ELSE IF (NPROB .EQ. 9) THEN C C This problem is taken from page 111 of Hock and Schittkowski's C book Test Examples for Nonlinear Programming Codes. It is their C test problem Number 100. C PRINT 90 90 FORMAT (/7X,'Output from test problem 9 (Hock and ', 1 'Schittkowski 100)') N=7 M=4 XOPT(1)=2.330499 XOPT(2)=1.951372 XOPT(3)=-0.4775414 XOPT(4)=4.365726 XOPT(5)=-0.624487 XOPT(6)=1.038131 XOPT(7)=1.594227 ELSE IF (NPROB .EQ. 10) THEN C C This problem is taken from page 415 of Luenberger's book Applied C Nonlinear Programming. It is to maximize the area of a hexagon of C unit diameter. C PRINT 100 100 FORMAT (/7X,'Output from test problem 10 (Hexagon area)') N=9 M=14 END IF DO 160 ICASE=1,2 DO 120 I=1,N 120 X(I)=1.0 RHOBEG=0.5 RHOEND=0.001 IF (ICASE .EQ. 2) RHOEND=0.0001 IPRINT=1 MAXFUN=2000 CALL COBYLA (N,M,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W,IACT) IF (NPROB .EQ. 10) THEN TEMPA=X(1)+X(3)+X(5)+X(7) TEMPB=X(2)+X(4)+X(6)+X(8) TEMPC=0.5/SQRT(TEMPA*TEMPA+TEMPB*TEMPB) TEMPD=TEMPC*SQRT(3.0) XOPT(1)=TEMPD*TEMPA+TEMPC*TEMPB XOPT(2)=TEMPD*TEMPB-TEMPC*TEMPA XOPT(3)=TEMPD*TEMPA-TEMPC*TEMPB XOPT(4)=TEMPD*TEMPB+TEMPC*TEMPA DO 130 I=1,4 130 XOPT(I+4)=XOPT(I) END IF TEMP=0.0 DO 140 I=1,N 140 TEMP=TEMP+(X(I)-XOPT(I))**2 PRINT 150, SQRT(TEMP) 150 FORMAT (/5X,'Least squares error in variables =',1PE16.6) 160 CONTINUE PRINT 170 170 FORMAT (2X,'----------------------------------------------', 1 '--------------------') 180 CONTINUE STOP END C------------------------------------------------------------------------------ SUBROUTINE CALCFC (N,M,X,F,CON) COMMON NPROB DIMENSION X(*),CON(*) IF (NPROB .EQ. 1) THEN C C Test problem 1 (Simple quadratic) C F=10.0*(X(1)+1.0)**2+X(2)**2 ELSE IF (NPROB .EQ. 2) THEN C C Test problem 2 (2D unit circle calculation) C F=X(1)*X(2) CON(1)=1.0-X(1)**2-X(2)**2 ELSE IF (NPROB .EQ. 3) THEN C C Test problem 3 (3D ellipsoid calculation) C F=X(1)*X(2)*X(3) CON(1)=1.0-X(1)**2-2.0*X(2)**2-3.0*X(3)**2 ELSE IF (NPROB .EQ. 4) THEN C C Test problem 4 (Weak Rosenbrock) C F=(X(1)**2-X(2))**2+(1.0+X(1))**2 ELSE IF (NPROB .EQ. 5) THEN C C Test problem 5 (Intermediate Rosenbrock) C F=10.0*(X(1)**2-X(2))**2+(1.0+X(1))**2 ELSE IF (NPROB .EQ. 6) THEN C C Test problem 6 (Equation (9.1.15) in Fletcher's book) C F=-X(1)-X(2) CON(1)=X(2)-X(1)**2 CON(2)=1.0-X(1)**2-X(2)**2 ELSE IF (NPROB .EQ. 7) THEN C C Test problem 7 (Equation (14.4.2) in Fletcher's book) C F=X(3) CON(1)=5.0*X(1)-X(2)+X(3) CON(2)=X(3)-X(1)**2-X(2)**2-4.0*X(2) CON(3)=X(3)-5.0*X(1)-X(2) ELSE IF (NPROB .EQ. 8) THEN C C Test problem 8 (Rosen-Suzuki) C F=X(1)**2+X(2)**2+2.0*X(3)**2+X(4)**2-5.0*X(1)-5.0*X(2) 1 -21.0*X(3)+7.0*X(4) CON(1)=8.0-X(1)**2-X(2)**2-X(3)**2-X(4)**2-X(1)+X(2) 1 -X(3)+X(4) CON(2)=10.0-X(1)**2-2.0*X(2)**2-X(3)**2-2.0*X(4)**2+X(1)+X(4) CON(3)=5.0-2.0*X(1)**2-X(2)**2-X(3)**2-2.0*X(1)+X(2)+X(4) ELSE IF (NPROB .EQ. 9) THEN C C Test problem 9 (Hock and Schittkowski 100) C F=(X(1)-10.0)**2+5.0*(X(2)-12.0)**2+X(3)**4+3.0*(X(4)-11.0)**2 