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PowellMinimize.f90
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!####### constrained and unconstrained minimization ######
!F90 translated version of plato.asu.edu/ftp/other_software/bobyqa.zip and newuoa
!Powell 2009 Method http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf
!AL Sept 2012/2020: translated to F90, changed to input "funkk" class function argument
!to minimize rather than calfun subroutine as original
!Typical Usage (typically Npt= 2*dimension+1), to get best fit in params(:) starting from guess
! Type(TBOBYQA) :: Minimize !bounded minimization
! if (Minimize%BOBYQA(this_class, class_function, dimension, Npt, params,param_min, &
! param_max, step_radius, tolerance, DebugLevel, max_eval)) ...
!
!Type(TNEWUOA) :: Minimize !unbounded minimization
! if (Minimize%NEWUOA(this_class, class_function, dimension, Npt, param_guess, step_radius, &
! tolerance, DebugLevel, max_eval)) ...
!
! The class_function is of the form
!
! real(dp) function func(this, x)
! class(TMyType) :: this
! real(dp), intent(in) :: x(:)
! func = ...f(x)...
! end function func
MODULE Powell
implicit none
private
INTEGER, PARAMETER :: dp = KIND(1.d0) !SELECTED_REAL_KIND(12, 60)
INTEGER, PARAMETER :: func_dp = dp !SELECTED_REAL_KIND(12, 60)
INTEGER, PARAMETER :: Powell_CO_prec = dp
! the type of function minimized, a class procedure taking in an array of values
interface
FUNCTION obj_vec_function(obj, x)
use precision
class(*) :: obj
real(dl) :: obj_vec_function
real(dl) :: x(:)
END FUNCTION obj_vec_function
end interface
type TMinimizer
REAL(dp) :: Last_bestfit
procedure(obj_vec_function), pointer, nopass :: funkk
class(*), pointer :: obj
end type
type, extends(TMinimizer) :: TBOBYQA
REAL(dp) :: FVAL_Converge_difference = 0._dp
contains
procedure :: BOBYQA
end type
type, extends(TMinimizer) :: TNEWUOA
contains
procedure :: NEWUOA
procedure, private :: newuob
end type
PUBLIC :: TBOBYQA , TNEWUOA, Powell_CO_prec
CONTAINS
function BOBYQA (this, obj, funk, N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,MAXFUN)
!Main function for bounded minimization
use iso_c_binding
class(TBOBYQA) this
class(*), target :: obj
real(dp), external :: funk !a class function f(obj,x) where obj is any class instance
integer, intent(in)::n, npt, maxfun, iPrint
real(dp), intent(in) :: RHOBEG, RHOEND
logical BOBYQA
INTEGER ibmat , id , ifv , igo , ihq , ipq , isl , isu , &
& ivl , iw , ixa , ixb , ixn , ixo , ixp , izmat , j , jsl ,&
& jsu, ndim , np
REAL(dp) temp , zero
REAL(dp) X(*),XL(*),XU(*)
real(dp), allocatable :: W(:)
! This subroutine seeks the least value of a function of many variables,
! by applying a trust region method that forms quadratic models by
! interpolation. There is usually some freedom in the interpolation
! conditions, which is taken up by minimizing the Frobenius norm of
! the change to the second derivative of the model, beginning with the
! zero matrix. The values of the variables are constrained by upper and
! lower bounds. The arguments of the subroutine are as follows.
!
! N must be set to the number of variables and must be at least two.
! NPT is the number of interpolation conditions. Its value must be in
! the interval [N+2,(N+1)(N+2)/2]. Choices that exceed 2*N+1 are not
! recommended.
! Initial values of the variables must be set in X(1),X(2),...,X(N). They
! will be changed to the values that give the least calculated F.
! For I=1,2,...,N, XL(I) and XU(I) must provide the lower and upper
! bounds, respectively, on X(I). The construction of quadratic models
! requires XL(I) to be strictly less than XU(I) for each I. Further,
! the contribution to a model from changes to the I-th variable is
! damaged severely by rounding errors if XU(I)-XL(I) is too small.
! RHOBEG and RHOEND must be set to the initial and final values of a trust
! region radius, so both must be positive with RHOEND no greater than
! RHOBEG. Typically, RHOBEG should be about one tenth of the greatest
! expected change to a variable, while RHOEND should indicate the
! accuracy that is required in the final values of the variables. An
! error return occurs if any of the differences XU(I)-XL(I), I=1,...,N,
! is less than 2*RHOBEG.
! The value of IPRINT should be set to 0, 1, 2 or 3, which controls the
! amount of printing. Specifically, there is no output if IPRINT=0 and
! there is output only at the return if IPRINT=1. Otherwise, each new
! value of RHO is printed, with the best vector of variables so far and
! the corresponding value of the objective function. Further, each new
! value of F with its variables are output if IPRINT=3.
! MAXFUN must be set to an upper bound on the number of calls of CALFUN.
! The array W will be used for working space. Its length must be at least
! (NPT+5)*(NPT+N)+3*N*(N+5)/2.
!
