!***************************************************************************************** !> ! BOBYQA: **B**ound **O**ptimization **BY** **Q**uadratic **A**pproximation ! ! The purpose of BOBYQA is to seek the least value of a function F of several ! variables, when derivatives are not available. The constraints are the lower ! and upper bounds on every variable, which can be set to huge values for ! unconstrained variables. ! ! The algorithm is intended to change the variables to values that are close ! to a local minimum of F. The user, however, should assume responsibility for ! finding out if the calculations are satisfactory, by considering carefully ! the values of F that occur. ! !# References ! * "[The BOBYQA algorithm for bound constrained optimization without ! derivatives](http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf)". ! !# History ! * M.J.D. Powell (January 5th, 2009) -- There are no restrictions on or charges ! for the use of the software. I hope that the time and effort I have spent on ! developing the package will be helpful to much research and to many applications. ! * Jacob Williams, July 2015 : refactoring of the code into modern Fortran. module bobyqa_module use kind_module, only: wp private abstract interface subroutine func (n, x, f) !! calfun interface import :: wp implicit none integer,intent(in) :: n real(wp),dimension(:),intent(in) :: x real(wp),intent(out) :: f end subroutine func end interface public :: bobyqa public :: bobyqa_test contains !***************************************************************************************** !***************************************************************************************** !> ! 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. ! ! In addition to providing CALFUN, an initial vector of variables and ! the lower and upper bounds, the user has to set the values of the parameters ! ```RHOBEG```, ```RHOEND``` and ```NPT```. After scaling the individual variables ! if necessary, so that the magnitudes of their expected changes are similar, ! ```RHOBEG``` is the initial steplength for changes to the variables, a reasonable choice ! being the mesh size of a coarse grid search. Further, ```RHOEND``` should be suitable for ! a search on a very fine grid. Typically, the software calculates a vector ! of variables that is within distance ```10*RHOEND``` of a local minimum. Another ! consideration is that every trial vector of variables is forced to satisfy ! the lower and upper bounds, but there has to be room to make a search in all ! directions. Therefore an error return occurs if the difference between the ! bounds on any variable is less than ```2*RHOBEG```. The parameter ```NPT``` specifies ! the number of interpolation conditions on each quadratic model, the value ! ```NPT=2*N+1``` being recommended for a start, where ```N``` is the number of ! variables. It is often worthwhile to try other choices too, but much larger values ! tend to be inefficient, because the amount of routine work of each iteration is ! of magnitude ```NPT**2```, and because the achievement of adequate accuracy in some ! matrix calculations becomes more difficult. Some excellent numerical results ! have been found in the case ```NPT=N+6``` even with more than 100 variables. subroutine bobyqa (n, npt, x, xl, xu, rhobeg, rhoend, iprint, maxfun, calfun) implicit none integer,intent(in) :: n !! number of variables (must be at least two) integer,intent(in) :: npt !! 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. real(wp),dimension(:),intent(inout) :: x !! 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. real(wp),dimension(:),intent(in) :: xl !! lower bounds on x. 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. real(wp),dimension(:),intent(in) :: xu !! upper bounds on x. 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. real(wp),intent(in) :: rhobeg !! RHOBEG must be set to the initial value of a trust region radius. !! It must be positive, and typically should be about one tenth of the greatest !! expected change to a variable. An error return occurs if any of !! the differences XU(I)-XL(I), I=1,...,N, is less than 2*RHOBEG. real(wp),intent(in) :: rhoend !! RHOEND must be set to the final value of a trust !! region radius. It must be positive with RHOEND no greater than !! RHOBEG. Typically, RHOEND should indicate the !! accuracy that is required in the final values of the variables. integer,intent(in) :: iprint !! 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. integer,intent(in) :: maxfun !! an upper bound on the number of calls of CALFUN. procedure (func) :: calfun !! SUBROUTINE CALFUN (N,X,F) 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. integer :: ibmat,id,ifv,igo,ihq,ipq,isl,isu,ivl,iw,ixa,& ixb,ixn,ixo,ixp,izmat,j,jsl,jsu,ndim,np real(wp),dimension(:),allocatable :: w real(wp) :: temp real(wp),parameter :: zero = 0.0_wp ! The array W will be used for working space. allocate( w((NPT+5)*(NPT+N)+3*N*(N+5)/2) ) ! ! Return if the value of NPT is unacceptable. ! np = n + 1 if (npt < n+2 .or. npt > ((n+2)*np)/2) then write(*,'(/4X,A)') & 'Return from BOBYQA because NPT is not in the required interval' return end if ! ! 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. ! do j = 1, n temp = xu (j) - xl (j) if (temp < rhobeg+rhobeg) then write(*,'(/4X,A)') & 'Return from BOBYQA because one of the differences '//& 'XU(I)-XL(I) 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) = max (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) = min (xl(j)-x(j),-rhobeg) w (jsu) = rhobeg end if end if end do ! ! Make the call of BOBYQB. ! call bobyqb (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), calfun) deallocate(w) end subroutine bobyqa subroutine bobyqb (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, calfun) implicit real (wp) (a-h, o-z) dimension x (*), xl (*), xu (*), xbase (*), xpt (npt,*), fval (*), xopt (*), & gopt (*), hq (*), pq (*), bmat (ndim,*), zmat (npt,*), sl (*), su (*), & xnew (*), xalt (*), d (*), vlag (*), w (*) procedure (func) :: calfun ! ! 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 = 0.5_wp one = 1.0_wp ten = 10.0_wp tenth = 0.1_wp two = 2.0_wp zero = 0.0_wp np = n + 1 nptm = npt - np nh = (n*np) / 2 ! ! 