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qlcpd.f
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qlcpd.f
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christen this file qlcpd.f
cut here >>>>>>>>>>>>>>>>>
c Copyright (C) 2010 Roger Fletcher
c Current version dated 18 April 2013
c THE ACCOMPANYING PROGRAM IS PROVIDED UNDER THE TERMS OF THE ECLIPSE PUBLIC
c LICENSE ("AGREEMENT"). ANY USE, REPRODUCTION OR DISTRIBUTION OF THE PROGRAM
c CONSTITUTES RECIPIENT'S ACCEPTANCE OF THIS AGREEMENT
subroutine qlcpd(n,m,k,kmax,maxg,a,la,x,bl,bu,f,fmin,g,r,w,e,ls,
* alp,lp,mlp,peq,ws,lws,v,nv,linear,rgtol,m0de,ifail,mxgr,iprint,
* nout)
implicit double precision (a-h,r-z), integer (i-q)
logical linear
c This routine finds a KT point for the Quadratic LCP (QP) problem
c minimize f(x) = ct.x + xt.G.x/2
c subject to l <= [I : A]t.x <= u (t = transpose)
c where x and c are n-vectors, G is a symmetric n*n matrix, and A is an
c n*m matrix. If G is also positive semi-definite then the KT point is a
c global solution, else usually a local solution. The method may also be
c used efficiently to solve an LP problem (G=0). A recursive form of an
c active set method is used, using Wolfe's method to resolve degeneracy.
c A limited memory sweep method is used for minimization in the null space.
c Matrix information is made available and processed by calls to external
c subroutines. Details of these are given in an auxiliary file named either
c 'denseL.f' or 'schurQR.f'. (schurQR.f is a more recent replacement for
c the file sparseL.f)
c parameter list (variables in a line starting with C must be set on entry)
c **************
C n number of variables
C m number of general constraints (columns of A)
c k dimension of the null space obtained by eliminating the active
c constraints (only to be set if mode>=2). The number of constraints in
c the active set is n-k
C kmax maximum value of k (kmax <= n)
C maxg max number of reduced gradient vectors stored in sweep method:
C (1 < maxg <= kmax+1), typically maxg = min(6,kmax+1)
C a(*) storage of reals associated with c and A. This storage may be provided
C in either dense or sparse format. Refer to either denseA.f or sparseA.f
C for information on how to set a(*) and la(*).
C la(*) storage of integers associated with c and A
C x(n) contains the vector of variables. Initially an estimate of the solution
C must be set, replaced by the solution (if it exists) on exit.
C bl(n+m) vector of lower bounds for variables and general constraints
C bu(n+m) vector of upper bounds (use numbers less than about 1.e30, and
C where possible supply realistic bounds on the x variables)
c f returns the value of f(x) when x is a feasible solution
c Otherwise f stores the sum of constraint infeasibilities
C fmin set a strict lower bound on f(x) (used to identify an unbounded LCP)
c g(n) returns the gradient vector of f(x) when x is feasible
c r(n+m) workspace: stores constraint residuals (or multipliers if the
c constraint is active). The sign convention is such that these are
c nonnegative at a solution (except multipliers of equality constraints)
c w(n+m) workspace: stores denominators for ratio tests
c e(n+m) stores steepest-edge normalization coefficients: if mode>2 then
c information in this vector from a previous call should not be changed.
c (In mode 3 these values provide approximate coefficients)
c ls(n+m) stores indices of the active constraints in locations 1:n and of
c the inactive constraints in locations n+1:n+m. The simple bounds
c on the variables are indexed by 1:n and the general constraints by
c n+1:n+m. The sign of ls(j) indicates whether the lower bound (+) or
c the upper bound (-) of constraint ls(j) is currently significant.
c Within the set of active constraints, locations 1:peq store the indices
c of any equality constraints, locations peq+1:n-k store the indices of
c any inequality constraints, and locations n-k+1 store the indices of
c any free variables (variables not on a bound, which are used to
c parametrise the null space: ls(j) is always positive in this range)
c If mode>=2, the first n-k elements of ls must be set on entry
c alp(mlp) workspace associated with recursion
c lp(mlp) list of pointers to recursion information in ls
C mlp maximum number of levels of recursion allowed (mlp>2: typically
C mlp=20 would usually be adequate but mlp=m is an upper bound)
c peq pointer to the end of equality constraint indices in ls
c ws(*) real workspace for gdotx (see below), qlcpd and denseL.f (or schurQR.f)
c Set the total number in mxws (see "Common" below).
