/*
* csbobyqa
*
* The MIT License
*
* Copyright (c) 2012 Anders Gustafsson, Cureos AB.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files
* (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge,
* publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so,
* subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE
* FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
* WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*
* Remarks:
*
* The original Fortran 77 version of this code was developed by Michael Powell (mjdp@cam.ac.uk) and can be downloaded from this location:
* http://plato.asu.edu/ftp/other_software/bobyqa.zip
*/
using System;
using System.IO;
namespace Cureos.Numerics
{
/// <summary>
/// Representation of supported exit statuses from the Bobyqa algorithm.
/// </summary>
public enum BobyqaExitStatus
{
TooFewVariables,
VariableBoundsArrayTooShort,
InvalidBoundsSpecification,
Normal,
InvalidInterpolationConditionNumber,
BoundsRangeTooSmall,
DenominatorCancellation,
TrustRegionStepReductionFailure,
MaximumIterationsReached
}
/// <summary>
/// C# implementation of Powell�s nonlinear derivative�free bound constrained optimization that uses a quadratic
/// approximation approach. The algorithm applies 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.
/// </summary>
public static class Bobyqa
{
#region FIELDS
private const double INF = 1.0e60;
private const double INFMIN = -1.0e60;
private const double ONEMIN = -1.0;
private const double ZERO = 0.0;
private const double TENTH = 0.1;
private const double HALF = 0.5;
private const double ONE = 1.0;
private const double TWO = 2.0;
private const double TEN = 10.0;
private static readonly string LF = Environment.NewLine;
private static readonly string InvalidNptText = LF + "Return from BOBYQA because NPT is not in the required interval";
private static readonly string TooSmallBoundRangeText = LF + "Return from BOBYQA because one of the differences XU[I]-XL[I] is less than 2*RHOBEG.";
private static readonly string DenominatorCancellationText = LF + "Return from BOBYQA because of much cancellation in a denominator.";
private static readonly string MaxIterationsText = LF + "Return from BOBYQA because CALFUN has been called MAXFUN times.";
private static readonly string TrustRegionStepFailureText = LF + "Return from BOBYQA because a trust region step has failed to reduce Q.";
private static readonly string IterationOutputFormat = LF + "Function number {0,6} F ={1,18:E10}" + LF + "The corresponding X is: {2}";
private static readonly string StageCompleteOutputFormat = LF + "Least value of F ={0,15:E9}" + LF + "The corresponding X is: {1}";
private static readonly string RhoUpdatedFormat = LF + "New RHO ={0,11:E4}" + LF + "Number of function values ={1,6}";
private static readonly string FinalNumberEvaluationsFormat = LF + "At the return from BOBYQA Number of function values = {0}";
#endregion
#region PUBLIC METHODS
/// <summary>
/// Find a (local) minimum of the objective function <paramref name="calfun"/>, potentially subject to variable
/// bounds <paramref name="xl"/> and <paramref name="xu"/>.
/// </summary>
/// <param name="calfun">Objective function subject to minimization.</param>
/// <param name="n">Number of optimization variables, must be at least two.</param>
/// <param name="x">On entry, initial estimates of the variables. On exit, optimized variable values corresponding
/// to the minimization of <paramref name="calfun"/>. The variable array is zero-based.</param>
/// <param name="xl">Lower bounds on the variables. Array is zero-based. If set to null, all variables
/// are treated as downwards unbounded.</param>
/// <param name="xu">Upper bounds on the variables. Array is zero-based. If set to null, all variables
/// are treated as upwards unbounded.</param>
/// <param name="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.</param>
/// <param name="rhobeg">Initial value of a trust region radius, must be positive and greater than <paramref name="rhoend"/>.
/// Typically, value should be about one tenth of the greatest expected change to a variable.</param>
/// <param name="rhoend">Final values of a trust region radius, must be positive and less than <paramref name="rhobeg"/>.
/// Indicate the accuracy that is required in the final values of the variables.</param>
/// <param name="iprint">Should be set to 0, 1, 2 or 3, to control the amount of printing. Specifically, there is
/// no output for 0 and there is output only at the return for 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 the objective function with its variables are output for value 3.</param>
/// <param name="maxfun">Maximum number of objective function evaluations.</param>
/// <param name="logger">If defined, text writer to which log output should be directed.</param>
/// <returns>Exit status of the objective function minimization.</returns>
/// <remarks>The construction of quadratic models requires <paramref name="xl"/> to be strictly less than
/// <paramref name="xu"/> for each index I. Further, the contribution to a model from changes to the I-th variable is
/// damaged severely by rounding errors if difference between upper and lower bound is too small.</remarks>
public static BobyqaExitStatus FindMinimum(Func<int, double[], double> calfun, int n, double[] x, double[] xl = null,
double[] xu = null, int npt = -1, double rhobeg = -1.0, double rhoend = -1.0, int iprint = 1, int maxfun = 10000,
TextWriter logger = null)
{
// Verify that the number of variables is greater than 1; BOBYQA does not support 1-D optimization.
if (n < 2) return BobyqaExitStatus.TooFewVariables;
// Verify that the number of variables, and bounds if defined, in the respective array is sufficient.
if (x.Length < n || (xl != null && xl.Length < n) || (xu != null && xu.Length < n))
{
return BobyqaExitStatus.VariableBoundsArrayTooShort;
}
// C# arrays are zero-based, whereas BOBYQA methods expect one-based arrays. Therefore define internal matrices
// to be dispatched to the private BOBYQA methods.
var ix = new double[1 + n];
Array.Copy(x, 0, ix, 1, n);
// If xl and/or xu are null, this is interpreted as that the optimization variables are all unbounded downwards and/or upwards.
// In that case, assign artificial +/- infinity values to the bounds array(s).
var ixl = new double[1 + n];
if (xl == null)
for (var i = 1; i <= n; ++i) ixl[i] = INFMIN;
else
Array.Copy(xl, 0, ixl, 1, n);
var ixu = new double[1 + n];
if (xu == null)
for (var i = 1; i <= n; ++i) ixu[i] = INF;
else
Array.Copy(xu, 0, ixu, 1, n);
// Verify that all lower bounds are less than upper bounds.