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 (n, npt, x, xl, xu, rhobeg, iprint, maxfun, xbase, xpt, fval, gopt, & & hq, pq, bmat, zmat, ndim, sl, su, nf, kopt, calfun) 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) write(*,'(/4X,A)') & 'Return from BOBYQA because CALFUN has been called MAXFUN times.' go to 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 /= 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 = min (delta, sqrt(dsq)) if (dnorm < half*rho) then ntrits = - 1 distsq = (ten*rho) ** 2 if (nf <= nfsav+2) go to 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 = max (diffa, diffb, diffc) frhosq = 0.125_wp * rho * rho if (crvmin > zero .and. errbig > frhosq*crvmin) go to 650 bdtol = errbig / rho do j = 1, n bdtest = bdtol if (xnew(j) == sl(j)) bdtest = w (j) if (xnew(j) == 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) go to 650 end if end do go to 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 <= 1.0e-3_wp*xoptsq) then fracsq = 0.25_wp * 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 == 0) go to 210 go to 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 (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), & & calfun) ! ! 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 /= 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) write(*,'(/4X,A)') & 'Return from BOBYQA because CALFUN has been called MAXFUN times.' go to 720 end if nresc = nf if (nfsav < nf) then nfsav = nf go to 20 end if if (ntrits > 0) go to 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 == 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) go to 190 if (iprint > 0) write(*,'(/5X,A)') & 'Return from BOBYQA because of much cancellation in a denominator.' go to 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 == kopt) cycle 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 = max (one, (distsq/delsq)**2) if (temp*den > scaden) then scaden = temp * den knew = k denom = den end if biglsq = max (biglsq, temp*vlag(k)**2) end do if (scaden <= half*biglsq) then if (nf > nresc) go to 190 if (iprint > 0) write(*,'(/5X,A)') & 'Return from BOBYQA because of much cancellation in a denominator.' go to 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) = min (max(xl(i), xbase(i)+xnew(i)), xu(i)) if (xnew(i) == sl(i)) x (i) = xl (i) if (xnew(i) == su(i)) x (i) = xu (i) end do if (nf >= maxfun) then if (iprint > 0) write(*,'(/4X,A)') & 'Return from BOBYQA because CALFUN has been called MAXFUN times.' go to 720 end if nf = nf + 1 call calfun (n, x(1:n), f) if (iprint == 3) then ! print 400, nf, f, (x(i), i=1, n) ! 400 format (/ 4 x, 'Function number', i6, ' F =', 1 pd18.10,& ! ' The corresponding X is:' / (2 x, 5d15.6)) write(*,'(/4x,a,i6,a,1pd18.10,a/(2x,5d15.6))') & 'Function number', nf, ' F =', f, & ' The corresponding X is:', (x(i), i=1, n) end if if (ntrits ==-1) then fsave = f go to 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 == 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 = abs (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) write(*,'(/4x,a)') 'Return from BOBYQA because a trust'//& ' region step has failed to reduce Q.' go to 720 end if ratio = (f-fopt) / vquad if (ratio <= tenth) then delta = min (half*delta, dnorm) else if (ratio <= 0.7_wp) then delta = max (half*delta, dnorm) else delta = max (half*delta, dnorm+dnorm) end if if (delta <= 1.5_wp*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 = max (one, (distsq/delsq)**2) if (temp*den > scaden) then scaden = temp * den knew = k denom = den end if biglsq = max (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 FOPT. ! 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) == sl(i)) then gqsq = gqsq + min (zero, gopt(i)) ** 2 gisq = gisq + min (zero, sum) ** 2 else if (xopt(i) == su(i)) then gqsq = gqsq + max (zero, gopt(i)) ** 2 gisq = gisq + max (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, max (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 == 0) go to 60 if (f <= fopt+tenth*vquad) go to 60 ! ! Alternatively, find out if the interpolation points are close enough ! to the best point so far. ! distsq = max ((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 = sqrt (distsq) if (ntrits ==-1) then delta = min (tenth*delta, half*dist) if (delta <= 1.5_wp*rho) delta = rho end if ntrits = 0 adelt = max (min(tenth*dist, delta), rho) dsq = adelt * adelt go to 90 end if if (ntrits ==-1) go to 680 if (ratio > zero) go to 60 if (max(delta, dnorm) > rho) go to 60 ! ! The calculations with the current value of RHO are complete. Pick the ! next values of RHO and DELTA. ! 680 if (rho > rhoend) then delta = half * rho ratio = rho / rhoend if (ratio <= 16.0_wp) then rho = rhoend else if (ratio <= 250.0_wp) then rho = sqrt (ratio) * rhoend else rho = tenth * rho end if delta = max (delta, rho) if (iprint >= 2) then if (iprint >= 3) write(*,'(5x)') '' print 700, rho, nf 700 format (/ 4 x, 'New RHO =', 1 pd11.4, 5 x, 'Number of',& ' function values =', i6) print 710, fval (kopt), (xbase(i)+xopt(i), i=1, n) 710 format (4 x, 'Least value of F =', 1 pd23.15, 9 x,& 'The corresponding X is:'/(2 x, 5d15.6)) end if ntrits = 0 nfsav = nf go to 60 end if ! ! Return from the calculation, after another Newton-Raphson step, if ! it is too short to have been tried before. ! if (ntrits ==-1) go to 360 720 if (fval(kopt) <= fsave) then do i = 1, n x (i) = min (max(xl(i), xbase(i)+xopt(i)), xu(i)) if (xopt(i) == sl(i)) x (i) = xl (i) if (xopt(i) == su(i)) x (i) = xu (i) end do f = fval (kopt) end if if (iprint >= 1) then print 740, nf 740 format (/ 4 x, 'At the return from BOBYQA', 5 x,& 'Number of function values =', i6) print 710, f, (x(i), i=1, n) end if return end subroutine bobyqb subroutine altmov (n, npt, xpt, xopt, bmat, zmat, ndim, sl, su, kopt, knew, adelt, & & xnew, xalt, alpha, cauchy, glag, hcol, w) implicit real (wp) (a-h, o-z) dimension 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 ! the Cauchy step within the trust region. An exception is that XALT is ! not calculated if all components of GLAG (see below) are zero. ! ALPHA will be set to the KNEW-th diagonal element of the H matrix. ! CAUCHY will be set to the square of the KNEW-th Lagrange function at ! the step XALT-XOPT from XOPT for the vector XALT that is returned, ! except that CAUCHY is set to zero if XALT is not calculated. ! GLAG is a working space vector of length N for the gradient of the ! KNEW-th Lagrange function at XOPT. ! HCOL