c lws(*) integer workspace for gdotx, qlcpd and denseL.f (or schurQR.f).
c Set the total number in mxlws (see "Common" below).
c The storage maps for ws and lws are set by the routine stmapq below
C v(maxg) set nv estimates of the eigenvalues of the reduced Hessian of f(x)
C (for example from a previous run of qlcpd). Set nv=1 and v(1)=1.D0
C in absence of other information. New values of v are left on exit
C nv Number of estimates in v
C linear (logical) set linear = .true. iff the problem is an LP
C rgtol required accuracy in the reduced gradient l2 norm: it is advisable not
C to seek too high accuracy - rgtol may be increased by the code if it
c is deemed to be too small, see the definition of sgnf below
C m0de mode of operation (larger numbers imply extra information):
C 0 = cold start (no other information available, takes simple
C bounds for the initial active set)
C 1 = as 0 but includes all equality constraints in initial active set
C 2 = user sets n-k active constraint indices in ls(j), j=1,..,n-k.
c For a general constraint the sign of ls(j) indicates which
c bound to use. For a simple bound the current value of x is used
C 3 = takes active set and other information from a previous call.
C Steepest edge weights are approximated using previous values.
C 4 = as 3 but it is also assumed that A is unchanged so
C that factors of the basis matrix stored in ws and lws are valid
C (changes in the vectors c, l, u and the matrix G are allowed)
C A local copy (mode) of m0de is made and may be changed by qlcpd
c ifail outcome of the process
c 0 = solution obtained
c 1 = unbounded problem detected (f(x)<=fmin would occur)
c 2 = bl(i) > bu(i) for some i
c 3 = infeasible problem detected in Phase 1
c 4 = line search cannot improve f (possibly increase rgtol)
c 5 = mxgr gradient calls exceeded (this test is only carried
c out at the start of each iteration)
c 6 = incorrect setting of m, n, kmax, maxg, mlp, mode or tol
c 7 = not enough space in ws or lws
c 8 = not enough space in lp (increase mlp)
c 9 = dimension of reduced space too large (increase kmax)
c 10 = maximum number of unsuccessful restarts taken
c > 10 = crash in pivot call
C mxgr maximum number of gradient evaluations
C iprint switch for diagnostic printing (0 = off, 1 = summary,
C 2 = scalar information, 3 = verbose)
C nout channel number for output
c Common
c ******
c User information about the lengths of ws and lws is supplied to qlcpd in
c common/wsc/kk,ll,kkk,lll,mxws,mxlws
c kk and ll refer to the length of ws and lws needed by gdotx.
c kkk and lll are the numbers of locations used by qlcpd and are set by qlcpd.
c the rest of ws and lws is used by the files denseL.f or schurQR.f
c mxws and mxlws must be set to the total length of ws and lws available: a
c message will be given if more storage is needed.
c User subroutine
c ***************
c The user must provide a subroutine to calculate the vector v := G.x from a
c given vector x. The header of the routine is
c subroutine gdotx(n,x,ws,lws,v)
c implicit double precision(a-h,o-z)
c dimension x(*),ws(*),lws(*),v(*)
c In the case that linear is .true. the subroutine is not called by qlcpd and
c a dummy routine may be supplied. Otherwise the user may use the parameters
c ws and lws (see above) for passing real or integer arrays relating to G.
c Locations ws(1),...,ws(kk) are available to the user for storing any real
c information to be used by gdotx. Likewise locations lws(1),...,lws(ll) are
c available for storing any integer information. Default values are kk=ll=0.
c Any other setting is made by changing common/wsc/kk,ll,kkk,lll,mxws,mxlws
c Tolerances and accuracy
c ***********************
c qlcpd uses tolerance and accuracy information stored in
c common/epsc/eps,tol,emin
c common/repc/sgnf,nrep,npiv,nres
c common/refactorc/mc,mxmc
c common/infoc/rgnorm,vstep,iter,npv,ngr
c eps must be set to the machine precision (unit round-off) and tol is a
c tolerance such that numbers whose absolute value is less than tol are
c truncated to zero. This tolerance strategy in the code assumes that the
c problem is well-scaled and a pre-scaling routine cscale is supplied in
c denseA.f or sparseA.f. The parameter sgnf is used to measure the maximum
c allowable relative error in gradient values. If at any stage the accuracy
c requirement rgtol < sgnf*rgnorm then rgtol is increased to sgnf*rgnorm
c The code allows one or more refinement steps after the
c calculation has terminated, to improve the accuracy of the solution,
c and a fixed number nrep of such repeats is allowed. However the code
c terminates without further repeats if no more than npiv pivots are taken.