1 +10.0*X(5)**6+7.0*X(6)**2+X(7)**4-4.0*X(6)*X(7)-10.0*X(6) 2 -8.0*X(7) CON(1)=127.0-2.0*X(1)**2-3.0*X(2)**4-X(3)-4.0*X(4)**2-5.0*X(5) CON(2)=282.0-7.0*X(1)-3.0*X(2)-10.0*X(3)**2-X(4)+X(5) CON(3)=196.0-23.0*X(1)-X(2)**2-6.0*X(6)**2+8.0*X(7) CON(4)=-4.0*X(1)**2-X(2)**2+3.0*X(1)*X(2)-2.0*X(3)**2-5.0*X(6) 1 +11.0*X(7) ELSE IF (NPROB .EQ. 10) THEN C C Test problem 10 (Hexagon area) C F=-0.5*(X(1)*X(4)-X(2)*X(3)+X(3)*X(9)-X(5)*X(9)+X(5)*X(8) 1 -X(6)*X(7)) CON(1)=1.0-X(3)**2-X(4)**2 CON(2)=1.0-X(9)**2 CON(3)=1.0-X(5)**2-X(6)**2 CON(4)=1.0-X(1)**2-(X(2)-X(9))**2 CON(5)=1.0-(X(1)-X(5))**2-(X(2)-X(6))**2 CON(6)=1.0-(X(1)-X(7))**2-(X(2)-X(8))**2 CON(7)=1.0-(X(3)-X(5))**2-(X(4)-X(6))**2 CON(8)=1.0-(X(3)-X(7))**2-(X(4)-X(8))**2 CON(9)=1.0-X(7)**2-(X(8)-X(9))**2 CON(10)=X(1)*X(4)-X(2)*X(3) CON(11)=X(3)*X(9) CON(12)=-X(5)*X(9) CON(13)=X(5)*X(8)-X(6)*X(7) CON(14)=X(9) END IF RETURN END ------------------------------------------------------------------------------- Output from test problem 1 (Simple quadratic) Normal return from subroutine COBYLA NFVALS = 37 F = 1.809750E-05 MAXCV = 0.000000E+00 X =-1.000879E+00 3.220609E-03 Least squares error in variables = 3.338389E-03 Normal return from subroutine COBYLA NFVALS = 65 F = 1.153291E-07 MAXCV = 0.000000E+00 X =-9.999341E-01 2.682342E-04 Least squares error in variables = 2.762020E-04 ------------------------------------------------------------------ Output from test problem 2 (2D unit circle calculation) Normal return from subroutine COBYLA NFVALS = 37 F =-4.999994E-01 MAXCV = 2.026558E-06 X = 7.062163E-01 -7.079976E-01 Least squares error in variables = 1.259601E-03 Normal return from subroutine COBYLA NFVALS = 44 F =-5.000000E-01 MAXCV = 5.960464E-08 X = 7.071500E-01 -7.070636E-01 Least squares error in variables = 6.107080E-05 ------------------------------------------------------------------ Output from test problem 3 (3D ellipsoid calculation) Normal return from subroutine COBYLA NFVALS = 52 F =-7.856688E-02 MAXCV = 6.288290E-06 X = 5.780203E-01 4.069204E-01 -3.340311E-01 Least squares error in variables = 1.642899E-03 Normal return from subroutine COBYLA NFVALS = 65 F =-7.856742E-02 MAXCV = 8.940697E-08 X = 5.773363E-01 4.082997E-01 -3.332995E-01 Least squares error in variables = 6.312904E-05 ------------------------------------------------------------------ Output from test problem 4 (Weak Rosenbrock) Normal return from subroutine COBYLA NFVALS = 100 F = 3.125441E-05 MAXCV = 0.000000E+00 X =-9.958720E-01 9.879909E-01 Least squares error in variables = 1.269875E-02 Normal return from subroutine COBYLA NFVALS = 173 F = 6.409362E-07 MAXCV = 0.000000E+00 X =-9.994782E-01 9.983495E-01 Least squares error in variables = 1.730967E-03 ------------------------------------------------------------------ Output from test problem 5 (Intermediate Rosenbrock) Normal return from subroutine COBYLA NFVALS = 347 F = 4.008353E-03 MAXCV = 0.000000E+00 X =-9.366965E-01 