! funkk(N,X) has to be provided by the user. It must set
! F to the value of the objective function for the current values of the
! variables X(1),X(2),...,X(N), which are generated automatically in a
! way that satisfies the bounds given in XL and XU.
!
! Return if the value of NPT is unacceptable.
call C_F_PROCPOINTER(c_funloc(funk), this%funkk)
this%obj => obj
BOBYQA = .false.
this%Last_bestfit = 1d30
NP=N+1
IF (NPT < N+2 .OR. NPT > ((N+2)*NP)/2) THEN
PRINT 10
10 FORMAT (/4X,'Return from BOBYQA because NPT is not in',&
& ' the required interval')
return
END IF
allocate(W((NPT+5)*(NPT+N)+3*N*(N+5)/2))
! Partition the working space array, so that different parts of it can
! be treated separately during the calculation of BOBYQB. The partition
! requires the first (NPT+2)*(NPT+N)+3*N*(N+5)/2 elements of W plus the
! space that is taken by the last array in the argument list of BOBYQB.
!
NDIM=NPT+N
IXB=1
IXP=IXB+N
IFV=IXP+N*NPT
IXO=IFV+NPT
IGO=IXO+N
IHQ=IGO+N
IPQ=IHQ+(N*NP)/2
IBMAT=IPQ+NPT
IZMAT=IBMAT+NDIM*N
ISL=IZMAT+NPT*(NPT-NP)
ISU=ISL+N
IXN=ISU+N
IXA=IXN+N
ID=IXA+N
IVL=ID+N
IW=IVL+NDIM
!
! Return if there is insufficient space between the bounds. Modify the
! initial X if necessary in order to avoid conflicts between the bounds
! and the construction of the first quadratic model. The lower and upper
! bounds on moves from the updated X are set now, in the ISL and ISU
! partitions of W, in order to provide useful and exact information about
! components of X that become within distance RHOBEG from their bounds.
!
ZERO=0.0D0
DO J=1,N
TEMP=XU(J)-XL(J)
IF (TEMP < RHOBEG+RHOBEG) THEN
PRINT 20
20 FORMAT (/4X,'Return from BOBYQA because one of the',&
& ' differences XU(I)-XL(I)'/6X,' is less than 2*RHOBEG.')
return
END IF
JSL=ISL+J-1
JSU=JSL+N
W(JSL)=XL(J)-X(J)
W(JSU)=XU(J)-X(J)
IF (W(JSL) >= -RHOBEG) THEN
IF (W(JSL) >= ZERO) THEN
X(J)=XL(J)
W(JSL)=ZERO
W(JSU)=TEMP
ELSE
X(J)=XL(J)+RHOBEG
W(JSL)=-RHOBEG
W(JSU)=DMAX1(XU(J)-X(J),RHOBEG)
END IF
ELSE IF (W(JSU) <= RHOBEG) THEN
IF (W(JSU) <= ZERO) THEN
X(J)=XU(J)
W(JSL)=-TEMP
W(JSU)=ZERO
ELSE
X(J)=XU(J)-RHOBEG
W(JSL)=DMIN1(XL(J)-X(J),-RHOBEG)
W(JSU)=RHOBEG
END IF
END IF
END DO
!
! Make the call of BOBYQB.
!
BOBYQA = BOBYQB (this,N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,MAXFUN,W(IXB),&
& W(IXP),W(IFV),W(IXO),W(IGO),W(IHQ),W(IPQ),W(IBMAT),W(IZMAT),&
& NDIM,W(ISL),W(ISU),W(IXN),W(IXA),W(ID),W(IVL),W(IW))
end function BOBYQA
function BOBYQB (this, N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,&
& MAXFUN,XBASE,XPT,FVAL,XOPT,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,&
& SL,SU,XNEW,XALT,D,VLAG,W)
class(TBOBYQA) this
integer NDIM, NPT
REAL(dp) X(*),XL(*),XU(*),XBASE(*),XPT(NPT,*),FVAL(*),&
& XOPT(*),GOPT(*),HQ(*),PQ(*),BMAT(NDIM,*),ZMAT(NPT,*),&
& SL(*),SU(*),XNEW(*),XALT(*),D(*),VLAG(*),W(*)
logical BOBYQB
REAL(dp) adelt , alpha , bdtest , bdtol , beta , biglsq , &
& bsum , cauchy , crvmin , curv , delsq , delta , den , &
& denom , densav , diff , diffa , diffb
REAL(dp) diffc , dist , distsq , dnorm , dsq , dx , errbig , f , &
& fopt , fracsq , frhosq , fsave , gisq , &
& gqsq , half , hdiag , one
REAL(dp) pqold , ratio , rho , Rhobeg , Rhoend , scaden , &
& sum , suma , sumb , sumpq , sumw , sumz , temp , ten ,&
& tenth , two
REAL(dp) vquad , xoptsq , zero
INTEGER i , ih , ip , Iprint , itest , j , jj , jp , k , kbase , &
& knew , kopt , ksav , Maxfun , N , nf , nfsav , nh , np
INTEGER nptm , nresc , ntrits
! The arguments N, NPT, X, XL, XU, RHOBEG, RHOEND, IPRINT and MAXFUN
! are identical to the corresponding arguments in SUBROUTINE BOBYQA.