c In case of any breakdown, the code is restarted, usually in mode 2.
c The maximum number of unsuccessful restarts allowed is set in nres.
c The basis matrix may be refactorised on occasions, for example to prevent
c build-up of round-off in the factors or (when using schurQR.f) to limit
c the growth in the Schur complement. The maximum interval between
c refactorizations (or size of Schur complement) is set in mxmc.
c Default values are set in block data but can be reset by the user.
c infoc returns information about the progress of the method: rgnorm is the
c norm of the reduced gradient on exit, and vstep is the length of the vertical
c step in the warm start process. iter is the total number of iterations taken,
c npv is the number of pivots, and ngr is the number of calls of gdotx.
parameter (ainfty=1.D100)
dimension a(*),x(*),bl(*),bu(*),g(*),r(*),w(*),e(*),alp(*),ws(*),
* la(*),ls(*),lp(*),lws(*),v(*)
character*32 spaces
common/lcpdc/na,na1,nb,nb1,krg,krg1,kr,kr1,
* ka,ka1,kb,kb1,kc,kc1,kd,kd1,ke,ke1,lu1,ll1
common/epsc/eps,t0l,emin
common/infoc/rgnorm,vstep,iter,npv,ngr
common/repc/sgnf,nrep,npiv,nres
common/wsc/kk,ll,kkk,lll,mxws,mxlws
common/refactorc/mc,mxmc
common/alphac/alpha,rp,pj,qqj,qqj1
logical plus
1 format(A,15I5)
2 format(A,6E15.7)
3 format(A/(15I5))
4 format(A/(5E15.7))
5 format((6E15.7))
6 format(A,I5,2E15.7)
spaces=' '
mode=m0de
tol=t0l
iter=0
npv=0
if(m.lt.0.or.n.le.0.or.mlp.lt.2.or.mode.lt.0.or.mode.gt.4.or.
* kmax.lt.0.or.(kmax.gt.0.and.maxg.le.1).or.tol.le.D0)then
ifail=6
return
endif
rgt0l=rgtol
n1=n+1
nm=n+m
nmi=nm
ngr=0
nv0=nv
if(iprint.ge.3)then
write(nout,1000)'lower bounds',(bl(i),i=1,nm)
write(nout,1000)'upper bounds',(bu(i),i=1,nm)
endif
irep=0
ires=0
mres=0
bestf=ainfty
do i=1,nm
t=bu(i)-bl(i)
if(t.lt.-tol)then
ifail=2
return
endif
if(t.le.tol.and.t.gt.0.D0)then
bl(i)=5.D-1*(bl(i)+bu(i))
bu(i)=bl(i)
endif
enddo
vmax=0.D0
do i=1,n
x(i)=min(bu(i),max(bl(i),x(i)))
vmax=max(vmax,bu(i)-bl(i))
enddo
if(mode.le.2)then
call stmapq(n,nm,kmax,maxg)
if(mode.eq.0)then
nk=0
elseif(mode.eq.1)then
c collect equality c/s
nk=0
do i=1,nm
if(bu(i).eq.bl(i))then
nk=nk+1
ls(nk)=i
endif
enddo
c write(nout,*)'number of eqty c/s =',nk
else
nk=n-k
endif
endif
c restarts loop
7 continue
lp(1)=nm
lev=1
if(mode.le.3)then
c set up factors of basis matrix and permutation vectors
ifail=mode