// If any start value is outside bounds, adjust this value to be within bounds.
var rng = new double[1 + n];
var minrng = Double.MaxValue;
var maxabsx = 0.0;
for (var i = 1; i <= n; ++i)
{
if ((rng[i] = ixu[i] - ixl[i]) <= 0.0)
return BobyqaExitStatus.InvalidBoundsSpecification;
minrng = Math.Min(rng[i], minrng);
if (ix[i] < ixl[i]) ix[i] = ixl[i];
if (ix[i] > ixu[i]) ix[i] = ixu[i];
maxabsx = Math.Max(Math.Abs(ix[i]), maxabsx);
}
// If rhobeg is non-positive, set rhobeg based on the absolute values of the variables' start values,
// using same strategy as R-project BOBYQA wrapper.
if (rhobeg <= 0.0) rhobeg = maxabsx > 0.0 ? Math.Min(0.95, 0.2 * maxabsx) : 0.95;
// Required that rhobeg is less than half the minimum bounds range; adjust rhobeg if necessary.
if (rhobeg > 0.5 * minrng) rhobeg = 0.2 * minrng;
// If rhoend is non-negative, set rhoend to one millionth of the rhobeg value (R-project strategy).
if (rhoend <= 0.0) rhoend = 1.0e-6 * rhobeg;
// If npt is non-positive, apply default value 2 * n + 1.
var inpt = npt > 0 ? npt : 2 * n + 1;
// Define internal calfun to account for that the x vector in the function invocation is one-based.
var icalfun = new Func<int, double[], double>((nn, ixx) =>
{
var xx = new double[n];
Array.Copy(ixx, 1, xx, 0, n);
return calfun(nn, xx);
});
// Invoke optimization. After completed optimization, transfer the optimized internal variable array to the
// variable array in the method call.
var status = BOBYQA(icalfun, n, inpt, ix, ixl, ixu, rhobeg, rhoend, iprint, maxfun, logger);
Array.Copy(ix, 1, x, 0, n);
return status;
}
#endregion
#region PRIVATE BOBYQA ALGORITHM METHODS
private static BobyqaExitStatus BOBYQA(Func<int, double[], double> calfun, int n, int npt, double[] x,
double[] xl, double[] xu, double rhobeg, double rhoend, int iprint, int maxfun, TextWriter logger)
{
// This subroutine seeks the least value of a function of many variables,
// by applying a trust region method that forms quadratic models by
// interpolation. There is usually some freedom in the interpolation
// conditions, which is taken up by minimizing the Frobenius norm of
// the change to the second derivative of the model, beginning with the
// zero matrix. The values of the variables are constrained by upper and
// lower bounds. The arguments of the subroutine are as follows.
//
// N must be set to the number of variables and must be at least two.
// NPT is the number of interpolation conditions. Its value must be in
// the interval [N+2,(N+1)(N+2)/2]. Choices that exceed 2*N+1 are not
// recommended.
// Initial values of the variables must be set in X(1),X(2),...,X(N). They
// will be changed to the values that give the least calculated F.
// For I=1,2,...,N, XL[I] and XU[I] must provide the lower and upper
// bounds, respectively, on X[I]. The construction of quadratic models
// requires XL[I] to be strictly less than XU[I] for each I. Further,
// the contribution to a model from changes to the I-th variable is
// damaged severely by rounding errors if XU[I]-XL[I] is too small.
// RHOBEG and RHOEND must be set to the initial and final values of a trust
// region radius, so both must be positive with RHOEND no greater than
// RHOBEG. Typically, RHOBEG should be about one tenth of the greatest
// expected change to a variable, while RHOEND should indicate the
// accuracy that is required in the final values of the variables. An
// error return occurs if any of the differences XU[I]-XL[I], I=1,...,N,
// is less than 2*RHOBEG.
// The value of IPRINT should be set to 0, 1, 2 or 3, which controls the
// amount of printing. Specifically, there is no output if IPRINT=0 and
// there is output only at the return if IPRINT=1. Otherwise, each new
// value of RHO is printed, with the best vector of variables so far and
// the corresponding value of the objective function. Further, each new
// value of F with its variables are output if IPRINT=3.
// MAXFUN must be set to an upper bound on the number of calls of CALFUN.
//
// 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.
// Return if the value of NPT is unacceptable.
var np = n + 1;
if (npt < n + 2 || npt > ((n + 2) * np) / 2)
{
if (logger != null) logger.WriteLine(InvalidNptText);
return BobyqaExitStatus.InvalidInterpolationConditionNumber;
}
var ndim = npt + n;
var sl = new double[1 + n];
var su = new double[1 + n];
// 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.
for (var j = 1; j <= n; ++j)
{
double temp = xu[j] - xl[j];
if (temp < rhobeg + rhobeg)
{
if (logger != null) logger.WriteLine(TooSmallBoundRangeText);
return BobyqaExitStatus.BoundsRangeTooSmall;
}
sl[j] = xl[j] - x[j];
su[j] = xu[j] - x[j];
if (sl[j] >= -rhobeg)
{
if (sl[j] >= ZERO)
{
x[j] = xl[j];
sl[j] = ZERO;
su[j] = temp;
}
else
{
x[j] = xl[j] + rhobeg;
sl[j] = -rhobeg;
su[j] = Math.Max(xu[j] - x[j], rhobeg);
}
}
else if (su[j] <= rhobeg)
{
if (su[j] <= ZERO)
{
x[j] = xu[j];
sl[j] = -temp;
su[j] = ZERO;
}
else
{
x[j] = xu[j] - rhobeg;
sl[j] = Math.Min(xl[j] - x[j], -rhobeg);
su[j] = rhobeg;
}
}
}
// Make the call of BOBYQB.
return BOBYQB(calfun, n, npt, x, xl, xu, rhobeg, rhoend, iprint, maxfun, ndim, sl, su, logger);
}
private static BobyqaExitStatus BOBYQB(Func<int, double[], double> calfun, int n, int npt, double[] x, double[] xl,
double[] xu, double rhobeg, double rhoend, int iprint, int maxfun, int ndim, double[] sl, double[] su,
TextWriter logger)
{
// The arguments N, NPT, X, XL, XU, RHOBEG, RHOEND, IPRINT and MAXFUN
// are identical to the corresponding arguments in SUBROUTINE BOBYQA.
// 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.
// Set some constants.
var np = n + 1;
var nptm = npt - np;
var nh = (n * np) / 2;
// 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.
// 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).
var xbase = new double[1 + n];
var xpt = new double[1 + npt,1 + n];
var fval = new double[1 + npt];
var xopt = new double[1 + n];
var gopt = new double[1 + n];
var hq = new double[1 + n * np / 2];
var pq = new double[1 + npt];
var bmat = new double[1 + ndim,1 + n];
var zmat = new double[1 + npt,1 + npt - np];
var xnew = new double[1 + n];
var xalt = new double[1 + n];
var d = new double[1 + n];
var vlag = new double[1 + ndim];
var wn = new double[1 + n];
var w2npt = new double[1 + 2 * npt];
var knew = 0;
var adelt = 0.0;
var alpha = 0.0;
var beta = 0.0;
var cauchy = 0.0;
var denom = 0.0;
var diffc = 0.0;
var ratio = 0.0;
var f = 0.0;
double distsq;
BobyqaExitStatus status;
// 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.
int nf, kopt;
PRELIM(calfun,n, npt, x, xl, xu, rhobeg, iprint, maxfun, xbase, xpt, fval, gopt, hq, pq, bmat, zmat, ndim, sl, su,
out nf, out kopt, logger);
var xoptsq = ZERO;
for (var i = 1; i <= n; ++i)
{
xopt[i] = xpt[kopt, i];
xoptsq += xopt[i] * xopt[i];
}
var fsave = fval[1];
if (nf < npt)
{
if (iprint > 0 && logger != null) logger.WriteLine(MaxIterationsText);
status = BobyqaExitStatus.MaximumIterationsReached;
goto L_720;
}
var kbase = 1;
// Complete the settings that are required for the iterative procedure.
var rho = rhobeg;
var delta = rho;
var nresc = nf;
var ntrits = 0;
var diffa = ZERO;
var diffb = ZERO;
var itest = 0;
var nfsav = nf;
// Update GOPT if necessary before the first iteration and after each
// call of RESCUE that makes a call of CALFUN.