! XBASE holds a shift of origin that should reduce the contributions
! from rounding errors to values of the model and Lagrange functions.
! XPT is a two-dimensional array that holds the coordinates of the
! interpolation points relative to XBASE.
! FVAL holds the values of F at the interpolation points.
! XOPT is set to the displacement from XBASE of the trust region centre.
! GOPT holds the gradient of the quadratic model at XBASE+XOPT.
! HQ holds the explicit second derivatives of the quadratic model.
! PQ contains the parameters of the implicit second derivatives of the
! quadratic model.
! BMAT holds the last N columns of H.
! ZMAT holds the factorization of the leading NPT by NPT submatrix of H,
! this factorization being ZMAT times ZMAT^T, which provides both the
! correct rank and positive semi-definiteness.
! NDIM is the first dimension of BMAT and has the value NPT+N.
! SL and SU hold the differences XL-XBASE and XU-XBASE, respectively.
! All the components of every XOPT are going to satisfy the bounds
! SL(I) .LEQ. XOPT(I) .LEQ. SU(I), with appropriate equalities when
! XOPT is on a constraint boundary.
! XNEW is chosen by SUBROUTINE TRSBOX or ALTMOV. Usually XBASE+XNEW is the
! vector of variables for the next call of CALFUN. XNEW also satisfies
! the SL and SU constraints in the way that has just been mentioned.
! XALT is an alternative to XNEW, chosen by ALTMOV, that may replace XNEW
! in order to increase the denominator in the updating of UPDATE.
! D is reserved for a trial step from XOPT, which is usually XNEW-XOPT.
! VLAG contains the values of the Lagrange functions at a new point X.
! They are part of a product that requires VLAG to be of length NDIM.
! W is a one-dimensional array that is used for working space. Its length
! must be at least 3*NDIM = 3*(NPT+N).
!
! Set some constants.
!
HALF=0.5D0
ONE=1.0D0
TEN=10.0D0
TENTH=0.1D0
TWO=2.0D0
ZERO=0.0D0
NP=N+1
NPTM=NPT-NP
NH=(N*NP)/2
BOBYQB = .true.
!
! The call of PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ,
! BMAT and ZMAT for the first iteration, with the corresponding values of
! of NF and KOPT, which are the number of calls of CALFUN so far and the
! index of the interpolation point at the trust region centre. Then the
! initial XOPT is set too. The branch to label 720 occurs if MAXFUN is
! less than NPT. GOPT will be updated if KOPT is different from KBASE.
!
CALL PRELIM (this,N,NPT,X,XL,XU,RHOBEG,IPRINT,MAXFUN,XBASE,XPT,&
& FVAL,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,SL,SU,NF,KOPT)
XOPTSQ=ZERO
DO I=1,N
XOPT(I)=XPT(KOPT,I)
XOPTSQ=XOPTSQ+XOPT(I)**2
END DO
FSAVE=FVAL(1)
IF (NF < NPT) THEN
IF (IPRINT > 0) PRINT 390
GOTO 720
END IF
KBASE=1
!
! Complete the settings that are required for the iterative procedure
!
RHO=RHOBEG
DELTA=RHO
NRESC=NF
NTRITS=0
DIFFA=ZERO
DIFFB=ZERO
ITEST=0
NFSAV=NF
!
! Update GOPT if necessary before the first iteration and after each
! call of RESCUE that makes a call of CALFUN.
!
20 IF (KOPT .NE. KBASE) THEN
IH=0
DO J=1,N
DO I=1,J
IH=IH+1
IF (I < J) GOPT(J)=GOPT(J)+HQ(IH)*XOPT(I)
GOPT(I)=GOPT(I)+HQ(IH)*XOPT(J)
END DO
END DO
IF (NF > NPT) THEN
DO K=1,NPT
TEMP=ZERO
DO J=1,N
TEMP=TEMP+XPT(K,J)*XOPT(J)
END DO
TEMP=PQ(K)*TEMP
DO I=1,N
GOPT(I)=GOPT(I)+TEMP*XPT(K,I)
END DO
END DO
END IF
END IF
!
! Generate the next point in the trust region that provides a small value
! of the quadratic model subject to the constraints on the variables.
! The integer NTRITS is set to the number "trust region" iterations that
! have occurred since the last "alternative" iteration. If the length
! of XNEW-XOPT is less than HALF*RHO, however, then there is a branch to
! label 650 or 680 with NTRITS=-1, instead of calculating F at XNEW.
!
60 CALL TRSBOX (N,NPT,XPT,XOPT,GOPT,HQ,PQ,SL,SU,DELTA,XNEW,D,&
& W,W(NP),W(NP+N),W(NP+2*N),W(NP+3*N),DSQ,CRVMIN)
DNORM=DMIN1(DELTA,DSQRT(DSQ))
IF (DNORM < HALF*RHO) THEN
NTRITS=-1
DISTSQ=(TEN*RHO)**2
IF (NF <= NFSAV+2) GOTO 650
!