call start_up(n,nm,nmi,a,la,nk,e,ls,ws(lu1),lws(ll1),mode,ifail)
if(ifail.gt.0)return
endif
8 continue
peq=0
ig=0
c refinement step loop
mpiv=iter+npiv
ninf=0
do i=1,n
g(i)=0.D0
enddo
if(mode.gt.0)then
call warm_start(n,nm,a,la,x,bl,bu,r,ls,ws(lu1),
* lws(ll1),ws(na1),vstep)
c print *,'vstep,vmax',vstep,vmax
if(vstep.gt.2.D0*vmax)then
mpiv=0
mode=0
nk=0
do i=1,n
x(i)=min(bu(i),max(bl(i),x(i)))
enddo
goto7
endif
if(vstep.gt.tol)mpiv=0
endif
k=0
c collect free variables
do j=n,1,-1
i=abs(ls(j))
if(i.le.n.and.x(i).gt.bl(i).and.x(i).lt.bu(i))then
call iexch(ls(j),ls(n-k))
k=k+1
endif
enddo
if(mode.eq.0)then
do j=1,n-k
i=ls(j)
if(x(i).eq.bu(i))ls(j)=-i
enddo
lp(1)=n
goto9
endif
phase=0
c move inactive general c/s to the end
do j=nm,n1,-1
i=abs(ls(j))
if(i.gt.n)then
call iexch(ls(j),ls(lp(1)))
lp(1)=lp(1)-1
endif
enddo
call residuals(n,n1,lp(1),a,la,x,bl,bu,r,ls,f,g,ninf)
if(ninf.gt.0)then
gnorm=sqrt(dble(ninf))
gtol=sgnf*gnorm
rgtol=max(rgt0l,gtol)
goto15
endif
9 continue
c enter phase 1
phase=1
c collect active equality c/s
do j=1,n-k
i=abs(ls(j))
if(bu(i).eq.bl(i))then
peq=peq+1
call iexch(ls(j),ls(peq))
endif
enddo
call residuals(n,lp(1)+1,nm,a,la,x,bl,bu,r,ls,f,g,ninf)
lp(1)=nm
if(ninf.gt.0)then
gnorm=sqrt(scpr(0.D0,g,g,n))
gtol=sgnf*gnorm
rgtol=max(rgt0l,gtol)
goto15
endif
10 continue
phase=2
if(iprint.ge.1)write(nout,*)'FEASIBILITY OBTAINED at level 1'
n_inf=0
call setfg2(n,linear,a,la,x,f,g,ws,lws)
fbase=f
f=0.D0
ngr=ngr+1
c write(nout,4)'g =',(g(i),i=1,n)
call newg
gnorm=sqrt(scpr(0.D0,g,g,n))
gtol=sgnf*gnorm
rgtol=max(rgt0l,gtol)
ig=0
if(iprint.ge.1)write(nout,'(''pivots ='',I5,
* '' level = 1 f ='',E16.8)')npv,fbase
goto16
c start of major iteration
15 continue
if(iprint.ge.1)then
if(ninf.eq.0)then
if(k.gt.0)then
c write(nout,'(''pivots ='',I5,
c * '' level = 1 df ='',E16.8,'' k ='',I4)')npv,f,k
write(nout,'(''pivots ='',I5,
* '' level = 1 df ='',E16.8,'' rg ='',E12.4,
* '' k ='',I4)')npv,f,rgnorm,k
else
write(nout,'(''pivots ='',I5,
* '' level = 1 f ='',E16.8)')npv,fbase+f
endif
elseif(phase.eq.0)then
write(nout,'(''pivots ='',I5,'' level = 1 f ='',
* E16.8,'' ninfb ='',I4)')npv,f,ninf
else
write(nout,'(''pivots ='',I5,'' level = 1 f ='',
* E16.8,'' ninf ='',I4)')npv,f,ninf
endif
endif
16 continue
c calculate multipliers
do i=1,nm
w(i)=0.D0
enddo
c write(nout,4)'g =',(g(i),i=1,n)
call fbsub(n,1,n,a,la,0,g,w,ls,ws(lu1),lws(ll1),.true.)