L_20:
if (kopt != kbase)
{
var ih = 0;
for (var j = 1; j <= n; ++j)
{
for (var i = 1; i <= j; ++i)
{
ih = ih + 1;
if (i < j) gopt[j] += hq[ih] * xopt[i];
gopt[i] += hq[ih] * xopt[j];
}
}
if (nf > npt)
{
for (var k = 1; k <= npt; ++k)
{
var temp = ZERO;
for (var j = 1; j <= n; ++j) temp += xpt[k, j] * xopt[j];
temp *= pq[k];
for (var i = 1; i <= n; ++i) gopt[i] += temp * xpt[k, i];
}
}
}
// 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.
L_60:
var gnew = new double[1 + n];
double dsq, crvmin;
TRSBOX(n, npt, xpt, xopt, gopt, hq, pq, sl, su, delta, xnew, d, gnew, out dsq, out crvmin);
var dnorm = Math.Min(delta, Math.Sqrt(dsq));
if (dnorm < HALF * rho)
{
ntrits = -1;
distsq = TEN * TEN * rho * rho;
if (nf <= nfsav + 2) goto L_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.
var errbig = Math.Max(diffa, Math.Max(diffb, diffc));
var frhosq = 0.125 * rho * rho;
if (crvmin > ZERO && errbig > frhosq * crvmin) goto L_650;
var bdtol = errbig / rho;
for (var j = 1; j <= n; ++j)
{
var bdtest = bdtol;
if (xnew[j] == sl[j]) bdtest = gnew[j];
if (xnew[j] == su[j]) bdtest = -gnew[j];
if (bdtest < bdtol)
{
var curv = hq[(j + j * j) / 2];
for (var k = 1; k <= npt; ++k)
{
curv = curv + pq[k] * xpt[k, j] * xpt[k, j];
}
bdtest = bdtest + HALF * curv * rho;
if (bdtest < bdtol) goto L_650;
}
}
goto L_680;
}
++ntrits;
// 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.
L_90:
if (dsq <= 1.0E-3 * xoptsq)
{
var fracsq = 0.25 * xoptsq;
var sumpq = ZERO;
for (var k = 1; k <= npt; ++k)
{
sumpq += pq[k];
var sum = -HALF * xoptsq;
for (var i = 1; i <= n; ++i) sum += xpt[k, i] * xopt[i];
w2npt[k] = sum;
var temp = fracsq - HALF * sum;
for (var i = 1; i <= n; ++i)
{
wn[i] = bmat[k, i];
vlag[i] = sum * xpt[k, i] + temp * xopt[i];
var ip = npt + i;
for (var j = 1; j <= i; ++j) bmat[ip, j] += wn[i] * vlag[j] + vlag[i] * wn[j];
}
}
// Then the revisions of BMAT that depend on ZMAT are calculated.
for (var jj = 1; jj <= nptm; ++jj)
{
var sumz = ZERO;
var sumw = ZERO;
for (var k = 1; k <= npt; ++k)
{
sumz += zmat[k, jj];
vlag[k] = w2npt[k] * zmat[k, jj];
sumw += vlag[k];
}
for (var j = 1; j <= n; ++j)
{
var sum = (fracsq * sumz - HALF * sumw) * xopt[j];
for (var k = 1; k <= npt; ++k) sum += vlag[k] * xpt[k, j];
wn[j] = sum;
for (var k = 1; k <= npt; ++k) bmat[k, j] += sum * zmat[k, jj];
}
for (var i = 1; i <= n; ++i)
{
var ip = i + npt;
var temp = wn[i];
for (var j = 1; j <= i; ++j) bmat[ip, j] += temp * wn[j];
}
}
// The following instructions complete the shift, including the changes
// to the second derivative parameters of the quadratic model.
var ih = 0;
for (var j = 1; j <= n; ++j)
{
wn[j] = -HALF * sumpq * xopt[j];
for (var k = 1; k <= npt; ++k)
{
wn[j] += pq[k] * xpt[k, j];
xpt[k, j] -= xopt[j];
}
for (var i = 1; i <= j; ++i)
{
hq[++ih] += wn[i] * xopt[j] + xopt[i] * wn[j];
bmat[npt + i, j] = bmat[npt + j, i];
}
}
for (var i = 1; i <= n; ++i)
{
xbase[i] += xopt[i];
xnew[i] -= xopt[i];
sl[i] -= xopt[i];
su[i] -= xopt[i];
xopt[i] = ZERO;
}
xoptsq = ZERO;
}
if (ntrits == 0) goto L_210;
goto L_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.
L_190:
nfsav = nf;
kbase = kopt;
RESCUE(calfun, n, npt, xl, xu, iprint, maxfun, xbase, xpt, fval, xopt, gopt, hq, pq, bmat, zmat, ndim, sl,
su, ref nf, delta, ref kopt, vlag, logger);
// 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)
{
for (var i = 1; i <= n; ++i)
{
xopt[i] = xpt[kopt, i];
xoptsq = xoptsq + xopt[i] * xopt[i];
}
}
if (nf < 0)
{
nf = maxfun;
if (iprint > 0 && logger != null) logger.WriteLine(MaxIterationsText);
status = BobyqaExitStatus.MaximumIterationsReached;
goto L_720;
}
nresc = nf;
if (nfsav < nf)
{
nfsav = nf;
goto L_20;
}
if (ntrits > 0) goto L_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.
L_210:
ALTMOV(n, npt, xpt, xopt, bmat, zmat, sl, su, kopt, knew, adelt, xnew, xalt, out alpha, out cauchy);
for (var i = 1; i <= n; ++i) d[i] = xnew[i] - xopt[i];
// 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.