! The following choice between labels 650 and 680 depends on whether or
! not our work with the current RHO seems to be complete. Either RHO is
! decreased or termination occurs if the errors in the quadratic model at
! the last three interpolation points compare favourably with predictions
! of likely improvements to the model within distance HALF*RHO of XOPT.
!
ERRBIG=DMAX1(DIFFA,DIFFB,DIFFC)
FRHOSQ=0.125D0*RHO*RHO
IF (CRVMIN > ZERO .AND. ERRBIG > FRHOSQ*CRVMIN)&
& GOTO 650
BDTOL=ERRBIG/RHO
DO J=1,N
BDTEST=BDTOL
IF (XNEW(J) .EQ. SL(J)) BDTEST=W(J)
IF (XNEW(J) .EQ. SU(J)) BDTEST=-W(J)
IF (BDTEST < BDTOL) THEN
CURV=HQ((J+J*J)/2)
DO K=1,NPT
CURV=CURV+PQ(K)*XPT(K,J)**2
END DO
BDTEST=BDTEST+HALF*CURV*RHO
IF (BDTEST < BDTOL) GOTO 650
END IF
END DO
GOTO 680
END IF
NTRITS=NTRITS+1
!
! Severe cancellation is likely to occur if XOPT is too far from XBASE.
! If the following test holds, then XBASE is shifted so that XOPT becomes
! zero. The appropriate changes are made to BMAT and to the second
! derivatives of the current model, beginning with the changes to BMAT
! that do not depend on ZMAT. VLAG is used temporarily for working space.
!
90 IF (DSQ <= 1.0D-3*XOPTSQ) THEN
FRACSQ=0.25D0*XOPTSQ
SUMPQ=ZERO
DO K=1,NPT
SUMPQ=SUMPQ+PQ(K)
SUM=-HALF*XOPTSQ
DO I=1,N
SUM=SUM+XPT(K,I)*XOPT(I)
END DO
W(NPT+K)=SUM
TEMP=FRACSQ-HALF*SUM
DO I=1,N
W(I)=BMAT(K,I)
VLAG(I)=SUM*XPT(K,I)+TEMP*XOPT(I)
IP=NPT+I
DO J=1,I
BMAT(IP,J)=BMAT(IP,J)+W(I)*VLAG(J)+VLAG(I)*W(J)
END DO
END DO
END DO
!
! Then the revisions of BMAT that depend on ZMAT are calculated.
!
DO JJ=1,NPTM
SUMZ=ZERO
SUMW=ZERO
DO K=1,NPT
SUMZ=SUMZ+ZMAT(K,JJ)
VLAG(K)=W(NPT+K)*ZMAT(K,JJ)
SUMW=SUMW+VLAG(K)
END DO
DO J=1,N
SUM=(FRACSQ*SUMZ-HALF*SUMW)*XOPT(J)
DO K=1,NPT
SUM=SUM+VLAG(K)*XPT(K,J)
END DO
W(J)=SUM
DO K=1,NPT
BMAT(K,J)=BMAT(K,J)+SUM*ZMAT(K,JJ)
END DO
END DO
DO I=1,N
IP=I+NPT
TEMP=W(I)
DO J=1,I
BMAT(IP,J)=BMAT(IP,J)+TEMP*W(J)
END DO
END DO
END DO
!
! The following instructions complete the shift, including the changes
! to the second derivative parameters of the quadratic model.
!
IH=0
DO J=1,N
W(J)=-HALF*SUMPQ*XOPT(J)
DO K=1,NPT
W(J)=W(J)+PQ(K)*XPT(K,J)
XPT(K,J)=XPT(K,J)-XOPT(J)
END DO
DO I=1,J
IH=IH+1
HQ(IH)=HQ(IH)+W(I)*XOPT(J)+XOPT(I)*W(J)
BMAT(NPT+I,J)=BMAT(NPT+J,I)
END DO
END DO
DO I=1,N
XBASE(I)=XBASE(I)+XOPT(I)
XNEW(I)=XNEW(I)-XOPT(I)
SL(I)=SL(I)-XOPT(I)
SU(I)=SU(I)-XOPT(I)
XOPT(I)=ZERO
END DO
XOPTSQ=ZERO
END IF
IF (NTRITS .EQ. 0) GOTO 210
GOTO 230
!
! XBASE is also moved to XOPT by a call of RESCUE. This calculation is
! more expensive than the previous shift, because new matrices BMAT and
! ZMAT are generated from scratch, which may include the replacement of
! interpolation points whose positions seem to be causing near linear
! dependence in the interpolation conditions. Therefore RESCUE is called
! only if rounding errors have reduced by at least a factor of two the
! denominator of the formula for updating the H matrix. It provides a
! useful safeguard, but is not invoked in most applications of BOBYQA.
!
190 NFSAV=NF
KBASE=KOPT
CALL RESCUE (this, N,NPT,XL,XU,IPRINT,MAXFUN,XBASE,XPT,FVAL,&
& XOPT,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,SL,SU,NF,DELTA,KOPT,&
& VLAG,W,W(N+NP),W(NDIM+NP))
!