call signst(n,r,w,ls)
c opposite bound or reset multiplier loop
20 continue
if(iprint.ge.3)then
write(nout,1001)'costs vector and indices',
* (ls(j),r(abs(ls(j))),j=1,n)
c write(nout,1000)'steepest edge coefficients',
c * (e(abs(ls(j))),j=1,n)
if(peq.gt.0.or.k.gt.0)write(nout,1)
* '# active equality c/s and free variables = ',peq,k
endif
c if(iphase.le.1)fbase=0.D0
c call checkq(n,lp(1),nmi,kmax,g,a,la,x,bl,bu,r,ls,ws(nb1),fbase+f,
c * ws,lws,ninf,peq,k,1,p,rp,linear)
21 continue
call optest(peq+1,n-k,r,e,ls,rp,pj)
if(phase.eq.0)then
c possibly choose an active general c/s to relax (marked by rp>0)
t=-1.D1*rp
do 13 j=1,n
i=abs(ls(j))
if(i.le.n)goto13
if(bu(i).eq.bl(i).and.r(i).lt.0.D0)then
r(i)=-r(i)
ls(j)=-ls(j)
endif
if(r(i)/e(i).le.t)goto13
rp=r(i)
t=rp/e(i)
pj=j
13 continue
endif
if(ig.eq.0)then
gg=0.D0
do j=n-k+1,n
i=ls(j)
gg=gg+r(i)**2
enddo
rgnorm=sqrt(gg)
endif
c print 2,'rgtol,rgnorm,rp',rgtol,rgnorm,rp
25 continue
if(rgnorm.le.rgtol.and.abs(rp).le.gtol)then
c allow for changes to norm(g)
gnorm=sqrt(scpr(0.D0,g,g,n))
gtol=sgnf*gnorm
rgtol=max(rgt0l,gtol)
endif
if((rgnorm.le.rgtol.and.abs(rp).le.gtol).or.ngr.gt.mxgr)then
c optimal at current level: first tidy up x
do j=peq+1,n-k
i=abs(ls(j))
if(i.le.n)then
if(ls(j).ge.0)then
x(i)=bl(i)
else
x(i)=bu(i)
endif
endif
enddo
do i=1,n
x(i)=max(min(x(i),bu(i)),bl(i))
enddo
do j=n1,nm
i=abs(ls(j))
if(r(i).eq.0.D0.and.i.le.n)then
if(ls(j).ge.0)then
x(i)=bl(i)
else
x(i)=bu(i)
endif
endif
enddo
if(ngr.gt.mxgr)then
f=fbase+f
ifail=5
return
endif
if(iprint.ge.2)then
write(nout,*)'OPTIMAL at level 1'
if(iprint.ge.3)then
c write(nout,1000)'x variables',(x(i),i=1,n)
write(nout,1001)'residual vector and indices',
* (ls(j),r(abs(ls(j))),j=n1,nm)
endif
endif
irep=irep+1
if(irep.le.nrep.and.iter.gt.mpiv)then
if(iprint.ge.1)write(nout,1)'refinement step #',irep
mode=4
goto8
endif
if(iprint.ge.2.and.nrep.gt.0)
* write(nout,*)'total number of restarts =',ires
if(ninf.gt.0)then
ifail=3
return
endif
nv=nv0
ifail=0
f=fbase+f
return
endif
if(rgnorm.ge.abs(rp))then
c ignore the multiplier of c/s p and set up or continue SD steps
p=0
else
p=abs(ls(pj))
if(iprint.ge.2)print 1,'CHOOSE p =',p
rp=r(p)
call iexch(ls(pj),ls(n-k))
pj=n-k
ig=0
endif
if(p.gt.0)then
c compute +/- Steepest Edge (SE) search direction s in an(.)
call tfbsub(n,a,la,p,ws(na1),ws(na1),ws(lu1),lws(ll1),
* e(p),.true.)
rp=scpr(0.D0,ws(na1),g,n)
if(ls(pj).lt.0)rp=-rp
if(rp*r(p).le.0.D0)then
r(p)=0.D0
goto21
endif
if(abs(rp-r(p)).gt.5.D-1*max(abs(rp),abs(r(p))))then
c if(abs(rp-r(p)).gt.1.D-1*gnorm)then
print 2,'1rp,r(p),rp-r(p)',rp,r(p),rp-r(p)
goto98
endif
snorm=e(p)
plus=ls(pj).ge.0.eqv.rp.lt.0.D0
f0=f
ig=0
else
if(ig.eq.0)then
c start up the limited memory sweep method
c if(p.gt.0)then
c transfer c/s p into Z
c if(ls(pj).lt.0)then
c r(p)=-r(p)
c ls(pj)=-ls(pj)
c endif
c k=k+1
c gg=gg+r(p)**2
c endif
ig=1
ngv=1
f0=f
ws(kb1)=gg
rgnorm=sqrt(gg)
c print 2,'initial rg =',(r(ls(j)),j=n-k+1,n)
if(k*ngv.gt.kmax*maxg)then
f=fbase+f
ifail=9
return
endif
call store_rg(k,ig,ws(krg1),r,ls(n-k+1))
endif
c compute Steepest Descent (SD) search direction s = -Z.rg in an(.)