L_230:
for (var k = 1; k <= npt; ++k)
{
var suma = ZERO;
var sumb = ZERO;
var sum = ZERO;
for (var j = 1; j <= n; ++j)
{
suma += xpt[k, j] * d[j];
sumb += xpt[k, j] * xopt[j];
sum += bmat[k, j] * d[j];
}
w2npt[k] = suma * (HALF * suma + sumb);
vlag[k] = sum;
w2npt[npt + k] = suma;
}
beta = ZERO;
for (var jj = 1; jj <= nptm; ++jj)
{
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += zmat[k, jj] * w2npt[k];
beta -= sum * sum;
for (var k = 1; k <= npt; ++k) vlag[k] += sum * zmat[k, jj];
}
dsq = ZERO;
var bsum = ZERO;
var dx = ZERO;
for (var j = 1; j <= n; ++j)
{
dsq = dsq + d[j] * d[j];
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += w2npt[k] * bmat[k, j];
bsum += sum * d[j];
var jp = npt + j;
for (var i = 1; i <= n; ++i) sum += bmat[jp, i] * d[i];
vlag[jp] = sum;
bsum += sum * d[j];
dx += d[j] * xopt[j];
}
beta += dx * dx + dsq * (xoptsq + dx + dx + HALF * dsq) - bsum;
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)
{
denom = vlag[knew] * vlag[knew] + alpha * beta;
if (denom < cauchy && cauchy > ZERO)
{
for (var i = 1; i <= n; ++i)
{
xnew[i] = xalt[i];
d[i] = xnew[i] - xopt[i];
}
cauchy = ZERO;
goto L_230;
}
if (denom <= HALF * vlag[knew] * vlag[knew])
{
if (nf > nresc) goto L_190;
if (iprint > 0 && logger != null) logger.WriteLine(DenominatorCancellationText);
status = BobyqaExitStatus.DenominatorCancellation;
goto L_720;
}
}
// 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
{
var delsq = delta * delta;
var scaden = ZERO;
var biglsq = ZERO;
knew = 0;
for (var k = 1; k <= npt; ++k)
{
if (k == kopt) continue;
var hdiag = ZERO;
for (var jj = 1; jj <= nptm; ++jj) hdiag += zmat[k, jj] * zmat[k, jj];
var den = beta * hdiag + vlag[k] * vlag[k];
distsq = ZERO;
for (var j = 1; j <= n; ++j) distsq += Math.Pow(xpt[k, j] - xopt[j], 2.0);
var temp = Math.Max(ONE, Math.Pow(distsq / delsq, 2.0));
if (temp * den > scaden)
{
scaden = temp * den;
knew = k;
denom = den;
}
biglsq = Math.Max(biglsq, temp * vlag[k] * vlag[k]);
}
if (scaden <= HALF * biglsq)
{
if (nf > nresc) goto L_190;
if (iprint > 0 && logger != null) logger.WriteLine(DenominatorCancellationText);
status = BobyqaExitStatus.DenominatorCancellation;
goto L_720;
}
}
// 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.
L_360:
for (var i = 1; i <= n; ++i)
{
x[i] = Math.Min(Math.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];
}
if (nf >= maxfun)
{
if (iprint > 0 && logger != null) logger.WriteLine(MaxIterationsText);
status = BobyqaExitStatus.MaximumIterationsReached;
goto L_720;
}
++nf;
f = calfun(n, x);
if (iprint == 3 && logger != null) logger.WriteLine(IterationOutputFormat, nf, f, x.ToString(n));
if (ntrits == -1)
{
fsave = f;
status = BobyqaExitStatus.Normal;
goto L_720;
}
// Use the quadratic model to predict the change in F due to the step D,
// and set DIFF to the error of this prediction.
var fopt = fval[kopt];
var vquad = ZERO;
{
var ih = 0;
for (var j = 1; j <= n; ++j)
{
vquad += d[j] * gopt[j];
for (var i = 1; i <= j; ++i)
vquad += hq[++ih] * (i == j ? HALF : ONE) * d[i] * d[j];
}
}
for (var k = 1; k <= npt; ++k) vquad += HALF * pq[k] * w2npt[npt + k] * w2npt[npt + k];
var diff = f - fopt - vquad;
diffc = diffb;
diffb = diffa;
diffa = Math.Abs(diff);
if (dnorm > rho) nfsav = nf;
// Pick the next value of DELTA after a trust region step.
if (ntrits > 0)
{
if (vquad >= ZERO)
{
if (iprint > 0 && logger != null) logger.WriteLine(TrustRegionStepFailureText);
status = BobyqaExitStatus.TrustRegionStepReductionFailure;
goto L_720;
}
ratio = (f - fopt) / vquad;
if (ratio <= TENTH)
delta = Math.Min(HALF * delta, dnorm);
else if (ratio <= 0.7)
delta = Math.Max(HALF * delta, dnorm);
else
delta = Math.Max(HALF * delta, dnorm + dnorm);
if (delta <= 1.5 * rho) delta = rho;
// Recalculate KNEW and DENOM if the new F is less than FOPT.
if (f < fopt)
{
var ksav = knew;
var densav = denom;
var delsq = delta * delta;
var scaden = ZERO;
var biglsq = ZERO;
knew = 0;
for (var k = 1; k <= npt; ++k)
{
var hdiag = ZERO;
for (var jj = 1; jj <= nptm; ++jj) hdiag += zmat[k, jj] * zmat[k, jj];
var den = beta * hdiag + vlag[k] * vlag[k];
distsq = ZERO;
for (var j = 1; j <= n; ++j) distsq += Math.Pow(xpt[k, j] - xnew[j], 2.0);
var temp = Math.Max(ONE, Math.Pow(distsq / delsq, 2.0));
if (temp * den > scaden)
{
scaden = temp * den;
knew = k;
denom = den;
}
biglsq = Math.Max(biglsq, temp * vlag[k] * vlag[k]);
}
if (scaden <= HALF * biglsq)
{
knew = ksav;
denom = densav;
}
}
}
// Update BMAT and ZMAT, so that the KNEW-th interpolation point can be
// moved. Also update the second derivative terms of the model.
var w = new double[1 + ndim];
UPDATE(n, npt, bmat, zmat, ndim, vlag, beta, denom, knew, w);
var pqold = pq[knew];
pq[knew] = ZERO;
{
var ih = 0;
for (var i = 1; i <= n; ++i)
{
var temp = pqold * xpt[knew, i];
for (var j = 1; j <= i; ++j) hq[++ih] += temp * xpt[knew, j];
}
}
for (var jj = 1; jj <= nptm; ++jj)
{
var temp = diff * zmat[knew, jj];
for (var k = 1; k <= npt; ++k) pq[k] += temp * zmat[k, jj];
}
// 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;
for (var i = 1; i <= n; ++i)
{
xpt[knew, i] = xnew[i];
wn[i] = bmat[knew, i];
}
for (var k = 1; k <= npt; ++k)
{
var suma = ZERO;
for (var jj = 1; jj <= nptm; ++jj) suma += zmat[knew, jj] * zmat[k, jj];
var sumb = ZERO;
for (var j = 1; j <= n; ++j) sumb += xpt[k, j] * xopt[j];
var temp = suma * sumb;
for (var i = 1; i <= n; ++i) wn[i] += temp * xpt[k, i];
}
for (var i = 1; i <= n; ++i) gopt[i] += diff * wn[i];
// Update XOPT, GOPT and KOPT if the new calculated F is less than FOPT.
if (f < fopt)
{
kopt = knew;
xoptsq = ZERO;
var ih = 0;
for (var j = 1; j <= n; ++j)
{
xopt[j] = xnew[j];
xoptsq += xopt[j] * xopt[j];
for (var i = 1; i <= j; ++i)
{
++ih;
if (i < j) gopt[j] += +hq[ih] * d[i];
gopt[i] += hq[ih] * d[j];
}
}
for (var k = 1; k <= npt; ++k)
{
var temp = ZERO;
for (var j = 1; j <= n; ++j) temp += xpt[k, j] * d[j];
temp *= pq[k];
for (var i = 1; i <= n; ++i) gopt[i] += temp * xpt[k, i];
}
}
// 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)
{
for (var k = 1; k <= npt; ++k)
{
vlag[k] = fval[k] - fval[kopt];
w2npt[k] = ZERO;
}
for (var j = 1; j <= nptm; ++j)
{
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += zmat[k, j] * vlag[k];
for (var k = 1; k <= npt; ++k) w2npt[k] = w2npt[k] + sum * zmat[k, j];
}
for (var k = 1; k <= npt; ++k)
{
var sum = ZERO;
for (var j = 1; j <= n; ++j) sum += xpt[k, j] * xopt[j];
w2npt[k + npt] = w2npt[k];
w2npt[k] *= sum;
}
var gqsq = ZERO;
var gisq = ZERO;
for (var i = 1; i <= n; ++i)
{
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += bmat[k, i] * vlag[k] + xpt[k, i] * w2npt[k];
if (xopt[i] == sl[i])
{
gqsq += Math.Pow(Math.Min(ZERO, gopt[i]), 2.0);
gisq += Math.Pow(Math.Min(ZERO, sum), 2.0);
}
else if (xopt[i] == su[i])
{
gqsq += Math.Pow(Math.Max(ZERO, gopt[i]), 2.0);
gisq += Math.Pow(Math.Max(ZERO, sum), 2.0);
}
else
{
gqsq += gopt[i] * gopt[i];
gisq += sum * sum;
}
vlag[npt + i] = sum;
}
// Test whether to replace the new quadratic model by the least Frobenius
// norm interpolant, making the replacement if the test is satisfied.