! XOPT is updated now in case the branch below to label 720 is taken.
! Any updating of GOPT occurs after the branch below to label 20, which
! leads to a trust region iteration as does the branch to label 60.
!
XOPTSQ=ZERO
IF (KOPT .NE. KBASE) THEN
DO I=1,N
XOPT(I)=XPT(KOPT,I)
XOPTSQ=XOPTSQ+XOPT(I)**2
END DO
END IF
IF (NF < 0) THEN
NF=MAXFUN
IF (IPRINT > 0) PRINT 390
GOTO 720
END IF
NRESC=NF
IF (NFSAV < NF) THEN
NFSAV=NF
GOTO 20
END IF
IF (NTRITS > 0) GOTO 60
!
! Pick two alternative vectors of variables, relative to XBASE, that
! are suitable as new positions of the KNEW-th interpolation point.
! Firstly, XNEW is set to the point on a line through XOPT and another
! interpolation point that minimizes the predicted value of the next
! denominator, subject to ||XNEW - XOPT|| .LEQ. ADELT and to the SL
! and SU bounds. Secondly, XALT is set to the best feasible point on
! a constrained version of the Cauchy step of the KNEW-th Lagrange
! function, the corresponding value of the square of this function
! being returned in CAUCHY. The choice between these alternatives is
! going to be made when the denominator is calculated.
!
210 CALL ALTMOV (N,NPT,XPT,XOPT,BMAT,ZMAT,NDIM,SL,SU,KOPT,&
& KNEW,ADELT,XNEW,XALT,ALPHA,CAUCHY,W,W(NP),W(NDIM+1))
DO I=1,N
D(I)=XNEW(I)-XOPT(I)
END DO
!
! Calculate VLAG and BETA for the current choice of D. The scalar
! product of D with XPT(K,.) is going to be held in W(NPT+K) for
! use when VQUAD is calculated.
!
230 DO K=1,NPT
SUMA=ZERO
SUMB=ZERO
SUM=ZERO
DO J=1,N
SUMA=SUMA+XPT(K,J)*D(J)
SUMB=SUMB+XPT(K,J)*XOPT(J)
SUM=SUM+BMAT(K,J)*D(J)
END DO
W(K)=SUMA*(HALF*SUMA+SUMB)
VLAG(K)=SUM
W(NPT+K)=SUMA
END DO
BETA=ZERO
DO JJ=1,NPTM
SUM=ZERO
DO K=1,NPT
SUM=SUM+ZMAT(K,JJ)*W(K)
END DO
BETA=BETA-SUM*SUM
DO K=1,NPT
VLAG(K)=VLAG(K)+SUM*ZMAT(K,JJ)
END DO
END DO
DSQ=ZERO
BSUM=ZERO
DX=ZERO
DO J=1,N
DSQ=DSQ+D(J)**2
SUM=ZERO
DO K=1,NPT
SUM=SUM+W(K)*BMAT(K,J)
END DO
BSUM=BSUM+SUM*D(J)
JP=NPT+J
DO I=1,N
SUM=SUM+BMAT(JP,I)*D(I)
END DO
VLAG(JP)=SUM
BSUM=BSUM+SUM*D(J)
DX=DX+D(J)*XOPT(J)
END DO
BETA=DX*DX+DSQ*(XOPTSQ+DX+DX+HALF*DSQ)+BETA-BSUM
VLAG(KOPT)=VLAG(KOPT)+ONE
!
! If NTRITS is zero, the denominator may be increased by replacing
! the step D of ALTMOV by a Cauchy step. Then RESCUE may be called if
! rounding errors have damaged the chosen denominator.
!
IF (NTRITS .EQ. 0) THEN
DENOM=VLAG(KNEW)**2+ALPHA*BETA
IF (DENOM < CAUCHY .AND. CAUCHY > ZERO) THEN
DO I=1,N
XNEW(I)=XALT(I)
D(I)=XNEW(I)-XOPT(I)
END DO
CAUCHY=ZERO
GO TO 230
END IF
IF (DENOM <= HALF*VLAG(KNEW)**2) THEN
IF (NF > NRESC) GOTO 190
IF (IPRINT > 0) PRINT 320
320 FORMAT (/5X,'Return from BOBYQA because of much',&
& ' cancellation in a denominator.')
GOTO 720
END IF
!
! Alternatively, if NTRITS is positive, then set KNEW to the index of
! the next interpolation point to be deleted to make room for a trust
! region step. Again RESCUE may be called if rounding errors have damaged
! the chosen denominator, which is the reason for attempting to select
! KNEW before calculating the next value of the objective function.
!
