c c Copyright (C) 1995-2010 Berwin A. Turlach <berwin@maths.uwa.edu.au> c c This program is free software; you can redistribute it and/or modify c it under the terms of the GNU General Public License as published by c the Free Software Foundation; either version 2 of the License, or c (at your option) any later version. c c This program is distributed in the hope that it will be useful, c but WITHOUT ANY WARRANTY; without even the implied warranty of c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the c GNU General Public License for more details. c c You should have received a copy of the GNU General Public License c along with this program; if not, write to the Free Software c Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, c USA. c c this routine uses the Goldfarb/Idnani algorithm to solve the c following minimization problem: c c minimize -d^T x + 1/2 * x^T D x c where A1^T x = b1 c A2^T x >= b2 c c the matrix D is assumed to be positive definite. Especially, c w.l.o.g. D is assumed to be symmetric. c c Input parameter: c dmat nxn matrix, the matrix D from above (dp) c *** WILL BE DESTROYED ON EXIT *** c The user has two possibilities: c a) Give D (ierr=0), in this case we use routines from LINPACK c to decompose D. c b) To get the algorithm started we need R^-1, where D=R^TR. c So if it is cheaper to calculate R^-1 in another way (D may c be a band matrix) then with the general routine, the user c may pass R^{-1}. Indicated by ierr not equal to zero. c dvec nx1 vector, the vector d from above (dp) c *** WILL BE DESTROYED ON EXIT *** c contains on exit the solution to the initial, i.e., c unconstrained problem c fddmat scalar, the leading dimension of the matrix dmat c n the dimension of dmat and dvec (int) c amat nxq matrix, the matrix A from above (dp) [ A=(A1 A2)^T ] c *** ENTRIES CORRESPONDING TO EQUALITY CONSTRAINTS MAY HAVE c CHANGED SIGNES ON EXIT *** c bvec qx1 vector, the vector of constants b in the constraints (dp) c [ b = (b1^T b2^T)^T ] c *** ENTRIES CORRESPONDING TO EQUALITY CONSTRAINTS MAY HAVE c CHANGED SIGNES ON EXIT *** c fdamat the first dimension of amat as declared in the calling program. c fdamat >= n !! c q integer, the number of constraints. c meq integer, the number of equality constraints, 0 <= meq <= q. c ierr integer, code for the status of the matrix D: c ierr = 0, we have to decompose D c ierr != 0, D is already decomposed into D=R^TR and we were c given R^{-1}. c c Output parameter: c sol nx1 the final solution (x in the notation above) c lagr qx1 the final Lagrange multipliers c crval scalar, the value of the criterion at the minimum c iact qx1 vector, the constraints which are active in the final c fit (int) c nact scalar, the number of constraints active in the final fit (int) c iter 2x1 vector, first component gives the number of "main" c iterations, the second one says how many constraints were c deleted after they became active c ierr integer, error code on exit, if c ierr = 0, no problems c ierr = 1, the minimization problem has no solution c ierr = 2, problems with decomposing D, in this case sol c contains garbage!! c c Working space: c work vector with length at least 2*n+r*(r+5)/2 + 2*q +1 c where r=min(n,q) c subroutine qpgen2(dmat, dvec, fddmat, n, sol, lagr, crval, amat, * bvec, fdamat, q, meq, iact, nact, iter, work, ierr) implicit none integer n, i, j, l, l1, * info, q, iact(*), iter(*), it1, * ierr, nact, iwzv, iwrv, iwrm, iwsv, iwuv, nvl, * r, fdamat, iwnbv, meq, fddmat double precision dmat(fddmat,*), dvec(*), lagr(*), sol(*), bvec(*) $ ,work(*), temp, sum, t1, tt, gc, gs, crval,nu, amat(fdamat,*) $ , vsmall, tmpa, tmpb logical t1inf, t2min r = min(n,q) l = 2*n + (r*(r+5))/2 + 2*q + 1 c c code gleaned from Powell's ZQPCVX routine to determine a small c number that can be assumed to be an upper bound on the relative c precision of the computer arithmetic. c vsmall = 1.0d-60 1 vsmall = vsmall + vsmall tmpa = 1.0d0 + 0.1d0*vsmall tmpb = 1.0d0 + 0.2d0*vsmall if( tmpa .LE. 1.0d0 ) goto 1 if( tmpb .LE. 1.0d0 ) goto 1 c c store the initial dvec to calculate below the unconstrained minima of c the critical value. c do 10 i=1,n work(i) = dvec(i) 10 continue do 11 i=n+1,l work(i) = 0.d0 11 continue do 12 i=1,q iact(i) = 0 lagr(i) = 0.d0 12 continue c c get the initial solution c if( ierr .EQ. 0 )then call dpofa(dmat,fddmat,n,info) if( info .NE. 0 )then ierr = 2 goto 999 endif call dposl(dmat,fddmat,n,dvec) call dpori(dmat,fddmat,n) else c c Matrix D is already factorized, so we have to multiply d first with c R^-T and then with R^-1. R^-1 is stored in the upper half of the c array dmat. c do 20 j=1,n sol(j) = 0.d0 do 21 i=1,j sol(j) = sol(j) + dmat(i,j)*dvec(i) 21 continue 20 continue do 22 j=1,n dvec(j) = 0.d0 do 23 i=j,n dvec(j) = dvec(j) + dmat(j,i)*sol(i) 23 continue 22 continue endif c c set lower triangular of dmat to zero, store dvec in sol and c calculate value of the criterion at unconstrained minima c crval = 0.d0 do 30 j=1,n sol(j) = dvec(j) crval = crval + work(j)*sol(j) work(j) = 0.d0 do 32 i=j+1,n dmat(i,j) = 0.d0 32 continue 30 continue crval = -crval/2.d0 ierr = 0 c c calculate some constants, i.e., from which index on the different c quantities are stored in the work matrix c iwzv = n iwrv = iwzv + n iwuv = iwrv + r iwrm = iwuv + r+1 iwsv = iwrm + (r*(r+1))/2 iwnbv = iwsv + q c c calculate the norm of each column of the A matrix c do 51 i=1,q sum = 0.d0 do 52 j=1,n sum = sum + amat(j,i)*amat(j,i) 52 continue work(iwnbv+i) = sqrt(sum) 51 continue nact = 0 iter(1) = 0 iter(2) = 0 50 continue c c start a new iteration c iter(1) = iter(1)+1 c c calculate all constraints and check which are still violated c for the equality constraints we have to check whether the normal c vector has to be negated (as well as bvec in that case) c l = iwsv do 60 i=1,q l = l+1 sum = -bvec(i) do 61 j = 1,n sum = sum + amat(j,i)*sol(j) 61 continue if ( abs(sum) .LT. vsmall ) then sum = 0.0d0 endif if (i .GT. meq) then work(l) = sum else work(l) = -abs(sum) if (sum .GT. 0.d0) then do 62 j=1,n amat(j,i) = -amat(j,i) 62 continue bvec(i) = -bvec(i) endif endif 60 continue c c as safeguard against rounding errors set already active constraints c explicitly to zero c do 70 i=1,nact work(iwsv+iact(i)) = 0.d0 70 continue c c we weight each violation by the number of non-zero elements in the c corresponding row of A. then we choose the violated constraint which c has maximal absolute value, i.e., the minimum. c by obvious commenting and uncommenting we can choose the strategy to c take always the first constraint which is violated. ;-) c nvl = 0 temp = 0.d0 do 71 i=1,q if (work(iwsv+i) .LT. temp*work(iwnbv+i)) then nvl = i temp = work(iwsv+i)/work(iwnbv+i) endif c if (work(iwsv+i) .LT. 0.d0) then c nvl = i c goto 72 c endif 71 continue 72 if (nvl .EQ. 0) then do 73 i=1,nact lagr(iact(i))=work(iwuv+i) 73 continue goto 999 endif c c calculate d=J^Tn^+ where n^+ is the normal vector of the violated c constraint. J is stored in dmat in this implementation!! c if we drop a constraint, we have to jump back here. c 55 continue do 80 i=1,n sum = 0.d0 do 81 j=1,n sum = sum + dmat(j,i)*amat(j,nvl) 81 continue work(i) = sum 80 continue c c Now calculate z = J_2 d_2 c l1 = iwzv do 90 i=1,n work(l1+i) =0.d0 90 continue do 92 j=nact+1,n do 93 i=1,n work(l1+i) = work(l1+i) + dmat(i,j)*work(j) 93 continue 92 continue c c and r = R^{-1} d_1, check also if r has positive elements (among the c entries corresponding to inequalities constraints). c t1inf = .TRUE. do 95 i=nact,1,-1 sum = work(i) l = iwrm+(i*(i+3))/2 l1 = l-i do 96 j=i+1,nact sum = sum - work(l)*work(iwrv+j) l = l+j 96 continue sum = sum / work(l1) work(iwrv+i) = sum if (iact(i) .LE. meq) goto 95 if (sum .LE. 0.d0) goto 95 7 t1inf = .FALSE. it1 = i 95 continue c c if r has positive elements, find the partial step length t1, which is c the maximum step in dual space without violating dual feasibility. c it1 stores in which component t1, the min of u/r, occurs. c if ( .NOT. t1inf) then t1 = work(iwuv+it1)/work(iwrv+it1) do 100 i=1,nact if (iact(i) .LE. meq) goto 100 if (work(iwrv+i) .LE. 0.d0) goto 100 temp = work(iwuv+i)/work(iwrv+i) if (temp .LT. t1) then t1 = temp it1 = i endif 100 continue endif c c test if the z vector is equal to zero c sum = 0.d0 do 110 i=iwzv+1,iwzv+n sum = sum + work(i)*work(i) 110 continue if (abs(sum) .LE. vsmall) then c c No step in primal space such that the new constraint becomes c feasible. Take step in dual space and drop a constant. c if (t1inf) then c c No step in dual space possible either, problem is not solvable c ierr = 1 goto 999 else c c we take a partial step in dual space and drop constraint it1, c that is, we drop the it1-th active constraint. c then we continue at step 2(a) (marked by label 55) c do 111 i=1,nact work(iwuv+i) = work(iwuv+i) - t1*work(iwrv+i) 111 continue work(iwuv+nact+1) = work(iwuv+nact+1) + t1 goto 700 endif else c c compute full step length t2, minimum step in primal space such that c the constraint becomes feasible. c keep sum (which is z^Tn^+) to update crval below! c sum = 0.d0 do 120 i = 1,n sum = sum + work(iwzv+i)*amat(i,nvl) 120 continue tt = -work(iwsv+nvl)/sum t2min = .TRUE. if (.NOT. t1inf) then if (t1 .LT. tt) then tt = t1 t2min = .FALSE. endif endif c c take step in primal and dual space c do 130 i=1,n sol(i) = sol(i) + tt*work(iwzv+i) 130 continue crval = crval + tt*sum*(tt/2.d0 + work(iwuv+nact+1)) do 131 i=1,nact work(iwuv+i) = work(iwuv+i) - tt*work(iwrv+i) 131 continue work(iwuv+nact+1) = work(iwuv+nact+1) + tt c c if it was a full step, then we check wheter further constraints are c violated otherwise we can drop the current constraint and iterate once c more if(t2min) then c c we took a full step. Thus add constraint nvl to the list of active c constraints and update J and R c nact = nact + 1 iact(nact) = nvl c c to update R we have to put the first nact-1 components of the d vector c into column (nact) of R c l = iwrm + ((nact-1)*nact)/2 + 1 do 150 i=1,nact-1 work(l) = work(i) l = l+1 150 continue c c if now nact=n, then we just have to add the last element to the new c row of R. c Otherwise we use Givens transformations to turn the vector d(nact:n) c into a multiple of the first unit vector. That multiple goes into the c last element of the new row of R and J is accordingly updated by the c Givens transformations. c if (nact .EQ. n) then work(l) = work(n) else do 160 i=n,nact+1,-1 c c we have to find the Givens rotation which will reduce the element c (l1) of d to zero. c if it is already zero we don't have to do anything, except of c decreasing l1 c if (work(i) .EQ. 0.d0) goto 160 gc = max(abs(work(i-1)),abs(work(i))) gs = min(abs(work(i-1)),abs(work(i))) temp = sign(gc*sqrt(1+gs*gs/(gc*gc)), work(i-1)) gc = work(i-1)/temp gs = work(i)/temp c c The Givens rotation is done with the matrix (gc gs, gs -gc). c If gc is one, then element (i) of d is zero compared with element c (l1-1). Hence we don't have to do anything. c If gc is zero, then we just have to switch column (i) and column (i-1) c of J. Since we only