call zprod(k,n,a,la,ws(na1),r,w,ls,ws(lu1),lws(ll1))
rp=scpr(0.D0,ws(na1),g,n)
if(abs(gg+rp).gt.5.D-1*max(gg,abs(rp)))then
c if(abs(gg+rp).gt.1.D-2*max(gg,abs(rp)))then
print 2,'gg,rp,gg+rp',gg,rp,gg+rp
goto98
endif
snorm=sqrt(scpr(0.D0,ws(na1),ws(na1),n))
plus=.true.
endif
c print 4,'s (or -s if .not.plus) =',(ws(i),i=na1,na+n)
c print *,'plus =',plus
c form At.s and denominators
call form_Ats(n1,lp(1),n,plus,a,la,ws(na1),w,ls,snorm*tol)
c return from degeneracy code
30 continue
if(iprint.ge.3)then
write(nout,1000)'x variables',(x(i),i=1,n)
write(nout,1001)'residual vector and indices',
* (ls(j),r(abs(ls(j))),j=n1,lp(1))
write(nout,1000)'denominators',(w(abs(ls(j))),j=n1,lp(1))
endif
c read *,i
40 continue
c level 1 ratio tests
amax=ainfty
qj=0
qj1=0
do 41 j=n-k+1,n
i=ls(j)
if(i.le.0)print *,'i.le.0'
if(i.le.0)stop
si=ws(na+i)
if(si.eq.0.D0)goto41
t=abs(si)
if(si.gt.0.D0.eqv.plus)then
z=bu(i)-x(i)
if(abs(z).lt.tol)then
z=0.D0
x(i)=bu(i)
else
z=z/t
endif
else
z=x(i)-bl(i)
if(abs(z).lt.tol)then
z=0.D0
x(i)=bl(i)
else
z=z/t
endif
endif
if(z.gt.amax)goto41
amax=z
qj=j
41 continue
if(ig.eq.0.and.rp.lt.0.D0.and.bu(p)-bl(p).lt.amax)then
amax=bu(p)-bl(p)
qj=pj
endif
if(ninf.gt.0)then
alpha1=ainfty
do 42 j=n1,lp(1)
i=abs(ls(j))
wi=w(i)
if(wi.eq.0.D0)goto42
ri=r(i)
if(wi.gt.0.D0)then
if(ri.lt.0.D0)goto42
z=(ri+tol)/wi
else
if(ri.lt.0.D0)then
z=ri/wi
if(z.lt.alpha1)then
alpha1=z
qj1=j
endif
endif
z=((bl(i)-bu(i))+ri-tol)/wi
endif
if(z.ge.amax)goto42
amax=z
qj=j
42 continue
if(qj1.gt.0.and.alpha1.le.amax)then
c find feasible step that zeros most infeasible c/s
do 43 j=n1,lp(1)
i=abs(ls(j))
wi=w(i)
if(wi.ge.0.D0)goto43
ri=r(i)
if(ri.lt.0.D0)then
z=ri/wi
if(z.gt.alpha1.and.z.le.amax)then
alpha1=z
qj1=j
endif
endif
43 continue
amax=alpha1
qj=qj1
else
qj1=0
endif
else
do 44 j=n1,lp(1)
i=abs(ls(j))
wi=w(i)
if(wi.eq.0.D0)goto44
ri=r(i)
if(wi.gt.0.D0)then
z=(ri+tol)/wi
else
z=(bl(i)-bu(i)+ri-tol)/wi
endif
if(z.ge.amax)goto44
amax=z
qj=j
44 continue
endif
q=abs(ls(qj))
if(iprint.ge.2.and.q.ne.p.and.qj.gt.n)
* write(nout,*)'q,r(q),w(q) =',q,r(q),w(q)
if(qj.gt.n.and.qj1.eq.0)then
if(w(q).gt.0.D0)then
amax=r(q)/w(q)
else
amax=(bl(q)-bu(q)+r(q))/w(q)
endif
endif
if(amax.eq.0.D0.and.rp.le.0.D0)then
alpha=0.D0
c potential degeneracy block at level 1
if(p.eq.0)goto65
if(bu(q).eq.bl(q))goto70
plev=n
do j=n1,lp(1)
i=abs(ls(j))
if(r(i).eq.0.D0)then
plev=plev+1
call iexch(ls(j),ls(plev))
if(bu(i).gt.bl(i))r(i)=1.D0
endif
enddo
if(plev.gt.n1)then
lp(2)=plev
lev=2
alp(1)=f
f=0.D0
qj=pj
q=p