ELSE
DELSQ=DELTA*DELTA
SCADEN=ZERO
BIGLSQ=ZERO
KNEW=0
DO K=1,NPT
IF (K .EQ. KOPT) continue
HDIAG=ZERO
DO JJ=1,NPTM
HDIAG=HDIAG+ZMAT(K,JJ)**2
END DO
DEN=BETA*HDIAG+VLAG(K)**2
DISTSQ=ZERO
DO J=1,N
DISTSQ=DISTSQ+(XPT(K,J)-XOPT(J))**2
END DO
TEMP=DMAX1(ONE,(DISTSQ/DELSQ)**2)
IF (TEMP*DEN > SCADEN) THEN
SCADEN=TEMP*DEN
KNEW=K
DENOM=DEN
END IF
BIGLSQ=DMAX1(BIGLSQ,TEMP*VLAG(K)**2)
END DO
IF (SCADEN <= HALF*BIGLSQ) THEN
IF (NF > NRESC) GOTO 190
IF (IPRINT > 0) PRINT 320
GOTO 720
END IF
END IF
!
! Put the variables for the next calculation of the objective function
! in XNEW, with any adjustments for the bounds.
!
!
! Calculate the value of the objective function at XBASE+XNEW, unless
! the limit on the number of calculations of F has been reached.
!
360 DO I=1,N
X(I)=DMIN1(DMAX1(XL(I),XBASE(I)+XNEW(I)),XU(I))
IF (XNEW(I) .EQ. SL(I)) X(I)=XL(I)
IF (XNEW(I) .EQ. SU(I)) X(I)=XU(I)
END DO
IF (NF >= MAXFUN) THEN
IF (IPRINT > 0) PRINT 390
390 FORMAT (/4X,'Return from BOBYQA because funkk has been',&
& ' called MAXFUN times.')
BOBYQB = .false.
GOTO 720
END IF
NF=NF+1
F = this%funkk(this%obj, X(1:N))
IF (IPRINT .EQ. 3) THEN
PRINT 400, NF,F,(X(I),I=1,N)
400 FORMAT (/4X,'Function number',I6,' F =',1PD18.10,&
& ' The corresponding X is:'/(2X,5D15.6))
END IF
IF (NTRITS .EQ. -1) THEN
FSAVE=F
GOTO 720
END IF
!
! Use the quadratic model to predict the change in F due to the step D,
! and set DIFF to the error of this prediction.
!
FOPT=FVAL(KOPT)
VQUAD=ZERO
IH=0
DO J=1,N
VQUAD=VQUAD+D(J)*GOPT(J)
DO I=1,J
IH=IH+1
TEMP=D(I)*D(J)
IF (I .EQ. J) TEMP=HALF*TEMP
VQUAD=VQUAD+HQ(IH)*TEMP
END DO
END DO
DO K=1,NPT
VQUAD=VQUAD+HALF*PQ(K)*W(NPT+K)**2
END DO
DIFF=F-FOPT-VQUAD
DIFFC=DIFFB
DIFFB=DIFFA
DIFFA=DABS(DIFF)
IF (DNORM > RHO) NFSAV=NF
!
! Pick the next value of DELTA after a trust region step.
!
IF (NTRITS > 0) THEN
IF (VQUAD >= ZERO) THEN
IF (IPRINT > 0) PRINT 430
430 FORMAT (/4X,'Return from BOBYQA because a trust',&
& ' region step has failed to reduce Q.')
BOBYQB = .false.
GOTO 720
END IF
RATIO=(F-FOPT)/VQUAD
IF (RATIO <= TENTH) THEN
DELTA=DMIN1(HALF*DELTA,DNORM)
ELSEIF (RATIO <= 0.7D0) THEN
DELTA=DMAX1(HALF*DELTA,DNORM)
ELSE
DELTA=DMAX1(HALF*DELTA,DNORM+DNORM)
END IF
IF (DELTA <= 1.5D0*RHO) DELTA=RHO
!
! Recalculate KNEW and DENOM if the new F is less than FOPT.
!
IF (F < FOPT) THEN
KSAV=KNEW
DENSAV=DENOM
DELSQ=DELTA*DELTA
SCADEN=ZERO
BIGLSQ=ZERO
KNEW=0
DO K=1,NPT
HDIAG=ZERO
DO JJ=1,NPTM
HDIAG=HDIAG+ZMAT(K,JJ)**2
END DO
DEN=BETA*HDIAG+VLAG(K)**2
DISTSQ=ZERO
DO J=1,N
DISTSQ=DISTSQ+(XPT(K,J)-XNEW(J))**2
END DO
TEMP=DMAX1(ONE,(DISTSQ/DELSQ)**2)
IF (TEMP*DEN > SCADEN) THEN
SCADEN=TEMP*DEN
KNEW=K
DENOM=DEN
END IF
BIGLSQ=DMAX1(BIGLSQ,TEMP*VLAG(K)**2)
END DO
IF (SCADEN <= HALF*BIGLSQ) THEN
KNEW=KSAV
DENOM=DENSAV
END IF
END IF
END IF
!
! Update BMAT and ZMAT, so that the KNEW-th interpolation point can be
! moved. Also update the second derivative terms of the model.
!