is a working space vector of length NPT for the second derivative ! coefficients of the KNEW-th Lagrange function. ! W is a working space vector of length 2N that is going to hold the ! constrained Cauchy step from XOPT of the Lagrange function, followed ! by the downhill version of XALT when the uphill step is calculated. ! ! Set the first NPT components of W to the leading elements of the ! KNEW-th column of the H matrix. ! half = 0.5_wp one = 1.0_wp zero = 0.0_wp const = one + sqrt (2.0_wp) do k = 1, npt hcol (k) = zero end do do j = 1, npt - n - 1 temp = zmat (knew, j) do k = 1, npt hcol (k) = hcol (k) + temp * zmat (k, j) end do end do alpha = hcol (knew) ha = half * alpha ! ! Calculate the gradient of the KNEW-th Lagrange function at XOPT. ! do i = 1, n glag (i) = bmat (knew, i) end do do k = 1, npt temp = zero do j = 1, n temp = temp + xpt (k, j) * xopt (j) end do temp = hcol (k) * temp do i = 1, n glag (i) = glag (i) + temp * xpt (k, i) end do end do ! ! Search for a large denominator along the straight lines through XOPT ! and another interpolation point. SLBD and SUBD will be lower and upper ! bounds on the step along each of these lines in turn. PREDSQ will be ! set to the square of the predicted denominator for each line. PRESAV ! will be set to the largest admissible value of PREDSQ that occurs. ! presav = zero do k = 1, npt if (k == kopt) cycle dderiv = zero distsq = zero do i = 1, n temp = xpt (k, i) - xopt (i) dderiv = dderiv + glag (i) * temp distsq = distsq + temp * temp end do subd = adelt / sqrt (distsq) slbd = - subd ilbd = 0 iubd = 0 sumin = min (one, subd) ! ! Revise SLBD and SUBD if necessary because of the bounds in SL and SU. ! do i = 1, n temp = xpt (k, i) - xopt (i) if (temp > zero) then if (slbd*temp < sl(i)-xopt(i)) then slbd = (sl(i)-xopt(i)) / temp ilbd = - i end if if (subd*temp > su(i)-xopt(i)) then subd = max (sumin, (su(i)-xopt(i))/temp) iubd = i end if else if (temp < zero) then if (slbd*temp > su(i)-xopt(i)) then slbd = (su(i)-xopt(i)) / temp ilbd = i end if if (subd*temp < sl(i)-xopt(i)) then subd = max (sumin, (sl(i)-xopt(i))/temp) iubd = - i end if end if end do ! ! Seek a large modulus of the KNEW-th Lagrange function when the index ! of the other interpolation point on the line through XOPT is KNEW. ! if (k == knew) then diff = dderiv - one step = slbd vlag = slbd * (dderiv-slbd*diff) isbd = ilbd temp = subd * (dderiv-subd*diff) if (abs(temp) > abs(vlag)) then step = subd vlag = temp isbd = iubd end if tempd = half * dderiv tempa = tempd - diff * slbd tempb = tempd - diff * subd if (tempa*tempb < zero) then temp = tempd * tempd / diff if (abs(temp) > abs(vlag)) then step = tempd / diff vlag = temp isbd = 0 end if end if ! ! Search along each of the other lines through XOPT and another point. ! else step = slbd vlag = slbd * (one-slbd) isbd = ilbd temp = subd * (one-subd) if (abs(temp) > abs(vlag)) then step = subd vlag = temp isbd = iubd end if if (subd > half) then if (abs(vlag) < 0.25_wp) then step = half vlag = 0.25_wp isbd = 0 end if end if vlag = vlag * dderiv end if ! ! Calculate PREDSQ for the current line search and maintain PRESAV. ! temp = step * (one-step) * distsq predsq = vlag * vlag * (vlag*vlag+ha*temp*temp) if (predsq > presav) then presav = predsq ksav = k stpsav = step ibdsav = isbd end if end do ! ! Construct XNEW in a way that satisfies the bound constraints exactly. ! do i = 1, n temp = xopt (i) + stpsav * (xpt(ksav, i)-xopt(i)) xnew (i) = max (sl(i), min(su(i), temp)) end do if (ibdsav < 0) xnew (-ibdsav) = sl (-ibdsav) if (ibdsav > 0) xnew (ibdsav) = su (ibdsav) ! ! Prepare for the iterative method that assembles the constrained Cauchy ! step in W. The sum of squares of the fixed components of W is formed in ! WFIXSQ, and the free components of W are set to BIGSTP. ! bigstp = adelt + adelt iflag = 0 100 wfixsq = zero ggfree = zero do i = 1, n w (i) = zero tempa = min (xopt(i)-sl(i), glag(i)) tempb = max (xopt(i)-su(i), glag(i)) if (tempa > zero .or. tempb < zero) then w (i) = bigstp ggfree = ggfree + glag (i) ** 2 end if end do if (ggfree == zero) then cauchy = zero return end if ! ! Investigate whether more components of W can be fixed. ! 120 temp = adelt * adelt - wfixsq if (temp > zero) then wsqsav = wfixsq step = sqrt (temp/ggfree) ggfree = zero do i = 1, n if (w(i) == bigstp) then temp = xopt (i) - step * glag (i) if (temp <= sl(i)) then w (i) = sl (i) - xopt (i) wfixsq = wfixsq + w (i) ** 2 else if (temp >= su(i)) then w (i) = su (i) - xopt (i) wfixsq = wfixsq + w (i) ** 2 else ggfree = ggfree + glag (i) ** 2 end if end if end do if (wfixsq > wsqsav .and. ggfree > zero) go to 120 end if ! ! Set the remaining free components of W and all components of XALT, ! except that W may be scaled later. ! gw = zero do i = 1, n if (w(i) == bigstp) then w (i) = - step * glag (i) xalt (i) = max (sl(i), min(su(i), xopt(i)+w(i))) else if (w(i) == zero) then xalt (i) = xopt (i) else if (glag(i) > zero) then xalt (i) = sl (i) else xalt (i) = su (i) end if gw = gw + glag (i) * w (i) end do ! ! Set CURV to the curvature of the KNEW-th Lagrange function along W. ! Scale W by a factor less than one if that can reduce the modulus of ! the Lagrange function at XOPT+W. Set CAUCHY to the final value of ! the square of this function. ! curv = zero do k = 1, npt temp = zero do j = 1, n temp = temp + xpt (k, j) * w (j) end do curv = curv + hcol (k) * temp * temp end do if (iflag == 1) curv = - curv if (curv >-gw .and. curv <-const*gw) then scale = - gw / curv do i = 1, n temp = xopt (i) + scale * w (i) xalt (i) = max (sl(i), min(su(i), temp)) end do cauchy = (half*gw*scale) ** 2 else cauchy = (gw+half*curv) ** 2 end if ! ! If IFLAG is zero, then XALT is calculated as before after reversing ! the sign of GLAG. Thus two XALT vectors become available. The one that ! is chosen is the one that gives the larger value of CAUCHY. ! if (iflag == 0) then do i = 1, n glag (i) = - glag (i) w (n+i) = xalt (i) end do csave = cauchy iflag = 1 go to 100 end if if (csave > cauchy) then do i = 1, n xalt (i) = w (n+i) end do cauchy = csave end if end subroutine altmov subroutine prelim (n, npt, x, xl, xu, rhobeg, iprint, maxfun, xbase, xpt, fval, gopt, & & hq, pq, bmat, zmat, ndim, sl, su, nf, kopt, calfun) implicit real (wp) (a-h, o-z) dimension x (*), xl (*), xu (*), xbase (*), xpt (npt,*), fval (*), gopt (*), hq & & (*), pq (*), bmat (ndim,*), zmat (npt,*), sl (*), su (*) procedure (func) :: calfun ! ! The arguments N, NPT, X, XL, XU, RHOBEG, IPRINT and MAXFUN are the ! same as the corresponding arguments in SUBROUTINE BOBYQA. ! The arguments XBASE, XPT, FVAL, HQ, PQ, BMAT, ZMAT, NDIM, SL and SU ! are the same as the corresponding arguments in BOBYQB, the elements ! of SL and SU being set in BOBYQA. ! GOPT is usually the gradient of the quadratic model at XOPT+XBASE, but ! it is set by PRELIM to the gradient of the quadratic model at XBASE. ! If XOPT is nonzero, BOBYQB will change it to its usual value later. ! NF is maintaned as the number of calls of CALFUN so far. ! KOPT will be such that the least calculated value of F so far is at ! the point XPT(KOPT,.)