switch columns in J, we have to be careful how we c update d depending on the sign of gs. c Otherwise we have to apply the Givens rotation to these columns. c The i-1 element of d has to be updated to temp. c if (gc .EQ. 1.d0) goto 160 if (gc .EQ. 0.d0) then work(i-1) = gs * temp do 170 j=1,n temp = dmat(j,i-1) dmat(j,i-1) = dmat(j,i) dmat(j,i) = temp 170 continue else work(i-1) = temp nu = gs/(1.d0+gc) do 180 j=1,n temp = gc*dmat(j,i-1) + gs*dmat(j,i) dmat(j,i) = nu*(dmat(j,i-1)+temp) - dmat(j,i) dmat(j,i-1) = temp 180 continue endif 160 continue c c l is still pointing to element (nact,nact) of the matrix R. c So store d(nact) in R(nact,nact) work(l) = work(nact) endif else c c we took a partial step in dual space. Thus drop constraint it1, c that is, we drop the it1-th active constraint. c then we continue at step 2(a) (marked by label 55) c but since the fit changed, we have to recalculate now "how much" c the fit violates the chosen constraint now. c sum = -bvec(nvl) do 190 j = 1,n sum = sum + sol(j)*amat(j,nvl) 190 continue if( nvl .GT. meq ) then work(iwsv+nvl) = sum else work(iwsv+nvl) = -abs(sum) if( sum .GT. 0.d0) then do 191 j=1,n amat(j,nvl) = -amat(j,nvl) 191 continue bvec(nvl) = -bvec(nvl) endif endif goto 700 endif endif goto 50 c c Drop constraint it1 c 700 continue c c if it1 = nact it is only necessary to update the vector u and nact c if (it1 .EQ. nact) goto 799 c c After updating one row of R (column of J) we will also come back here c 797 continue c c we have to find the Givens rotation which will reduce the element c (it1+1,it1+1) of R to zero. c if it is already zero we don't have to do anything except of updating c u, iact, and shifting column (it1+1) of R to column (it1) c l will point to element (1,it1+1) of R c l1 will point to element (it1+1,it1+1) of R c l = iwrm + (it1*(it1+1))/2 + 1 l1 = l+it1 if (work(l1) .EQ. 0.d0) goto 798 gc = max(abs(work(l1-1)),abs(work(l1))) gs = min(abs(work(l1-1)),abs(work(l1))) temp = sign(gc*sqrt(1+gs*gs/(gc*gc)), work(l1-1)) gc = work(l1-1)/temp gs = work(l1)/temp c c The Givens rotatin is done with the matrix (gc gs, gs -gc). c If gc is one, then element (it1+1,it1+1) of R is zero compared with c element (it1,it1+1). Hence we don't have to do anything. c if gc is zero, then we just have to switch row (it1) and row (it1+1) c of R and column (it1) and column (it1+1) of J. Since we swithc rows in c R and columns in J, we can ignore the sign of gs. c Otherwise we have to apply the Givens rotation to these rows/columns. c if (gc .EQ. 1.d0) goto 798 if (gc .EQ. 0.d0) then do 710 i=it1+1,nact temp = work(l1-1) work(l1-1) = work(l1) work(l1) = temp l1 = l1+i 710 continue do 711 i=1,n temp = dmat(i,it1) dmat(i,it1) = dmat(i,it1+1) dmat(i,it1+1) = temp 711 continue else nu = gs/(1.d0+gc) do 720 i=it1+1,nact temp = gc*work(l1-1) + gs*work(l1) work(l1) = nu*(work(l1-1)+temp) - work(l1) work(l1-1) = temp l1 = l1+i 720 continue do 721 i=1,n temp = gc*dmat(i,it1) + gs*dmat(i,it1+1) dmat(i,it1+1) = nu*(dmat(i,it1)+temp) - dmat(i,it1+1) dmat(i,it1) = temp 721 continue endif c c shift column (it1+1) of R to column (it1) (that is, the first it1 c elements). The posit1on of element (1,it1+1) of R was calculated above c and stored in l. c 798 continue l1 = l-it1 do 730 i=1,it1 work(l1)=work(l) l = l+1 l1 = l1+1 730 continue c c update vector u and iact as necessary c Continue with updating the matrices J and R c work(iwuv+it1) = work(iwuv+it1+1) iact(it1) = iact(it1+1) it1 = it1+1 if (it1 .LT. nact) goto 797 799 work(iwuv+nact) = work(iwuv+nact+1) work(iwuv+nact+1) = 0.d0 iact(nact) = 0 nact = nact-1 iter(2) = iter(2)+1 goto 55 999 continue return end

Generated by Doxygen 1.6.0 Back to index