CALL UPDATE (N,NPT,BMAT,ZMAT,NDIM,VLAG,BETA,DENOM,KNEW,W)
IH=0
PQOLD=PQ(KNEW)
PQ(KNEW)=ZERO
DO I=1,N
TEMP=PQOLD*XPT(KNEW,I)
DO J=1,I
IH=IH+1
HQ(IH)=HQ(IH)+TEMP*XPT(KNEW,J)
END DO
END DO
DO JJ=1,NPTM
TEMP=DIFF*ZMAT(KNEW,JJ)
DO K=1,NPT
PQ(K)=PQ(K)+TEMP*ZMAT(K,JJ)
END DO
END DO
!
! Include the new interpolation point, and make the changes to GOPT at
! the old XOPT that are caused by the updating of the quadratic model.
!
FVAL(KNEW)=F
DO I=1,N
XPT(KNEW,I)=XNEW(I)
W(I)=BMAT(KNEW,I)
END DO
DO K=1,NPT
SUMA=ZERO
DO JJ=1,NPTM
SUMA=SUMA+ZMAT(KNEW,JJ)*ZMAT(K,JJ)
END DO
SUMB=ZERO
DO J=1,N
SUMB=SUMB+XPT(K,J)*XOPT(J)
END DO
TEMP=SUMA*SUMB
DO I=1,N
W(I)=W(I)+TEMP*XPT(K,I)
END DO
END DO
DO I=1,N
GOPT(I)=GOPT(I)+DIFF*W(I)
END DO
!
! Update XOPT, GOPT and KOPT if the new calculated F is less than FO
!
IF (F < FOPT) THEN
KOPT=KNEW
XOPTSQ=ZERO
IH=0
DO J=1,N
XOPT(J)=XNEW(J)
XOPTSQ=XOPTSQ+XOPT(J)**2
DO I=1,J
IH=IH+1
IF (I < J) GOPT(J)=GOPT(J)+HQ(IH)*D(I)
GOPT(I)=GOPT(I)+HQ(IH)*D(J)
END DO
END DO
DO K=1,NPT
TEMP=ZERO
DO J=1,N
TEMP=TEMP+XPT(K,J)*D(J)
END DO
TEMP=PQ(K)*TEMP
DO I=1,N
GOPT(I)=GOPT(I)+TEMP*XPT(K,I)
END DO
END DO
END IF
!
! Calculate the parameters of the least Frobenius norm interpolant to
! the current data, the gradient of this interpolant at XOPT being put
! into VLAG(NPT+I), I=1,2,...,N.
!
IF (NTRITS > 0) THEN
DO K=1,NPT
VLAG(K)=FVAL(K)-FVAL(KOPT)
W(K)=ZERO
END DO
DO J=1,NPTM
SUM=ZERO
DO K=1,NPT
SUM=SUM+ZMAT(K,J)*VLAG(K)
END DO
DO K=1,NPT
W(K)=W(K)+SUM*ZMAT(K,J)
END DO
END DO
DO K=1,NPT
SUM=ZERO
DO J=1,N
SUM=SUM+XPT(K,J)*XOPT(J)
END DO
W(K+NPT)=W(K)
W(K)=SUM*W(K)
END DO
GQSQ=ZERO
GISQ=ZERO
DO I=1,N
SUM=ZERO
DO K=1,NPT
SUM=SUM+BMAT(K,I)*VLAG(K)+XPT(K,I)*W(K)
END DO
IF (XOPT(I) .EQ. SL(I)) THEN
GQSQ=GQSQ+DMIN1(ZERO,GOPT(I))**2
GISQ=GISQ+DMIN1(ZERO,SUM)**2
ELSE IF (XOPT(I) .EQ. SU(I)) THEN
GQSQ=GQSQ+DMAX1(ZERO,GOPT(I))**2
GISQ=GISQ+DMAX1(ZERO,SUM)**2
ELSE
GQSQ=GQSQ+GOPT(I)**2
GISQ=GISQ+SUM*SUM
END IF
VLAG(NPT+I)=SUM
END DO
!
! Test whether to replace the new quadratic model by the least Frobenius
! norm interpolant, making the replacement if the test is satisfied.
!
ITEST=ITEST+1
IF (GQSQ < TEN*GISQ) ITEST=0
IF (ITEST >= 3) THEN
DO I=1,MAX0(NPT,NH)
IF (I <= N) GOPT(I)=VLAG(NPT+I)
IF (I <= NPT) PQ(I)=W(NPT+I)
IF (I <= NH) HQ(I)=ZERO
ITEST=0
END DO
END IF
END IF
!
! If a trust region step has provided a sufficient decrease in F, then
! branch for another trust region calculation. The case NTRITS=0 occurs
! when the new interpolation point was reached by an alternative step.
!
IF (NTRITS .EQ. 0) GOTO 60
IF (F <= FOPT+TENTH*VQUAD) GOTO 60
!
! Alternatively, find out if the interpolation points are close enough
! to the best point so far.
!
DISTSQ=DMAX1((TWO*DELTA)**2,(TEN*RHO)**2)
650 KNEW=0
DO K=1,NPT
SUM=ZERO
DO J=1,N
SUM=SUM+(XPT(K,J)-XOPT(J))**2
END DO
IF (SUM > DISTSQ) THEN
KNEW=K
DISTSQ=SUM
END IF
END DO
!