+XBASE in the space of the variables. ! ! SUBROUTINE PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ, ! BMAT and ZMAT for the first iteration, and it maintains the values of ! NF and KOPT. The vector X is also changed by PRELIM. ! ! Set some constants. ! half = 0.5_wp one = 1.0_wp two = 2.0_wp zero = 0.0_wp rhosq = rhobeg * rhobeg recip = one / rhosq np = n + 1 ! ! Set XBASE to the initial vector of variables, and set the initial ! elements of XPT, BMAT, HQ, PQ and ZMAT to zero. ! do j = 1, n xbase (j) = x (j) do k = 1, npt xpt (k, j) = zero end do do i = 1, ndim bmat (i, j) = zero end do end do do ih = 1, (n*np) / 2 hq (ih) = zero end do do k = 1, npt pq (k) = zero do j = 1, npt - np zmat (k, j) = zero end do end do ! ! Begin the initialization procedure. NF becomes one more than the number ! of function values so far. The coordinates of the displacement of the ! next initial interpolation point from XBASE are set in XPT(NF+1,.). ! nf = 0 50 nfm = nf nfx = nf - n nf = nf + 1 if (nfm <= 2*n) then if (nfm >= 1 .and. nfm <= n) then stepa = rhobeg if (su(nfm) == zero) stepa = - stepa xpt (nf, nfm) = stepa else if (nfm > n) then stepa = xpt (nf-n, nfx) stepb = - rhobeg if (sl(nfx) == zero) stepb = min (two*rhobeg, su(nfx)) if (su(nfx) == zero) stepb = max (-two*rhobeg, sl(nfx)) xpt (nf, nfx) = stepb end if else itemp = (nfm-np) / n jpt = nfm - itemp * n - n ipt = jpt + itemp if (ipt > n) then itemp = jpt jpt = ipt - n ipt = itemp end if xpt (nf, ipt) = xpt (ipt+1, ipt) xpt (nf, jpt) = xpt (jpt+1, jpt) end if ! ! Calculate the next value of F. The least function value so far and ! its index are required. ! do j = 1, n x (j) = min (max(xl(j), xbase(j)+xpt(nf, j)), xu(j)) if (xpt(nf, j) == sl(j)) x (j) = xl (j) if (xpt(nf, j) == su(j)) x (j) = xu (j) end do call calfun (n, x(1:n), f) if (iprint == 3) then print 70, nf, f, (x(i), i=1, n) 70 format (/ 4 x, 'Function number', i6, ' F =', 1 pd18.10,& ' The corresponding X is:' / (2 x, 5d15.6)) end if fval (nf) = f if (nf == 1) then fbeg = f kopt = 1 else if (f < fval(kopt)) then kopt = nf end if ! ! Set the nonzero initial elements of BMAT and the quadratic model in the ! cases when NF is at most 2*N+1. If NF exceeds N+1, then the positions ! of the NF-th and (NF-N)-th interpolation points may be switched, in ! order that the function value at the first of them contributes to the ! off-diagonal second derivative terms of the initial quadratic model. ! if (nf <= 2*n+1) then if (nf >= 2 .and. nf <= n+1) then gopt (nfm) = (f-fbeg) / stepa if (npt < nf+n) then bmat (1, nfm) = - one / stepa bmat (nf, nfm) = one / stepa bmat (npt+nfm, nfm) = - half * rhosq end if else if (nf >= n+2) then ih = (nfx*(nfx+1)) / 2 temp = (f-fbeg) / stepb diff = stepb - stepa hq (ih) = two * (temp-gopt(nfx)) / diff gopt (nfx) = (gopt(nfx)*stepb-temp*stepa) / diff if (stepa*stepb < zero) then if (f < fval(nf-n)) then fval (nf) = fval (nf-n) fval (nf-n) = f if (kopt == nf) kopt = nf - n xpt (nf-n, nfx) = stepb xpt (nf, nfx) = stepa end if end if bmat (1, nfx) = - (stepa+stepb) / (stepa*stepb) bmat (nf, nfx) = - half / xpt (nf-n, nfx) bmat (nf-n, nfx) = - bmat (1, nfx) - bmat (nf, nfx) zmat (1, nfx) = sqrt (two) / (stepa*stepb) zmat (nf, nfx) = sqrt (half) / rhosq zmat (nf-n, nfx) = - zmat (1, nfx) - zmat (nf, nfx) end if ! ! Set the off-diagonal second derivatives of the Lagrange functions and ! the initial quadratic model. ! else ih = (ipt*(ipt-1)) / 2 + jpt zmat (1, nfx) = recip zmat (nf, nfx) = recip zmat (ipt+1, nfx) = - recip zmat (jpt+1, nfx) = - recip temp = xpt (nf, ipt) * xpt (nf, jpt) hq (ih) = (fbeg-fval(ipt+1)-fval(jpt+1)+f) / temp end if if (nf < npt .and. nf < maxfun) go to 50 end subroutine prelim subroutine rescue (n, npt, xl, xu, iprint, maxfun, xbase, xpt, fval, xopt, gopt, hq, & & pq, bmat, zmat, ndim, sl, su, nf, delta, kopt, vlag, ptsaux, ptsid, w, calfun) implicit real (wp) (a-h, o-z) dimension xl (*), xu (*), xbase (*), xpt (npt,*), fval (*), xopt (*), gopt (*), & & hq (*), pq (*), bmat (ndim,*), zmat (npt,*), sl (*), su (*), vlag (*), ptsaux & & (2,*), ptsid (*), w (*) procedure (func) :: calfun ! ! The arguments N, NPT, XL, XU, IPRINT, MAXFUN, XBASE, XPT, FVAL, XOPT, ! GOPT, HQ, PQ, BMAT, ZMAT, NDIM, SL and SU have the same meanings as ! the corresponding arguments of BOBYQB on the entry to RESCUE. ! NF is maintained as the number of calls of CALFUN so far, except that ! NF is set to -1 if the value of MAXFUN prevents further progress. ! KOPT is maintained so that FVAL(KOPT) is the least calculated function ! value. Its correct value must be given on entry. It is updated if a ! new least function value is found, but the corresponding changes to ! XOPT and GOPT have to be made later by the calling program. ! DELTA is the current trust region radius. ! VLAG is a working space vector that will be used for the values of the ! provisional Lagrange functions at each of the interpolation points. ! They are part of a product that requires VLAG to be of length NDIM. ! PTSAUX is also a working space array. For J=1,2,...,N, PTSAUX(1,J) and ! PTSAUX(2,J) specify the two positions of provisional interpolation ! points when a nonzero step is taken along e_J (the J-th coordinate ! direction) through XBASE+XOPT, as specified below. Usually these ! steps have length DELTA, but other lengths are chosen if necessary ! in order to satisfy the given bounds on the variables. ! PTSID is also a working space array. It has NPT components that denote ! provisional new positions of the original interpolation points, in ! case changes are needed to restore the linear independence of the ! interpolation conditions. The K-th point is a candidate for change ! if and only if PTSID(K) is nonzero. In this case let p and q be the ! integer parts of PTSID(K) and (PTSID(K)-p) multiplied by N+1. If p ! and q are both positive, the step from XBASE+XOPT to the new K-th ! interpolation point is PTSAUX(1,p)*e_p + PTSAUX(1,q)*e_q. Otherwise ! the step is PTSAUX(1,p)*e_p or PTSAUX(2,q)*e_q in the cases q=0 or ! p=0, respectively. ! The first NDIM+NPT elements of the array W are used for working space. ! The final elements of BMAT and ZMAT are set in a well-conditioned way ! to the values that are appropriate for the new interpolation points. ! The elements of GOPT, HQ and PQ are also revised to the values that are ! appropriate to the final quadratic model. ! ! Set some constants. ! half = 0.5_wp one = 1.0_wp zero = 0.0_wp np = n + 1 sfrac = half / real (np, wp) nptm = npt - np ! ! Shift the interpolation points so that XOPT becomes the origin, and set ! the elements of ZMAT to zero. The value of SUMPQ is required in the ! updating of HQ below. The squares of the distances from XOPT to the ! other interpolation points are set at the end of W. Increments of WINC ! may be added later to these squares to balance the consideration of ! the choice of point that is going to become current. ! sumpq = zero winc = zero do k = 1, npt distsq = zero do j = 1, n xpt (k, j) = xpt (k, j) - xopt (j) distsq = distsq + xpt (k, j) ** 2 end do sumpq = sumpq + pq (k) w (ndim+k) = distsq winc = max (winc, distsq) do j = 1, nptm zmat (k, j) = zero end do end do ! ! Update HQ so that HQ and PQ define the second derivatives of the model ! after XBASE has been shifted to the trust region centre. ! 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) end do do i = 1, j ih = ih + 1 hq (ih) = hq (ih) + w (i) * xopt (j) + w (j) * xopt (i) end do end do ! ! Shift XBASE, SL, SU and XOPT. Set the elements of BMAT to zero, and ! also set the elements of PTSAUX. ! do j = 1, n xbase (j) = xbase (j) + xopt (j) sl (j) = sl (j) - xopt (j) su (j) = su (j) - xopt (j) xopt (j) = zero ptsaux (1, j) = min (delta, su(j)) ptsaux (2, j) = max (-delta, sl(j)) if (ptsaux(1, j)+ptsaux(2, j) < zero) then temp = ptsaux (1, j) ptsaux (1, j) = ptsaux (2, j) ptsaux (2, j) = temp end if if (abs(ptsaux(2, j)) < half*abs(ptsaux(1, j))) then ptsaux (2, j) = half * ptsaux (1, j) end if do i = 1, ndim bmat (i, j) = zero end do end do fbase = fval (kopt) ! ! Set the identifiers of the artificial interpolation points that are ! along a coordinate direction from XOPT, and set the corresponding ! nonzero elements of BMAT and ZMAT. ! ptsid (1) = sfrac do j = 1, n jp = j + 1 jpn = jp + n ptsid (jp) = real (j, wp) + sfrac if (jpn <= npt) then ptsid (jpn) = real (j, wp) / real (np, wp) + sfrac temp = one / (ptsaux(1, j)-ptsaux(2, j)) bmat (jp, j) = - temp + one / ptsaux (1, j) bmat (jpn, j) = temp + one / ptsaux (2, j) bmat (1, j) = - bmat (jp, j) - bmat (jpn, j) zmat (1, j) = sqrt (2.0_wp) / abs (ptsaux(1, j)*ptsaux(2, j)) zmat (jp, j) = zmat (1, j) * ptsaux (2, j) * temp zmat (jpn, j) = - zmat (1, j) * ptsaux (1, j) * temp else bmat (1, j) = - one / ptsaux (1, j) bmat (jp, j) = one / ptsaux (1, j) bmat (j+npt, j) = - half * ptsaux (1, j) ** 2 end if end do ! ! Set any remaining identifiers with their nonzero elements of ZMAT. ! if (npt >= n+np) then do k = 2 * np, npt iw = (real(k-np, wp)-half) / real (n, wp) ip = k - np - iw * n iq = ip + iw if (iq > n) iq = iq - n ptsid (k) = real (ip, wp) + real (iq, wp) / real (np, wp) + sfrac temp = one / (ptsaux(1, ip)*ptsaux(1, iq)) zmat (1, k-np) = temp zmat (ip+1, k-np) = - temp zmat (iq+1, k-np) = - temp zmat (k, k-np) = temp end do end if nrem = npt kold = 1 knew = kopt ! ! Reorder the provisional points in the way that exchanges PTSID(KOLD) ! with PTSID(KNEW). ! 80 do j = 1, n temp = bmat (kold, j) bmat (kold, j) = bmat (knew, j) bmat (knew, j) = temp end do do j = 1, nptm temp = zmat (kold, j) zmat (kold, j) = zmat (knew, j) zmat (knew, j) = temp end do ptsid (kold) = ptsid (knew) ptsid (knew) = zero w (ndim+knew) = zero nrem = nrem - 1 if (knew /= kopt) then temp = vlag (kold) vlag (kold) = vlag (knew) vlag (knew) = temp ! ! Update the BMAT and ZMAT matrices so that the status of the KNEW-th ! interpolation point can be changed from provisional to original. The ! subroutine returns if all the original points are reinstated. ! The nonnegative values of W(NDIM+K) are required in the search below. ! call update (n, npt, bmat, zmat, ndim, vlag, beta, denom, knew, w) if (nrem == 0) return do k = 1, npt w (ndim+k) = abs (w(ndim+k)) end do end if ! ! Pick the index KNEW of an original interpolation point that has not ! yet replaced one of the provisional interpolation points, giving ! attention to the closeness to XOPT and to previous tries with KNEW. ! 120 dsqmin = zero do k = 1, npt if (w(ndim+k) > zero) then if (dsqmin == zero .or. w(ndim+k) < dsqmin) then knew = k dsqmin = w (ndim+k) end if end if end do if (dsqmin == zero) go to 260 ! ! Form the W-vector of the chosen original interpolation point. ! do j = 1, n w (npt+j) = xpt (knew, j) end do do k = 1, npt sum = zero if (k == kopt) then continue else if (ptsid(k) == zero) then do j = 1, n sum = sum + w (npt+j) * xpt (k, j) end do else ip = ptsid (k) if (ip > 0) sum = w (npt+ip) * ptsaux (1, ip) iq = real (np, wp) * ptsid (k) - real (ip*np, wp) if (iq > 0) then iw = 1 if (ip == 0) iw = 2 sum = sum + w (npt+iq) * ptsaux (iw, iq) end if end if w (k) = half * sum * sum end do ! ! Calculate VLAG and BETA for the required updating of the H matrix if ! XPT(KNEW,.) is reinstated in the set of interpolation points. ! do k = 1, npt sum = zero do j = 1, n sum = sum + bmat (k, j) * w (npt+j) end do vlag (k) = sum end do beta = zero do j = 1, nptm sum = zero do k = 1, npt sum = sum + zmat (k, j) * w (k) end do beta = beta - sum * sum do k = 1, npt vlag (k) = vlag (k) + sum * zmat (k, j) end do end do bsum = zero distsq = zero do j = 1, n sum = zero do k = 1, npt sum = sum + bmat (k, j) * w (k) end do jp = j + npt bsum = bsum + sum * w (jp) do ip = npt + 1, ndim sum = sum + bmat (ip, j) * w (ip) end do bsum = bsum + sum * w (jp) vlag (jp) = sum distsq = distsq + xpt (knew, j) ** 2 end do beta = half * distsq * distsq + beta - bsum vlag (kopt) = vlag (kopt) + one ! ! KOLD is set to the index of the provisional interpolation