++itest;
if (gqsq < TEN * gisq) itest = 0;
if (itest >= 3)
{
for (var i = 1; i <= Math.Max(npt, nh); ++i)
{
if (i <= n) gopt[i] = vlag[npt + i];
if (i <= npt) pq[i] = w2npt[npt + i];
if (i <= nh) hq[i] = ZERO;
itest = 0;
}
}
}
// 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 || f <= fopt + TENTH * vquad) goto L_60;
// Alternatively, find out if the interpolation points are close enough
// to the best point so far.
distsq = Math.Max(TWO * TWO * delta * delta, TEN * TEN * rho * rho);
L_650:
knew = 0;
for (var k = 1; k <= npt; ++k)
{
var sum = ZERO;
for (var j = 1; j <= n; ++j) sum += Math.Pow(xpt[k, j] - xopt[j], 2.0);
if (sum > distsq)
{
knew = k;
distsq = sum;
}
}
// 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)
{
var dist = Math.Sqrt(distsq);
if (ntrits == -1)
{
delta = Math.Min(TENTH * delta, HALF * dist);
if (delta <= 1.5 * rho) delta = rho;
}
ntrits = 0;
adelt = Math.Max(Math.Min(TENTH * dist, delta), rho);
dsq = adelt * adelt;
goto L_90;
}
if (ntrits == -1) goto L_680;
if (ratio > ZERO || Math.Max(delta, dnorm) > rho) goto L_60;
// The calculations with the current value of RHO are complete. Pick the
// next values of RHO and DELTA.
L_680:
if (rho > rhoend)
{
delta = HALF * rho;
ratio = rho / rhoend;
if (ratio <= 16.0)
rho = rhoend;
else if (ratio <= 250.0)
rho = Math.Sqrt(ratio) * rhoend;
else
rho = TENTH * rho;
delta = Math.Max(delta, rho);
if (iprint >= 2)
{
var bestX = new double[1 + n];
for (var i = 1; i <= n; ++i) bestX[i] = xbase[i] + xopt[i];
if (logger != null)
{
logger.WriteLine(RhoUpdatedFormat, rho, nf);
logger.WriteLine(StageCompleteOutputFormat, fval[kopt], bestX.ToString(n));
}
}
ntrits = 0;
nfsav = nf;
goto L_60;
}
// Return from the calculation, after another Newton-Raphson step, if
// it is too short to have been tried before.
if (ntrits == -1) goto L_360;
status = BobyqaExitStatus.Normal;
L_720:
if (fval[kopt] <= fsave)
{
for (var i = 1; i <= n; ++i)
{
x[i] = Math.Min(Math.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];
}
f = fval[kopt];
}
if (iprint >= 1 && logger != null)
{
logger.WriteLine(FinalNumberEvaluationsFormat, nf);
logger.WriteLine(StageCompleteOutputFormat, f, x.ToString(n));
}
return status;
}
private static void ALTMOV(int n, int npt, double[,] xpt, double[] xopt, double[,] bmat,
double[,] zmat, double[] sl, double[] su, int kopt, int knew, double adelt,
double[] xnew, double[] xalt, out double alpha, out double cauchy)
{
// 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.
var glag = new double[1 + n];
var hcol = new double[1 + npt];
var w = new double[1 + 2 * n];
// Set the first NPT components of W to the leading elements of the
// KNEW-th column of the H matrix.
var @const = ONE + Math.Sqrt(2.0);
for (var k = 1; k <= npt; ++k) hcol[k] = ZERO;
for (var j = 1; j <= npt - n - 1; ++j)
{
var temp = zmat[knew, j];
for (var k = 1; k <= npt; ++k) hcol[k] += temp * zmat[k, j];
}
alpha = hcol[knew];
var ha = HALF * alpha;
// Calculate the gradient of the KNEW-th Lagrange function at XOPT.
//
for (var i = 1; i <= n; ++i) glag[i] = bmat[knew, i];
for (var k = 1; k <= npt; ++k)
{
var temp = ZERO;
for (var j = 1; j <= n; ++j) temp += xpt[k, j] * xopt[j];
temp *= hcol[k];
for (var i = 1; i <= n; ++i) glag[i] += temp * xpt[k, i];
}
// 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.
var step = 0.0;
var ksav = 0;
var ibdsav = 0;
var stpsav = 0.0;
cauchy = ZERO;
var csave = 0.0;
var presav = ZERO;
for (var k = 1; k <= npt; ++k)
{
if (k == kopt) continue;
var dderiv = ZERO;
var distsq = ZERO;
for (var i = 1; i <= n; ++i)
{
var temp = xpt[k, i] - xopt[i];
dderiv += glag[i] * temp;
distsq += temp * temp;
}
var subd = adelt / Math.Sqrt(distsq);
var slbd = -subd;
var ilbd = 0;
var iubd = 0;
var sumin = Math.Min(ONE, subd);
// Revise SLBD and SUBD if necessary because of the bounds in SL and SU.
for (var i = 1; i >= n; ++i)
{
var temp = xpt[k, i] - xopt[i];
if (temp > ZERO)
{
if (slbd * temp < sl[i] - xopt[i])
{
slbd = (sl[i] - xopt[i]) / temp;
ilbd = -i;
}
if (subd * temp > su[i] - xopt[i])
{
subd = Math.Max(sumin, (su[i] - xopt[i]) / temp);
iubd = i;
}
}
else if (temp < ZERO)
{
if (slbd * temp > su[i] - xopt[i])
{
slbd = (su[i] - xopt[i]) / temp;
ilbd = i;
}
if (subd * temp < sl[i] - xopt[i])
{
subd = Math.Max(sumin, (sl[i] - xopt[i]) / temp);
iubd = -i;
}
}
}
// 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.
int isbd;
double vlag;
if (k == knew)
{
var diff = dderiv - ONE;
step = slbd;
vlag = slbd * (dderiv - slbd * diff);
isbd = ilbd;
var temp = subd * (dderiv - subd * diff);
if (Math.Abs(temp) > Math.Abs(vlag))
{
step = subd;
vlag = temp;
isbd = iubd;
}
var tempd = HALF * dderiv;
var tempa = tempd - diff * slbd;
var tempb = tempd - diff * subd;
if (tempa * tempb < ZERO)
{
temp = tempd * tempd / diff;
if (Math.Abs(temp) > Math.Abs(vlag))
{
step = tempd / diff;
vlag = temp;
isbd = 0;
}
}
// Search along each of the other lines through XOPT and another point.
