R/pdhessmin.R

In YafeiXu/CopulaModel: CopulaModel: Dependence Modeling with Copulas

# Minimization with modified Newton-Raphson iterations,
# Hessian is modified to be positive definite at each step.
# Algorithm and code produced by Pavel Krupskii (2013)
# see PhD thesis Krupskii (2014). UBC.

# param = starting point on entry
# objfn = function to be minimized with gradient and Hessian
# dstruct = list with  data set and other variables used by objfn
# lb = lower bound vector
# ub = upper bound vector
# mxiter = max number of iterations
# eps = tolerance for Newton-Raphson iterations
# iprint = control on amount of printing
#           F for no printing of iterations
#           T for printing x^{(k)} on each iteration
# bdd = bound on difference of 2 consecutive iterations
#  (useful is starting point is far from solution and func is far from convex)
# Outputs:
# fnval = function value at minimum
# parmin = param for minimum
# invh = inverse Hessian 
# iconv = 1 if converged, -1 for a boundary point, 0 otherwise
pdhessmin=function(param,objfn,dstruct,LB,UB,mxiter=30,eps=1.e-6,bdd=5,iprint=F)
{ np=length(param)
  iter=0; mxdif=1;
  while(mxdif>eps & iter<mxiter)
  { fn=objfn(param,dstruct,iprint);
    fnold= fn$fnval; 
    evd= eigen(fn$hess,F);
    eigval= evd$values;
    eigvec= evd$vectors;
    ind= (eigval<=0);
    #replace negative eigenvalues with a positive number, keeping eigenvectors
    if(sum(ind)>0 & iprint) cat("replacing eigenvalues in iteration ", iter,"\n")
    eigval[ind]=1e-1;    
    newhess= eigvec%*%diag(eigval)%*%t(eigvec);
    diff=solve(newhess,fn$grad); 
    mxdif=max(abs(diff))
    iter=iter+1
    param=param-diff
    outb=0;
    if(any(param<=LB) | any(param>=UB)) outb=1;
    if(is.infinite(fn$fnval) | is.nan(fn$fnval)) outb=1;
    if(any(is.infinite(fn$grad)) | any(is.nan(fn$grad))) outb=1;
    # modified NR to limit size of step to within bounds
    while(outb==1 | mxdif>bdd)
    { mxdif=mxdif/2.;
      outb=0;
      diff=diff/2; param=param+diff; 
      if(any(param<=LB) | any(param>=UB)) outb=1;
    }
    #i01 = (param<1e-4  & param>0);
    #i02 = (param>-1e-4 & param<0);
    #param[i01]=1e-4;
    #param[i02]=-1e-4;
    # find a step size to ensure the objective fn decreases
    # later to have option on objfn to not compute derivs
    fnnew= objfn(param,dstruct,F)$fnval; 
    while(fnold<=fnnew) 
    { mxdif=mxdif/2.;
      diff=diff/2; param=param+diff; 
      fnnew= objfn(param,dstruct,F)$fnval;   
    }
    if(iprint) { cat(iter, fnnew, mxdif,"\n") }
  }
  iconv=1;
  if(iter>=mxiter) { cat("did not converge\n"); iconv=0; }
  else { if(max(abs(fn$grad))>eps) iconv=-1 }
  # check for positive definite
  iposdef=isposdef(fn$hess)
  # inverse Hessian
  invh=solve(fn$hess)
  list(parmin=param,fnval=fn$fnval,invh=invh,iconv=iconv,iposdef=iposdef)
}

 
# Version with ifixed as argument
# param = starting point on entry
# objfn = function to be minimized with gradient and Hessian
# ifixed = vector of length(param) of True/False, such that
#      ifixed[i]=T iff param[i] is fixed at the given value
# dstruct = list with  data set and other variables used by objfn
# lb = lower bound vector
# ub = upper bound vector
# mxiter = max number of iterations
# eps = tolerance for Newton-Raphson iterations
# iprint = control on amount of printing
#           F for no printing of iterations
#           T for printing x^{(k)} on each iteration
# bdd = bound on difference of 2 consecutive iterations
#  (useful is starting point is far from solution and func is far from convex)
# Outputs:
# fnval = function value at minimum
# parmin = param for minimum
# invh = inverse Hessian 
# iconv = 1 if converged, -1 for a boundary point, 0 otherwise
pdhessminb=function(param,objfn,ifixed,dstruct,LB,UB,mxiter=30,eps=1.e-6,
  bdd=5,iprint=F) 
{ np=length(param)
  iter=0; mxdif=1; brk=0;
  while(mxdif>eps & iter<mxiter)
  { dstruct$pdf=0;
    fn=objfn(param,dstruct,iprint);
    fnold=fn$fnval;
    # get subset of grad, hess
    sgrad=fn$grad[!ifixed]; shess=fn$hess[!ifixed,!ifixed]
    evd= eigen(shess,F);
    eigval= evd$values;
    eigvec= evd$vectors;
    ind= (eigval<=0);
    #replace negative eigenvalues with a positive number, keeping eigenvectors
    if(sum(ind)>0 & iprint) cat("replacing eigenvalues in iteration ", iter,"\n")
    eigval[ind]=1e-1;    
    newhess= eigvec%*%diag(eigval)%*%t(eigvec);
    sdiff=try(solve(newhess,sgrad),T);
    if(is.vector(sdiff) == FALSE) { brk = 1; break; }
    diff=rep(0,np)
    diff[!ifixed]=sdiff
    mxdif=max(abs(diff))
    iter=iter+1
    param=param-diff
    outb=0;
    if(any(param<=LB) | any(param>=UB)) outb=1;
    if(is.infinite(fn$fnval) | is.nan(fn$fnval)) outb=1;
    if(any(is.infinite(fn$grad)) || any(is.nan(fn$grad))) outb=1;
    # modified NR to limit size of step to within bounds
    while(outb==1 | mxdif>bdd)
    { mxdif=mxdif/2.;
      outb=0;
      diff=diff/2; param=param+diff; 
      if(any(param<=LB) | any(param>=UB)) outb=1;
    }
    # find a step size to ensure the objective fn decreases
    LB.ind= param<=LB +0.008
    UB.ind= param>=UB -0.008
    ifixed[LB.ind]=T
    ifixed[UB.ind]=T
    dstruct$pdf=1; # positive definite
    fnnew=objfn(param,dstruct,F)$fnval;
    brkvr=0; 
    while(fnold<=fnnew) 
    { mxdif=mxdif/2.; 
      diff=diff/2; param=param+diff; 
      fnnew= objfn(param,dstruct,F)$fnval; 
      brkvr=brkvr+1;
      if(brkvr>10)
      { print("break: step size is very small"); 
        fnnew=fnold-1e-5; mxdif=eps-1e-5; 
      }   
    }
    if(iprint) { cat(iter, fn$fnval, mxdif,"\n") }
  }
  if(brk==0)
  { iconv=1;
    if(iter>=mxiter) { cat("did not converge\n"); iconv=0; }
    else { if(max(abs(fn$grad))>eps) iconv=-1 }
    # check for positive definite
    iposdef=isposdef(shess)
    # inverse Hessian
    invh=solve(shess)
  }
  if(brk==1)
  { iconv=2; #algorithm failed
    param=rep(1e5,np);
    fnnew=1e5;
    invh=NA;
    isposdef=FALSE;
  }
  list(parmin=param,fnval=fnnew,invh=invh,iconv=iconv,iposdef=iposdef)
}