point that is ! going to be deleted to make way for the KNEW-th original interpolation ! point. The choice of KOLD is governed by the avoidance of a small value ! of the denominator in the updating calculation of UPDATE. ! denom = zero vlmxsq = zero do k = 1, npt if (ptsid(k) /= zero) then hdiag = zero do j = 1, nptm hdiag = hdiag + zmat (k, j) ** 2 end do den = beta * hdiag + vlag (k) ** 2 if (den > denom) then kold = k denom = den end if end if vlmxsq = max (vlmxsq, vlag(k)**2) end do if (denom <= 1.0e-2_wp*vlmxsq) then w (ndim+knew) = - w (ndim+knew) - winc go to 120 end if go to 80 ! ! When label 260 is reached, all the final positions of the interpolation ! points have been chosen although any changes have not been included yet ! in XPT. Also the final BMAT and ZMAT matrices are complete, but, apart ! from the shift of XBASE, the updating of the quadratic model remains to ! be done. The following cycle through the new interpolation points begins ! by putting the new point in XPT(KPT,.) and by setting PQ(KPT) to zero, ! except that a RETURN occurs if MAXFUN prohibits another value of F. ! 260 do kpt = 1, npt if (ptsid(kpt) == zero) cycle if (nf >= maxfun) then nf = - 1 return end if ih = 0 do j = 1, n w (j) = xpt (kpt, j) xpt (kpt, j) = zero temp = pq (kpt) * w (j) do i = 1, j ih = ih + 1 hq (ih) = hq (ih) + temp * w (i) end do end do pq (kpt) = zero ip = ptsid (kpt) iq = real (np, wp) * ptsid (kpt) - real (ip*np, wp) if (ip > 0) then xp = ptsaux (1, ip) xpt (kpt, ip) = xp end if if (iq > 0) then xq = ptsaux (1, iq) if (ip == 0) xq = ptsaux (2, iq) xpt (kpt, iq) = xq end if ! ! Set VQUAD to the value of the current model at the new point. ! vquad = fbase if (ip > 0) then ihp = (ip+ip*ip) / 2 vquad = vquad + xp * (gopt(ip)+half*xp*hq(ihp)) end if if (iq > 0) then ihq = (iq+iq*iq) / 2 vquad = vquad + xq * (gopt(iq)+half*xq*hq(ihq)) if (ip > 0) then iw = max (ihp, ihq) - abs (ip-iq) vquad = vquad + xp * xq * hq (iw) end if end if do k = 1, npt temp = zero if (ip > 0) temp = temp + xp * xpt (k, ip) if (iq > 0) temp = temp + xq * xpt (k, iq) vquad = vquad + half * pq (k) * temp * temp end do ! ! Calculate F at the new interpolation point, and set DIFF to the factor ! that is going to multiply the KPT-th Lagrange function when the model ! is updated to provide interpolation to the new function value. ! do i = 1, n w (i) = min (max(xl(i), xbase(i)+xpt(kpt, i)), xu(i)) if (xpt(kpt, i) == sl(i)) w (i) = xl (i) if (xpt(kpt, i) == su(i)) w (i) = xu (i) end do nf = nf + 1 call calfun (n, w(1:n), f) if (iprint == 3) then print 300, nf, f, (w(i), i=1, n) 300 format (/ 4 x, 'Function number', i6, ' F =', 1 pd18.10,& ' The corresponding X is:' / (2 x, 5d15.6)) end if fval (kpt) = f if (f < fval(kopt)) kopt = kpt diff = f - vquad ! ! Update the quadratic model. The RETURN from the subroutine occurs when ! all the new interpolation points are included in the model. ! do i = 1, n gopt (i) = gopt (i) + diff * bmat (kpt, i) end do do k = 1, npt sum = zero do j = 1, nptm sum = sum + zmat (k, j) * zmat (kpt, j) end do temp = diff * sum if (ptsid(k) == zero) then pq (k) = pq (k) + temp else ip = ptsid (k) iq = real (np, wp) * ptsid (k) - real (ip*np, wp) ihq = (iq*iq+iq) / 2 if (ip == 0) then hq (ihq) = hq (ihq) + temp * ptsaux (2, iq) ** 2 else ihp = (ip*ip+ip) / 2 hq (ihp) = hq (ihp) + temp * ptsaux (1, ip) ** 2 if (iq > 0) then hq (ihq) = hq (ihq) + temp * ptsaux (1, iq) ** 2 iw = max (ihp, ihq) - abs (iq-ip) hq (iw) = hq (iw) + temp * ptsaux (1, ip) * ptsaux (1, iq) end if end if end if end do ptsid (kpt) = zero end do end subroutine rescue subroutine trsbox (n, npt, xpt, xopt, gopt, hq, pq, sl, su, delta, xnew, d, gnew, & & xbdi, s, hs, hred, dsq, crvmin) implicit real (wp) (a-h, o-z) dimension xpt (npt,*), xopt (*), gopt (*), hq (*), pq (*), sl (*), su (*), xnew & & (*), d (*), gnew (*), xbdi (*), s (*), hs (*), hred (*) ! ! The arguments N, NPT, XPT, XOPT, GOPT, HQ, PQ, SL and SU have the same ! meanings as the corresponding arguments of BOBYQB. ! DELTA is the trust region radius for the present calculation, which ! seeks a small value of the quadratic model within distance DELTA of ! XOPT subject to the bounds on the variables. ! XNEW will be set to a new vector of variables that is approximately ! the one that minimizes the quadratic model within the trust region ! subject to the SL and SU constraints on the variables. It satisfies ! as equations the bounds that become active during the calculation. ! D is the calculated trial step from XOPT, generated iteratively from an ! initial value of zero. Thus XNEW is XOPT+D after the final iteration. ! GNEW holds the gradient of the quadratic model at XOPT+D. It is updated ! when D is updated. ! XBDI is a working space vector. For I=1,2,...,N, the element XBDI(I) is ! set to -1.0, 0.0, or 1.0, the value being nonzero if and only if the ! I-th variable has become fixed at a bound, the bound being SL(I) or ! SU(I) in the case XBDI(I)=-1.0 or XBDI(I)=1.0, respectively. This ! information is accumulated during the construction of XNEW. ! The arrays S, HS and HRED are also used for working space. They hold the ! current search direction, and the changes in the gradient of Q along S ! and the reduced D, respectively, where the reduced D is the same as D, ! except that the components of the fixed variables are zero. ! DSQ will be set to the square of the length of XNEW-XOPT. ! CRVMIN is set to zero if D reaches the trust region boundary. Otherwise ! it is set to the least curvature of H that occurs in the conjugate ! gradient searches that are not restricted by any constraints. The ! value CRVMIN=-1.0D0 is set, however, if all of these searches are ! constrained. ! ! A version of the truncated conjugate gradient is applied. If a line ! search is restricted by a constraint, then the procedure is restarted, ! the values of the variables that are at their bounds being fixed. If ! the trust region boundary is reached, then further changes may be made ! to D, each one being in the two dimensional space that is spanned ! by the current D and the gradient of Q at XOPT+D, staying on the trust ! region boundary. Termination occurs when the reduction in Q seems to ! be close to the greatest reduction that can be achieved. ! ! Set some constants. ! half = 0.5_wp one = 1.0_wp onemin = - 1.0_wp zero = 0.0_wp ! ! The sign of GOPT(I) gives