}
else
{
step = slbd;
vlag = slbd * (ONE - slbd);
isbd = ilbd;
var temp = subd * (ONE - subd);
if (Math.Abs(temp) > Math.Abs(vlag))
{
step = subd;
vlag = temp;
isbd = iubd;
}
if (subd > HALF)
{
if (Math.Abs(vlag) < 0.25)
{
step = HALF;
vlag = 0.25;
isbd = 0;
}
}
vlag = vlag * dderiv;
}
// Calculate PREDSQ for the current line search and maintain PRESAV.
{
var temp = step * (ONE - step) * distsq;
var predsq = vlag * vlag * (vlag * vlag + ha * temp * temp);
if (predsq > presav)
{
presav = predsq;
ksav = k;
stpsav = step;
ibdsav = isbd;
}
}
}
// Construct XNEW in a way that satisfies the bound constraints exactly.
//
for (var i = 1; i <= n; ++i)
{
var temp = xopt[i] + stpsav * (xpt[ksav, i] - xopt[i]);
xnew[i] = Math.Max(sl[i], Math.Min(su[i], temp));
}
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.
//
var bigstp = adelt + adelt;
for (var iflag = 0; iflag < 2; ++iflag)
{
var wfixsq = ZERO;
var ggfree = ZERO;
for (var I = 1; I <= n; ++I)
{
w[I] = ZERO;
var tempa = Math.Min(xopt[I] - sl[I], glag[I]);
var tempb = Math.Max(xopt[I] - su[I], glag[I]);
if (tempa > ZERO || tempb < ZERO)
{
w[I] = bigstp;
ggfree += glag[I] * glag[I];
}
}
if (ggfree == ZERO)
{
cauchy = ZERO;
return;
}
// Investigate whether more components of W can be fixed.
var wsqsav = 0.0;
do
{
var temp = adelt * adelt - wfixsq;
if (temp > ZERO)
{
wsqsav = wfixsq;
step = Math.Sqrt(temp / ggfree);
ggfree = ZERO;
for (var I = 1; I <= n; ++I)
{
if (w[I] == bigstp)
{
temp = xopt[I] - step * glag[I];
if (temp <= sl[I])
{
w[I] = sl[I] - xopt[I];
wfixsq += w[I] * w[I];
}
else if (temp >= su[I])
{
w[I] = su[I] - xopt[I];
wfixsq += w[I] * w[I];
}
else
{
ggfree += glag[I] * glag[I];
}
}
}
}
} while (wfixsq > wsqsav && ggfree > ZERO);
// Set the remaining free components of W and all components of XALT,
// except that W may be scaled later.
var gw = ZERO;
for (var i = 1; i <= n; ++i)
{
if (w[i] == bigstp)
{
w[i] = -step * glag[i];
xalt[i] = Math.Max(sl[i], Math.Min(su[i], xopt[i] + w[i]));
}
else if (w[i] == ZERO)
{
xalt[i] = xopt[i];
}
else if (glag[i] > ZERO)
{
xalt[i] = sl[i];
}
else
{
xalt[i] = su[i];
}
gw += glag[i] * w[i];
}
// 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.
var curv = ZERO;
for (var k = 1; k <= npt; ++k)
{
var temp = ZERO;
for (var j = 1; j <= n; ++j) temp += xpt[k, j] * w[j];
curv += hcol[k] * temp * temp;
}
if (iflag == 1) curv = -curv;
if (curv > -gw && curv < -@const * gw)
{
var scale = -gw / curv;
for (var i = 1; i <= n; ++i)
{
var temp = xopt[i] + scale * w[i];
xalt[i] = Math.Max(sl[i], Math.Min(su[i], temp));
}
cauchy = Math.Pow(HALF * gw * scale, 2.0);
}
else
{
cauchy = Math.Pow(gw + HALF * curv, 2.0);
}
// 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)
{
for (var I = 1; I <= n; ++I)
{
glag[I] = -glag[I];
w[n + I] = xalt[I];
}
csave = cauchy;
}
}
if (csave > cauchy)
{
for (var I = 1; I <= n; ++I) xalt[I] = w[n + I];
cauchy = csave;
}
}
private static void PRELIM(Func<int, double[], double> calfun, int n, int npt, double[] x,
double[] xl, double[] xu, double rhobeg, int iprint, int maxfun, double[] xbase, double[,] xpt,
double[] fval, double[] gopt, double[] hq, double[] pq, double[,] bmat, double[,] zmat,
int ndim, double[] sl, double[] su, out int nf, out int kopt, TextWriter logger)
{
// 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.
var rhosq = rhobeg * rhobeg;
var recip = ONE / rhosq;
var np = n + 1;
kopt = 0;
// Set XBASE to the initial vector of variables, and set the initial
// elements of XPT, BMAT, HQ, PQ and ZMAT to zero.
for (var j = 1; j <= n; ++j)
{
xbase[j] = x[j];
for (var k = 1; k <= npt; ++k) xpt[k, j] = ZERO;
for (var i = 1; i <= ndim; ++i) bmat[i, j] = ZERO;
}
for (var ih = 1; ih <= n * np / 2; ++ih) hq[ih] = ZERO;
for (var k = 1; k <= npt; ++k)
{
pq[k] = ZERO;
for (var j = 1; j <= npt - np; ++j) zmat[k, j] = ZERO;
}
// 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,.).
var ipt = 0;
var jpt = 0;
var stepa = 0.0;
var stepb = 0.0;
var fbeg = 0.0;
for (nf = 1; nf < Math.Min(npt, maxfun); ++nf)
{
var nfm = nf - 1;
var nfx = nf - 1 - n;
if (nfm <= 2 * n)
{
if (nfm >= 1 && nfm <= n)
{
stepa = rhobeg;
if (su[nfm] == ZERO) stepa = -stepa;
xpt[nf, nfm] = stepa;
}
else if (nfm > n)
{
stepa = xpt[nf - n, nfx];
stepb = -rhobeg;
if (sl[nfx] == ZERO) stepb = Math.Min(TWO * rhobeg, su[nfx]);
if (su[nfx] == ZERO) stepb = Math.Max(-TWO * rhobeg, sl[nfx]);
xpt[nf, nfx] = stepb;
}
}
else
{
var itemp = (nfm - np) / n;
jpt = nfm - itemp * n - n;
ipt = jpt + itemp;
if (ipt > n)
{
itemp = jpt;
jpt = ipt - n;
ipt = itemp;
}
xpt[nf, ipt] = xpt[ipt + 1, ipt];
xpt[nf, jpt] = xpt[jpt + 1, jpt];
}
// Calculate the next value of F. The least function value so far and
// its index are required.