8.777190E-01 Least squares error in variables = 1.376952E-01 Normal return from subroutine COBYLA NFVALS = 698 F = 9.516375E-05 MAXCV = 0.000000E+00 X =-9.904447E-01 9.803594E-01 Least squares error in variables = 2.184159E-02 ------------------------------------------------------------------ Output from test problem 6 (Equation (9.1.15) in Fletcher) Normal return from subroutine COBYLA NFVALS = 30 F =-1.414216E+00 MAXCV = 2.950430E-06 X = 7.071948E-01 7.070208E-01 Least squares error in variables = 1.230355E-04 Normal return from subroutine COBYLA NFVALS = 40 F =-1.414214E+00 MAXCV = 0.000000E+00 X = 7.071791E-01 7.070344E-01 Least squares error in variables = 1.023325E-04 ------------------------------------------------------------------ Output from test problem 7 (Equation (14.4.2) in Fletcher) Normal return from subroutine COBYLA NFVALS = 28 F =-2.999881E+00 MAXCV = 0.000000E+00 X = 1.362514E-08 -2.999881E+00 -2.999881E+00 Least squares error in variables = 1.689246E-04 Normal return from subroutine COBYLA NFVALS = 32 F =-3.000046E+00 MAXCV = 4.673004E-05 X = 1.207445E-08 -3.000000E+00 -3.000046E+00 Least squares error in variables = 4.649224E-05 ------------------------------------------------------------------ Output from test problem 8 (Rosen-Suzuki) Normal return from subroutine COBYLA NFVALS = 66 F =-4.400000E+01 MAXCV = 3.099442E-06 X =-6.020486E-04 9.995968E-01 2.000546E+00 -9.994259E-01 Least squares error in variables = 1.073541E-03 Normal return from subroutine COBYLA NFVALS = 79 F =-4.400000E+01 MAXCV = 1.251698E-06 X =-2.246869E-04 9.996516E-01 2.000260E+00 -9.997578E-01 Least squares error in variables = 5.460466E-04 ------------------------------------------------------------------ Output from test problem 9 (Hock and Schittkowski 100) Normal return from subroutine COBYLA NFVALS = 237 F = 6.806309E+02 MAXCV = 0.000000E+00 X = 2.332463E+00 1.951341E+00 -4.587620E-01 4.364742E+00 -6.244753E-01 1.038812E+00 1.593632E+00 Least squares error in variables = 1.892897E-02 Normal return from subroutine COBYLA NFVALS = 248 F = 6.806310E+02 MAXCV = 1.907349E-05 X = 2.332446E+00 1.951307E+00 -4.577461E-01 4.364753E+00 -6.241184E-01 1.039491E+00 1.593760E+00 Least squares error in variables = 1.996995E-02 ------------------------------------------------------------------ Output from test problem 10 (Hexagon area) Normal return from subroutine COBYLA NFVALS = 150 F =-8.660254E-01 MAXCV = 1.192093E-06 X = 6.605685E-01 7.507660E-01 -3.188329E-01 9.478114E-01 6.614124E-01 7.500232E-01 -3.198982E-01 9.474520E-01 -6.671554E-11 Least squares error in variables = 1.124314E-03 Normal return from subroutine COBYLA NFVALS = 173 F =-8.660254E-01 MAXCV = 3.352761E-07 X = 6.606672E-01 7.506790E-01 -3.195507E-01 9.475691E-01 6.608437E-01 7.505235E-01 -3.197733E-01 9.474941E-01 -3.822812E-11 Least squares error in variables = 2.350494E-04 ------------------------------------------------------------------