the sign of the change to the I-th variable ! that will reduce Q from its value at XOPT. Thus XBDI(I) shows whether ! or not to fix the I-th variable at one of its bounds initially, with ! NACT being set to the number of fixed variables. D and GNEW are also ! set for the first iteration. DELSQ is the upper bound on the sum of ! squares of the free variables. QRED is the reduction in Q so far. ! iterc = 0 nact = 0 sqstp = zero do i = 1, n xbdi (i) = zero if (xopt(i) <= sl(i)) then if (gopt(i) >= zero) xbdi (i) = onemin else if (xopt(i) >= su(i)) then if (gopt(i) <= zero) xbdi (i) = one end if if (xbdi(i) /= zero) nact = nact + 1 d (i) = zero gnew (i) = gopt (i) end do delsq = delta * delta qred = zero crvmin = onemin ! ! Set the next search direction of the conjugate gradient method. It is ! the steepest descent direction initially and when the iterations are ! restarted because a variable has just been fixed by a bound, and of ! course the components of the fixed variables are zero. ITERMAX is an ! upper bound on the indices of the conjugate gradient iterations. ! 20 beta = zero 30 stepsq = zero do i = 1, n if (xbdi(i) /= zero) then s (i) = zero else if (beta == zero) then s (i) = - gnew (i) else s (i) = beta * s (i) - gnew (i) end if stepsq = stepsq + s (i) ** 2 end do if (stepsq == zero) go to 190 if (beta == zero) then gredsq = stepsq itermax = iterc + n - nact end if if (gredsq*delsq <= 1.0e-4_wp*qred*qred) go to 190 ! ! Multiply the search direction by the second derivative matrix of Q and ! calculate some scalars for the choice of steplength. Then set BLEN to ! the length of the the step to the trust region boundary and STPLEN to ! the steplength, ignoring the simple bounds. ! go to 210 50 resid = delsq ds = zero shs = zero do i = 1, n if (xbdi(i) == zero) then resid = resid - d (i) ** 2 ds = ds + s (i) * d (i) shs = shs + s (i) * hs (i) end if end do if (resid <= zero) go to 90 temp = sqrt (stepsq*resid+ds*ds) if (ds < zero) then blen = (temp-ds) / stepsq else blen = resid / (temp+ds) end if stplen = blen if (shs > zero) then stplen = min (blen, gredsq/shs) end if ! ! ! Reduce STPLEN if necessary in order to preserve the simple bounds, ! letting IACT be the index of the new constrained variable. ! iact = 0 do i = 1, n if (s(i) /= zero) then xsum = xopt (i) + d (i) if (s(i) > zero) then temp = (su(i)-xsum) / s (i) else temp = (sl(i)-xsum) / s (i) end if if (temp < stplen) then stplen = temp iact = i end if end if end do ! ! Update CRVMIN, GNEW and D. Set SDEC to the decrease that occurs in Q. ! sdec = zero if (stplen > zero) then iterc = iterc + 1 temp = shs / stepsq if (iact == 0 .and. temp > zero) then crvmin = min (crvmin, temp) if (crvmin == onemin) crvmin = temp end if ggsav = gredsq gredsq = zero do i = 1, n gnew (i) = gnew (i) + stplen * hs (i) if (xbdi(i) == zero) gredsq = gredsq + gnew (i) ** 2 d (i) = d (i) + stplen * s (i) end do sdec = max (stplen*(ggsav-half*stplen*shs), zero) qred = qred + sdec end if ! ! Restart the conjugate gradient method if it has hit a new bound. ! if (iact > 0) then nact = nact + 1 xbdi (iact) = one if (s(iact) < zero) xbdi (iact) = onemin delsq = delsq - d (iact) ** 2 if (delsq <= zero) go to 90 go to 20 end if ! ! If STPLEN is less than BLEN, then either apply another conjugate ! gradient iteration or RETURN. ! if (stplen < blen) then if (iterc == itermax) go to 190 if (sdec <= 0.01_wp*qred) go to 190 beta = gredsq / ggsav go to 30 end if 90 crvmin = zero ! ! Prepare for the alternative iteration by calculating some scalars and ! by multiplying the reduced D by the second derivative matrix of Q. ! 100 if (nact >= n-1) go to 190 dredsq = zero dredg = zero gredsq = zero do i = 1, n if (xbdi(i) == zero) then dredsq = dredsq + d (i) ** 2 dredg = dredg + d (i) * gnew (i) gredsq = gredsq + gnew (i) ** 2 s (i) = d (i) else s (i) = zero end if end do itcsav = iterc go to 210 ! ! Let the search direction S be a linear combination of the reduced D ! and the reduced G that is orthogonal to the reduced D. ! 120 iterc = iterc + 1 temp = gredsq * dredsq - dredg * dredg if (temp <= 1.0e-4_wp*qred*qred) go to 190 temp = sqrt (temp) do i = 1, n if (xbdi(i) == zero) then s (i) = (dredg*d(i)-dredsq*gnew(i)) / temp else s (i) = zero end if end do sredg = - temp ! ! By considering the simple bounds on the variables, calculate an upper ! bound on the tangent of half the angle of the alternative iteration, ! namely ANGBD, except that, if already a free variable has reached a ! bound, there is a branch back to label 100 after fixing that variable. ! angbd = one iact = 0 do i = 1, n if (xbdi(i) == zero) then tempa = xopt (i) + d (i) - sl (i) tempb = su (i) - xopt (i) - d (i) if (tempa <= zero) then nact = nact + 1 xbdi (i) = onemin go to 100 else if (tempb <= zero) then nact = nact + 1 xbdi (i) = one go to 100 end if ratio = one ssq = d (i) ** 2 + s (i) ** 2 temp = ssq - (xopt(i)-sl(i)) ** 2 if (temp > zero) then temp = sqrt (temp) - s (i) if (angbd*temp > tempa) then angbd = tempa / temp iact = i xsav = onemin end if end if temp = ssq - (su(i)-xopt(i)) ** 2 if (temp > zero) then temp = sqrt (temp) + s (i) if (angbd*temp > tempb) then angbd = tempb / temp iact = i xsav = one end if end if end if end do ! ! Calculate HHD and some curvatures for the alternative iteration. ! go to 210 150 shs = zero dhs = zero dhd = zero do i = 1, n if (xbdi(i) == zero) then shs = shs + s (i) * hs (i) dhs = dhs + d (i) * hs (i) dhd = dhd + d (i) * hred (i) end if end do ! ! Seek the greatest reduction in Q for a range of equally spaced values ! of ANGT in [0,ANGBD], where ANGT is the tangent of half the angle of ! the alternative iteration. ! redmax = zero isav = 0 redsav = zero iu = 17.0_wp * angbd + 3.1_wp do i = 1, iu angt = angbd * real (i, wp) / real (iu, wp) sth = (angt+angt) / (one+angt*angt) temp = shs + angt * (angt*dhd-dhs-dhs) rednew = sth * (angt*dredg-sredg-half*sth*temp) if (rednew > redmax) then redmax = rednew isav = i rdprev = redsav else if (i == isav+1) then rdnext = rednew end if redsav = rednew end do ! ! Return if the reduction is zero. Otherwise, set the sine and cosine ! of the angle of the alternative iteration, and calculate SDEC. ! if (isav == 0) go to 190 if (isav < iu) then temp = (rdnext-rdprev) / (redmax+redmax-rdprev-rdnext) angt = angbd * (real(isav, wp)+half*temp) / real (iu, wp) end if cth = (one-angt*angt) / (one+angt*angt) sth = (angt+angt) / (one+angt*angt) temp = shs + angt * (angt*dhd-dhs-dhs) sdec = sth * (angt*dredg-sredg-half*sth*temp) if (sdec <= zero) go to 190 ! ! Update GNEW, D and HRED. If the angle of the alternative iteration ! is restricted by a bound on a free variable, that variable is fixed ! at the bound. ! dredg = zero gredsq = zero do i = 1, n gnew (i) = gnew (i) + (cth-one) * hred (i) + sth * hs (i) if (xbdi(i) == zero) then d (i) = cth * d (i) + sth * s (i) dredg = dredg + d (i) * gnew (i) gredsq = gredsq + gnew (i) ** 2 end if hred (i) = cth * hred (i) + sth * hs (i) end do qred = qred + sdec if (iact > 0 .and. isav == iu) then nact = nact + 1 xbdi (iact) = xsav go to 100 end if ! ! If SDEC is sufficiently small, then RETURN after setting XNEW to ! XOPT+D, giving careful attention to the bounds. ! if (sdec > 0.01_wp*qred) go to 120 190 dsq = zero do i = 1, n xnew (i) = max (min(xopt(i)+d(i), su(i)), sl(i)) if (xbdi(i) == onemin) xnew (i) = sl (i) if (xbdi(i) == one) xnew (i) = su (i) d (i) = xnew (i) - xopt (i) dsq = dsq + d (i) ** 2 end do return ! ! The following instructions multiply the current S-vector by the second ! derivative matrix of the quadratic model, putting the product in HS. ! They are reached from three different parts of the software above and ! they can be regarded as an external subroutine. ! 210 ih = 0 do j = 1, n hs (j) = zero do i = 1, j ih = ih + 1 if (i < j) hs (j) = hs (j) + hq (ih) * s (i) hs (i) = hs (i) + hq (ih) * s (j) end do end do do k = 1, npt if (pq(k) /= zero) then temp = zero do j = 1, n temp = temp + xpt (k, j) * s (j) end do temp = temp * pq (k) do i = 1, n hs (i) = hs (i) + temp * xpt (k, i) end do end if end do if (crvmin /= zero) go to 50 if (iterc > itcsav) go to 150 do i = 1, n hred (i) = hs (i) end do go to 120 end subroutine trsbox subroutine update (n, npt, bmat, zmat, ndim, vlag, beta, denom, knew, w) implicit real (wp) (a-h, o-z) dimension bmat (ndim,*), zmat (npt,*), vlag (*), w (*) ! ! The arrays BMAT and ZMAT are updated, as required by the new position ! of the interpolation point that has the index KNEW. The vector VLAG has ! N+NPT components, set on entry to the first NPT and last N components ! of the product Hw in equation (4.11) of the Powell (2006) paper on ! NEWUOA. Further, BETA is set on entry to the value of the parameter ! with that name, and DENOM is set to the denominator of the updating ! formula. Elements of ZMAT may be treated as zero if their moduli are ! at most ZTEST. The first NDIM elements of W are used for working space. ! ! Set some constants. ! one = 1.0_wp zero = 0.0_wp nptm = npt - n - 1 ztest = zero do k = 1, npt do j = 1, nptm ztest = max (ztest, abs(zmat(k, j))) end do end do ztest = 1.0e-20_wp * ztest ! ! Apply the rotations that put zeros in the KNEW-th row of ZMAT. ! jl = 1 do j = 2, nptm if (abs(zmat(knew, j)) > ztest) then temp = sqrt (zmat(knew, 1)**2+zmat(knew, j)**2) tempa = zmat (knew, 1) / temp tempb = zmat (knew, j) / temp do i = 1, npt temp = tempa * zmat (i, 1) + tempb * zmat (i, j) zmat (i, j) = tempa * zmat (i, j) - tempb * zmat (i, 1) zmat (i, 1) = temp end do end if zmat (knew, j) = zero end do ! ! Put the first NPT components of the KNEW-th column of HLAG into W, ! and calculate the parameters of the updating formula. ! do i = 1, npt w (i) = zmat (knew, 1) * zmat (i, 1) end do alpha = w (knew) tau = vlag (knew) vlag (knew) = vlag (knew) - one ! ! Complete the updating of ZMAT. ! temp = sqrt (denom) tempb = zmat (knew, 1) / temp tempa = tau / temp do i = 1, npt zmat (i, 1) = tempa * zmat (i, 1) - tempb * vlag (i) end do ! ! Finally, update the matrix BMAT. ! do j = 1, n jp = npt + j w (jp) = bmat (knew, j) tempa = (alpha*vlag(jp)-tau*w(jp)) / denom tempb = (-beta*w(jp)-tau*vlag(jp)) / denom do i = 1, jp bmat (i, j) = bmat (i, j) + tempa * vlag (i) + tempb * w (i) if (i > npt) bmat (jp, i-npt) = bmat (i, j) end do end do end subroutine update !***************************************************************************************** !> ! Test problem for [[bobyqa]], the objective function being the sum of ! the reciprocals of all pairwise distances between the points P_I, ! I=1,2,...,M in two dimensions, where M=N/2 and where the components ! of P_I are X(2*I-1) and X(2*I). Thus each vector X of N variables ! defines the M points P_I. The initial X gives equally spaced points ! on a circle. Four different choices of the pairs (N,NPT) are tried, ! namely (10,16), (10,21), (20,26) and (20,41). Convergence to a local ! minimum that is not global occurs in both the N=10 cases. The details ! of the results are highly sensitive to computer rounding errors. The ! choice IPRINT=2 provides the current X and optimal F so far whenever ! RHO is reduced. The bound constraints of the problem require every ! component of X to be in the interval [-1,1]. subroutine bobyqa_test() implicit none real(wp),dimension(100) :: x, xl, xu integer :: i,j,m,n,jcase,npt real(wp) :: temp real(wp),parameter :: twopi = 8.0_wp * atan (1.0_wp) real(wp),parameter :: bdl = - 1.0_wp real(wp),parameter :: bdu = 1.0_wp integer,parameter :: iprint = 2 integer,parameter :: maxfun = 500000 real(wp),parameter :: rhobeg = 1.0e-1_wp real(wp),parameter :: rhoend = 1.0e-6_wp m = 5 do n = 2 * m do i = 1, n xl (i) = bdl xu (i) = bdu end do do jcase = 1, 2 npt = n + 6 if (jcase == 2) npt = 2 * n + 1 print 30, m, n, npt 30 format (/ / 5 x, '2D output with M =', i4, ', N =', i4, ' and NPT =', i4) do j = 1, m temp = real (j, wp) * twopi / real (m, wp) x (2*j-1) = cos (temp) x (2*j) = sin (temp) end do call bobyqa (n, npt, x, xl, xu, rhobeg, rhoend, iprint, maxfun, calfun) end do m = m + m if (m > 10) exit end do contains subroutine calfun (n, x, f) implicit none integer,intent(in) :: n real(wp),dimension(:),intent(in) :: x real(wp),intent(out) :: f integer :: i,j real(wp) :: temp f = 0.0_wp do i = 4, n, 2 do j = 2, i - 2, 2 temp = (x(i-1)-x(j-1)) ** 2 + (x(i)-x(j)) ** 2 temp = max (temp, 1.0e-6_wp) f = f + 1.0_wp / sqrt (temp) end do end do end subroutine calfun end subroutine bobyqa_test !***************************************************************************************** !***************************************************************************************** end module bobyqa_module !*****************************************************************************************