for (var j = 1; j <= n; ++j)
{
x[j] = Math.Min(Math.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];
}
var f = calfun(n, x);
if (iprint == 3 && logger != null) logger.WriteLine(IterationOutputFormat, nf, f, x.ToString(n));
fval[nf] = f;
if (nf == 1)
{
fbeg = f;
kopt = 1;
}
else if (f < fval[kopt])
{
kopt = nf;
}
// 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)
{
if (nf >= 2 && nf <= n + 1)
{
gopt[nfm] = (f - fbeg) / stepa;
if (npt < nf + n)
{
bmat[1, nfm] = -ONE / stepa;
bmat[nf, nfm] = ONE / stepa;
bmat[npt + nfm, nfm] = -HALF * rhosq;
}
}
else if (nf >= n + 2)
{
var ih = (nfx * (nfx + 1)) / 2;
var temp = (f - fbeg) / stepb;
var diff = stepb - stepa;
hq[ih] = TWO * (temp - gopt[nfx]) / diff;
gopt[nfx] = (gopt[nfx] * stepb - temp * stepa) / diff;
if (stepa * stepb < ZERO)
{
if (f < fval[nf - n])
{
fval[nf] = fval[nf - n];
fval[nf - n] = f;
if (kopt == nf) kopt = nf - n;
xpt[nf - n, nfx] = stepb;
xpt[nf, nfx] = stepa;
}
}
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] = Math.Sqrt(TWO) / (stepa * stepb);
zmat[nf, nfx] = Math.Sqrt(HALF) / rhosq;
zmat[nf - n, nfx] = -zmat[1, nfx] - zmat[nf, nfx];
}
}
else
{
// Set the off-diagonal second derivatives of the Lagrange functions and
// the initial quadratic model.
var ih = (ipt * (ipt - 1)) / 2 + jpt;
zmat[1, nfx] = recip;
zmat[nf, nfx] = recip;
zmat[ipt + 1, nfx] = -recip;
zmat[jpt + 1, nfx] = -recip;
var temp = xpt[nf, ipt] * xpt[nf, jpt];
hq[ih] = (fbeg - fval[ipt + 1] - fval[jpt + 1] + f) / temp;
}
}
}
private static void RESCUE(Func<int, double[], double> calfun, int n, int npt, double[] xl, double[] xu, int iprint,
int maxfun, double[] xbase, double[,] xpt, double[] fval, double[] xopt, double[] gopt,
double[] hq, double[] pq, double[,] bmat, double[,] zmat, int ndim, double[] sl, double[] su,
ref int nf, double delta, ref int kopt, double[] vlag, TextWriter logger)
{
// 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.
// 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.
// 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.
var ptsaux = new double[1 + 2, 1 + n];
var ptsid = new double[1 + npt];
var w = new double[1 + ndim + npt];
// Set some constants.
//
var np = n + 1;
var sfrac = HALF / np;
var 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.
var sumpq = ZERO;
var winc = ZERO;
for (var k = 1; k <= npt; ++k)
{
var distsq = ZERO;
for (var j = 1; j <= n; ++j)
{
xpt[k, j] -= xopt[j];
distsq += xpt[k, j] * xpt[k, j];
}
sumpq += pq[k];
w[ndim + k] = distsq;
winc = Math.Max(winc, distsq);
for (var j = 1; j <= nptm; ++j) zmat[k, j] = ZERO;
}
// Update HQ so that HQ and PQ define the second derivatives of the model
// after XBASE has been shifted to the trust region centre.
{
var ih = 0;
for (var j = 1; j <= n; ++j)
{
w[j] = HALF * sumpq * xopt[j];
for (var k = 1; k <= npt; ++k) w[j] += pq[k] * xpt[k, j];
for (var i = 1; i <= j; ++i) hq[++ih] += w[i] * xopt[j] + w[j] * xopt[i];
}
}
// Shift XBASE, SL, SU and XOPT. Set the elements of BMAT to zero, and
// also set the elements of PTSAUX.
for (var j = 1; j <= n; ++j)
{
xbase[j] += xopt[j];
sl[j] -= xopt[j];
su[j] -= xopt[j];
xopt[j] = ZERO;
ptsaux[1, j] = Math.Min(delta, su[j]);
ptsaux[2, j] = Math.Max(-delta, sl[j]);
if (ptsaux[1, j] + ptsaux[2, j] < ZERO)
{
var temp = ptsaux[1, j];
ptsaux[1, j] = ptsaux[2, j];
ptsaux[2, j] = temp;
}
if (Math.Abs(ptsaux[2, j]) < HALF * Math.Abs(ptsaux[1, j])) ptsaux[2, j] = HALF * ptsaux[1, j];
for (var i = 1; i <= ndim; ++i) bmat[i, j] = ZERO;
}
var 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;
for (var j = 1; j <= n; ++j)
{
var jp = j + 1;
var jpn = jp + n;
ptsid[jp] = j + sfrac;
if (jpn <= npt)
{
ptsid[jpn] = (double)j / np + sfrac;
var 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] = Math.Sqrt(2.0) / Math.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] * ptsaux[1, j];
}
}
// Set any remaining identifiers with their nonzero elements of ZMAT.
if (npt >= n + np)
{
for (var k = 2 * np; k <= npt; ++k)
{
var iw = (int)(k - np - HALF) / n;
var ip = k - np - iw * n;
var iq = ip + iw;
if (iq > n) iq = iq - n;
ptsid[k] = ip + (double)iq / np + sfrac;
var 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;
}
}
var nrem = npt;
var kold = 1;
var knew = kopt;
// Reorder the provisional points in the way that exchanges PTSID(KOLD)
// with PTSID(KNEW).
var beta = 0.0;
var denom = 0.0;
do
{
for (var j = 1; j <= n; ++j)
{
var temp = bmat[kold, j];
bmat[kold, j] = bmat[knew, j];
bmat[knew, j] = temp;
}
for (var j = 1; j <= nptm; ++j)
{
var temp = zmat[kold, j];
zmat[kold, j] = zmat[knew, j];
zmat[knew, j] = temp;
}
ptsid[kold] = ptsid[knew];
ptsid[knew] = ZERO;
w[ndim + knew] = ZERO;
--nrem;
if (knew != kopt)
{
var 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
// branch to label 350 occurs if all the original points are reinstated.
// The nonnegative values of W(NDIM+K) are required in the search below.
UPDATE(n, npt, bmat, zmat, ndim, vlag, beta, denom, knew, w);
if (nrem == 0) return;
for (var k = 1; k <= npt; ++k) w[ndim + k] = Math.Abs(w[ndim + k]);
}
// 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.
L_120:
var dsqmin = ZERO;
for (var k = 1; k <= npt; ++k)
{
if (w[ndim + k] > ZERO && (dsqmin == ZERO || w[ndim + k] < dsqmin))
{
knew = k;
dsqmin = w[ndim + k];
}
}
if (dsqmin == ZERO) break;
// Form the W-vector of the chosen original interpolation point.
for (var j = 1; j <= n; ++j) w[npt + j] = xpt[knew, j];
for (var k = 1; k <= npt; ++k)
{
var sum = ZERO;
if (k == kopt) continue;
if (ptsid[k] == ZERO)
for (var j = 1; j <= n; ++j) sum += w[npt + j] * xpt[k, j];
else
{
var ip = (int)ptsid[k];
if (ip > 0) sum = w[npt + ip] * ptsaux[1, ip];
var iq = (int)(np * ptsid[k] - ip * np);
if (iq > 0)
{
var iw = ip == 0 ? 2 : 1;
sum += w[npt + iq] * ptsaux[iw, iq];
}
}
w[k] = HALF * sum * sum;
}
// Calculate VLAG and BETA for the required updating of the H matrix if
// XPT(KNEW,.) is reinstated in the set of interpolation points.