if(iprint.ge.1)write(nout,'(''pivots ='',I5,'' level = 2'',
* '' f ='',E16.8)')npv,f
goto86
endif
qj=n1
r(q)=0.D0
c print *,'only one degenerate c/s'
goto70
endif
if(ninf.gt.0)then
alpha=amax
else
if(linear)then
alpha=amax
ff=f+alpha*rp
if(ff.lt.fmin)goto75
f=ff
goto60
endif
call gdotx(n,ws(na1),ws,lws,ws(nb1))
ngr=ngr+1
sgs=scpr(0.D0,ws(na1),ws(nb1),n)
c print 2,'rp,sgs',rp,sgs
ggo=gg
if(p.eq.0)then
t=v(nv)
if(t.le.0.D0)goto52
alpha=1.D0/t
if(alpha.ge.amax)goto52
nv=nv-1
ff=f+alpha*(rp+5.D-1*alpha*sgs)
if(ff.ge.f0)goto52
c print 2,'alphar =',alpha
c need to set f0 somewhere
if(iprint.ge.2)write(nout,*)'Ritz value step: alpha =',
* alpha,' p =',p
goto54
endif
52 continue
if(sgs.gt.0.D0)then
alpha=-rp/sgs
if(alpha.lt.amax)then
c accept Cauchy step
if(iprint.ge.2)write(nout,*)'Cauchy step: alpha =',
* alpha,' p =',p
ff=f+alpha*(rp+5.D-1*alpha*sgs)
nv=0
c print 2,'alphac =',alpha
goto54
endif
endif
c Cauchy step infeasible
alpha=amax
ff=f+alpha*(rp+5.D-1*alpha*sgs)
if(ff.lt.fmin)goto75
if(ff.ge.f)then
if(ires.lt.nres)goto98
f=fbase+f
if(iprint.ge.1)write(nout,'(''pivots ='',I5,
* '' level = 1 f ='',E16.8)')npv,f
ifail=4
return
endif
f=ff
if(plus)then
call mysaxpy(alpha,ws(nb1),g,n)
else
call mysaxpy(-alpha,ws(nb1),g,n)
endif
c print 4,'new g =',(g(i),i=1,n)
call newg
goto60
54 continue
if(ff.lt.fmin)goto75
if(ff.ge.f)then
if(ires.lt.nres)goto98
f=fbase+f
if(iprint.ge.1)write(nout,'(''pivots ='',I5,
* '' level = 1 f ='',E16.8)')npv,f
ifail=4
return
endif
f=ff
if(plus)then
call mysaxpy(alpha,ws(nb1),g,n)
else
call mysaxpy(-alpha,ws(nb1),g,n)
endif
c print 4,'new g =',(g(i),i=1,n)
call newg
if(ig.eq.0)goto60
ig1=ig+1
if(ig1.gt.maxg)ig1=1
call fbsub(n,1,n,a,la,0,g,w,ls,ws(lu1),lws(ll1),.true.)
c print 4,'new rg =',(w(ls(j)),j=n-k+1,n)
if(ngv.lt.maxg)ngv=ngv+1
if(k*ngv.gt.kmax*maxg)then
f=fbase+f
ifail=9
return
endif
call store_rg(k,ig1,ws(krg1),w,ls(n-k+1))
gpg=0.D0
gg=0.D0
do j=n-k+1,n
i=ls(j)
gpg=gpg+r(i)*w(i)
gg=gg+w(i)**2
enddo
rgnorm=sqrt(gg)
c print 2,'gpg,gg',gpg,gg
c print 2,'f =',f
call signst(n,r,w,ls)
ws(ka+ig)=1.D0/alpha
ws(kb+ig1)=gg
ws(kc+ig)=gpg
if(nv.eq.0.or.gg.gt.ggo)then
c compute new Ritz values
if(ngv.eq.2)then
nv=1
v(1)=1.D0/alpha
else
nv=min(ngv-1,k)
if(nv.le.0)print 1,'ngv,k,ig,nv =',ngv,k,ig,nv
if(nv.le.0)stop
c print 1,'ngv,k,ig,nv =',ngv,k,ig,nv
c print 4,'G =',(ws(krg+i),i=1,k*ngv)
c print 4,'a =',(ws(ka+i),i=1,ngv)
c print 4,'b =',(ws(kb+i),i=1,ngv+1)
c print 4,'c =',(ws(kc+i),i=1,ngv)
call formR(nv,k,ig,maxg,ws(ka1),ws(kb1),ws(kc1),ws(kd1),
* ws(ke1),ws(krg1),ws(kr1))