! If KNEW is positive, then ALTMOV finds alternative new positions for
! the KNEW-th interpolation point within distance ADELT of XOPT. It is
! reached via label 90. Otherwise, there is a branch to label 60 for
! another trust region iteration, unless the calculations with the
! current RHO are complete.
!
IF (KNEW > 0) THEN
DIST=DSQRT(DISTSQ)
IF (NTRITS .EQ. -1) THEN
DELTA=DMIN1(TENTH*DELTA,HALF*DIST)
IF (DELTA <= 1.5D0*RHO) DELTA=RHO
END IF
NTRITS=0
ADELT=DMAX1(DMIN1(TENTH*DIST,DELTA),RHO)
DSQ=ADELT*ADELT
GOTO 90
END IF
IF (NTRITS .EQ. -1) GOTO 680
IF (RATIO > ZERO) GOTO 60
IF (DMAX1(DELTA,DNORM) > RHO) GOTO 60
!
! The calculations with the current value of RHO are complete. Pick the
! next values of RHO and DELTA.
!
680 IF (RHO > RHOEND .and. (this%FVAL_Converge_difference==0._dp .or. &
abs(FVAL(KOPT)-this%Last_bestfit) > this%FVAL_Converge_difference)) THEN
DELTA=HALF*RHO
RATIO=RHO/RHOEND
IF (RATIO <= 16.0D0) THEN
RHO=RHOEND
ELSE IF (RATIO <= 250.0D0) THEN
RHO=DSQRT(RATIO)*RHOEND
ELSE
RHO=TENTH*RHO
END IF
DELTA=DMAX1(DELTA,RHO)
this%Last_bestfit = FVAL(KOPT)
IF (IPRINT >= 2) THEN
IF (IPRINT >= 3) PRINT 690
690 FORMAT (5X)
PRINT 700, RHO,NF
700 FORMAT (/4X,'New RHO =',1PD11.4,5X,'Number of',&
& ' function values =',I6)
PRINT 710, FVAL(KOPT),(XBASE(I)+XOPT(I),I=1,N)
710 FORMAT (4X,'Least value of F =',1PD23.15,9X,&
& 'The corresponding X is:'/(2X,5D15.6))
END IF
NTRITS=0
NFSAV=NF
GOTO 60
END IF
!
! Return from the calculation, after another Newton-Raphson step, if
! it is too short to have been tried before.
!
IF (NTRITS .EQ. -1) GOTO 360
720 IF (FVAL(KOPT) <= FSAVE) THEN
DO I=1,N
X(I)=DMIN1(DMAX1(XL(I),XBASE(I)+XOPT(I)),XU(I))
IF (XOPT(I) .EQ. SL(I)) X(I)=XL(I)
IF (XOPT(I) .EQ. SU(I)) X(I)=XU(I)
END DO
F=FVAL(KOPT)
END IF
IF (IPRINT >= 1) THEN
if (RHO > RHOEND) write(*,*) ' FVAL_Converge_difference reached'
PRINT 740, NF
740 FORMAT (/4X,'At the return from BOBYQA',5X,&
& 'Number of function values =',I6)
PRINT 710, F,(X(I),I=1,N)
END IF
end function BOBYQB
SUBROUTINE ALTMOV (N,NPT,XPT,XOPT,BMAT,ZMAT,NDIM,SL,SU,KOPT,&
& KNEW,ADELT,XNEW,XALT,ALPHA,CAUCHY,GLAG,HCOL,W)
REAL(dp) Adelt , Alpha , bigstp , Cauchy , const , csave , &
& curv , dderiv , diff , distsq , ggfree , gw , ha , &
& half , one , predsq , presav
REAL(dp) scale , slbd , step , stpsav , subd , sumin , &
& temp , tempa , tempb , tempd , vlag , wfixsq , wsqsav, zero
INTEGER i , ibdsav , iflag , ilbd , isbd , iubd , j , k , Knew , &
& Kopt , ksav , N , Ndim , Npt
REAL(dp) XPT(NPT,*),XOPT(*),BMAT(NDIM,*),ZMAT(NPT,*),SL(*),&
& SU(*),XNEW(*),XALT(*),GLAG(*),HCOL(*),W(*)
!
! The arguments N, NPT, XPT, XOPT, BMAT, ZMAT, NDIM, SL and SU all have
! the same meanings as the corresponding arguments of BOBYQB.
! KOPT is the index of the optimal interpolation point.
! KNEW is the index of the interpolation point that is going to be moved.
! ADELT is the current trust region bound.
! XNEW will be set to a suitable new position for the interpolation point
! XPT(KNEW,.). Specifically, it satisfies the SL, SU and trust region
! bounds and it should provide a large denominator in the next call of
! UPDATE. The step XNEW-XOPT from XOPT is restricted to moves along the
! straight lines through XOPT and another interpolation point.
! XALT also provides a large value of the modulus of the KNEW-th Lagrange
! function subject to the constraints that have been mentioned, its main
! difference from XNEW being that XALT-XOPT is a constrained version of