c call checkT(nv,maxg,ws(kr1),ws(ke1),ws(kd1))
call formT(nv,maxg,ws(kr1),v,ws(ke1))
c print 4,'T matrix',(v(i),i=1,nv)
c if(nv.gt.1)print 5,(ws(ke+i),i=1,nv-1)
call trid(v(1),ws(ke1),nv)
c print 4,'eigenvalues of T',(v(i),i=1,nv)
call insort(nv,v)
c print 4,'sorted eigenvalues of T',(v(i),i=1,nv)
endif
nv0=nv
f0=f
endif
ig=ig1
endif
60 continue
if(alpha.gt.0.D0)then
c update x
if(plus)then
call mysaxpy(alpha,ws(na1),x,n)
else
call mysaxpy(-alpha,ws(na1),x,n)
endif
c update r for inactive c/s
iter=iter+1
if(ninf.gt.0)then
n_inf=0
ff=f
f=0.D0
do 61 j=n1,lp(1)
i=abs(ls(j))
if(w(i).eq.0.D0)then
if(r(i).ge.0.D0)goto61
n_inf=n_inf+1
f=f-r(i)
goto61
endif
ri=r(i)-alpha*w(i)
if(abs(ri).le.tol)ri=0.D0
if(r(i).lt.0.D0)then
if(ri.ge.0.D0)then
c remove contribution to gradient
if(i.gt.n)then
call saipy(sign(1.D0,dble(ls(j))),a,la,i-n,g,n)
else
g(i)=0.D0
endif
else
n_inf=n_inf+1
f=f-ri
endif
endif
if(w(i).lt.0.D0)then
ro=(bu(i)-bl(i))-ri
if(abs(ro).le.tol)ro=0.D0
if(ro.lt.ri)then
ri=ro
ls(j)=-ls(j)
endif
endif
if(ri.eq.0.D0.and.i.le.n)then
if(ls(j).ge.0)then
x(i)=bl(i)
else
x(i)=bu(i)
endif
endif
r(i)=ri
61 continue
if(n_inf.ne.ninf)then
call iexch(ninf,n_inf)
call newg
c elseif(f.ge.ff)then
elseif(f.ge.eps*ff+ff)then
goto98
endif
else
n_inf=0
do 62 j=n1,lp(1)
i=abs(ls(j))
if(w(i).eq.0.D0)goto62
ri=r(i)-alpha*w(i)
if(w(i).lt.0.D0)then
ro=(bu(i)-bl(i))-ri
if(ro.lt.ri)then
ri=max(ro,0.D0)
w(i)=-w(i)
ls(j)=-ls(j)
endif
endif
if(ri.le.tol)then
ri=0.D0
if(i.le.n)then
if(ls(j).ge.0)then
x(i)=bl(i)
else
x(i)=bu(i)
endif
endif
endif
r(i)=ri
62 continue
endif
endif
if(alpha.lt.amax)then
if(ig.gt.0)then
c continue limited memory SD iterations
if(iprint.ge.1)write(nout,'(''pivots ='',I5,
* '' level = 1 df ='',E16.8,'' rg ='',E12.4,
* '' k ='',I4)')npv,f,rgnorm,k
if(alpha.gt.0.D0)goto20
print *,'alpha.le.0'
goto98
endif
c Cauchy step with SE iteration
k=k+1
if(p.le.n)then
ls(pj)=p
goto15
endif
c case p>n: find best inactive simple bound to replace p in ls(pj)
t=0.D0
do j=n1,lp(1)
i=abs(ls(j))
if(i.le.n)then
ti=abs(ws(na+i))
if(ti.gt.t)then
t=ti
qj=j
endif
endif
enddo
if(t.le.snorm*tol)then
print *,'no suitable simple bound available'
goto98
endif
q=abs(ls(qj))
ls(qj)=q
if(iprint.ge.2)write(nout,1)'New free variable',q
goto70
endif
65 continue
if(iprint.ge.2)
* write(nout,*)'New active c/s: alpha =',alpha,' q =',q
if(ig.gt.0)then
c case alpha=amax and SD step: find best free variable to relax
k=k-1
if(qj.le.n)then
c case: q is a free variable
if(ws(na+q).gt.0.D0)ls(qj)=-q
call iexch(ls(qj),ls(n-k))
ig=0