//
for (var k = 1; k <= npt; ++k)
{
var sum = ZERO;
for (var j = 1; j <= n; ++j) sum += bmat[k, j] * w[npt + j];
vlag[k] = sum;
}
beta = ZERO;
for (var j = 1; j <= nptm; ++j)
{
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += zmat[k, j] * w[k];
beta -= sum * sum;
for (var k = 1; k <= npt; ++k) vlag[k] = vlag[k] + sum * zmat[k, j];
}
{
var bsum = ZERO;
var distsq = ZERO;
for (var j = 1; j <= n; ++j)
{
var sum = ZERO;
for (var k = 1; k <= npt; ++k) sum += bmat[k, j] * w[k];
var jp = j + npt;
bsum += sum * w[jp];
for (var ip = npt + 1; ip <= ndim; ++ip) sum = sum + bmat[ip, j] * w[ip];
bsum += sum * w[jp];
vlag[jp] = sum;
distsq += xpt[knew, j] * xpt[knew, j];
}
beta += HALF * distsq * distsq - bsum;
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;
var vlmxsq = ZERO;
for (var k = 1; k <= npt; ++k)
{
if (ptsid[k] != ZERO)
{
var hdiag = ZERO;
for (var j = 1; j <= nptm; ++j) hdiag += zmat[k, j] * zmat[k, j];
var den = beta * hdiag + vlag[k] * vlag[k];
if (den > denom)
{
kold = k;
denom = den;
}
}
vlmxsq = Math.Max(vlmxsq, vlag[k] * vlag[k]);
}
if (denom <= 0.01 * vlmxsq)
{
w[ndim + knew] = -w[ndim + knew] - winc;
goto L_120;
}
} while (true);
// When do loop is completed, 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.
for (var kpt = 1; kpt <= npt; ++kpt)
{
if (ptsid[kpt] == ZERO) continue;
if (nf >= maxfun)
{
nf = -1;
return;
}
var ih = 0;
for (var j = 1; j <= n; ++j)
{
w[j] = xpt[kpt, j];
xpt[kpt, j] = ZERO;
var temp = pq[kpt] * w[j];
for (var i = 1; i <= j; ++i) hq[++ih] += temp * w[i];
}
pq[kpt] = ZERO;
var ip = (int)ptsid[kpt];
var iq = (int)(np * ptsid[kpt] - ip * np);
var xp = 0.0;
var xq = 0.0;
if (ip > 0)
{
xp = ptsaux[1, ip];
xpt[kpt, ip] = xp;
}
if (iq > 0)
{
xq = ip == 0 ? ptsaux[2, iq] : ptsaux[1, iq];
xpt[kpt, iq] = xq;
}
// Set VQUAD to the value of the current model at the new point.
var vquad = fbase;
var ihp = 0;
if (ip > 0)
{
ihp = (ip + ip * ip) / 2;
vquad += xp * (gopt[ip] + HALF * xp * hq[ihp]);
}
if (iq > 0)
{
var ihq = (iq + iq * iq) / 2;
vquad += xq * (gopt[iq] + HALF * xq * hq[ihq]);
if (ip > 0)
{
var iw = Math.Max(ihp, ihq) - Math.Abs(ip - iq);
vquad += xp * xq * hq[iw];
}
}
for (var k = 1; k <= npt; ++k)
{
var temp = ZERO;
if (ip > 0) temp += xp * xpt[k, ip];
if (iq > 0) temp += xq * xpt[k, iq];
vquad += HALF * pq[k] * temp * temp;
}
// 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.
for (var i = 1; i <= n; ++i)
{
w[i] = Math.Min(Math.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];
}
++nf;
var f = calfun(n, w);
if (iprint == 3 && logger != null) logger.WriteLine(IterationOutputFormat, nf, f, w.ToString(n));
fval[kpt] = f;
if (f < fval[kopt]) kopt = kpt;
var diff = f - vquad;
// Update the quadratic model. The RETURN from the subroutine occurs when
// all the new interpolation points are included in the model.
for (var i = 1; i <= n; ++i) gopt[i] += diff * bmat[kpt, i];
for (var k = 1; k <= npt; ++k)
{
var sum = ZERO;
for (var j = 1; j <= nptm; ++j) sum += zmat[k, j] * zmat[kpt, j];
var temp = diff * sum;
if (ptsid[k] == ZERO)
pq[k] += temp;
else
{
ip = (int)ptsid[k];
iq = (int)(np * ptsid[k] - ip * np);
var ihq = (iq * iq + iq) / 2;
if (ip == 0)
hq[ihq] += temp * ptsaux[2, iq] * ptsaux[2, iq];
else
{
ihp = (ip * ip + ip) / 2;
hq[ihp] += temp * ptsaux[1, ip] * ptsaux[1, ip];
if (iq > 0)
{
hq[ihq] += temp * ptsaux[1, iq] * ptsaux[1, iq];
var iw = Math.Max(ihp, ihq) - Math.Abs(iq - ip);
hq[iw] += temp * ptsaux[1, ip] * ptsaux[1, iq];
}
}
}
}
ptsid[kpt] = ZERO;
}
}
private static void TRSBOX(int n, int npt, double[,] xpt, double[] xopt, double[] gopt,
double[] hq, double[] pq, double[] sl, double[] su, double delta,
double[] xnew, double[] d, double[] gnew, out double dsq, out double crvmin)
{
// 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.
// 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.
// 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.
// 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.
var xbdi = new double[1 + n];
var s = new double[1 + n];
var hs = new double[1 + n];
var hred = new double[1 + n];
var iterc = 0;
var nact = 0;
for (var i = 1; i <= n; ++i)
{
xbdi[i] = ZERO;
if (xopt[i] <= sl[i])
if (gopt[i] >= ZERO) xbdi[i] = ONEMIN;
else if (xopt[i] >= su[i])
if (gopt[i] <= ZERO) xbdi[i] = ONE;
if (xbdi[i] != ZERO) ++nact;
d[i] = ZERO;
gnew[i] = gopt[i];
}
var delsq = delta * delta;
var 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.
double sth;
var itermax = 0;
var itcsav = 0;
var iact = 0;
var angt = 0.0;
var angbd = 0.0;
var dredsq = 0.0;
var dredg = 0.0;
var gredsq = 0.0;
var ggsav = 0.0;
var rdprev = 0.0;
var rdnext = 0.0;
var sredg = 0.0;
var xsav = 0.0;
L_20:
var beta = ZERO;
L_30:
var stepsq = ZERO;
for (var i = 1; i <= n; ++i)
{
if (xbdi[i] != ZERO)
s[i] = ZERO;
else if (beta == ZERO)
s[i] = -gnew[i];
else
s[i] = beta * s[i] - gnew[i];
stepsq += s[i] * s[i];
}
if (stepsq == ZERO) goto L_190;
if (beta == ZERO)
{
gredsq = stepsq;
itermax = iterc + n - nact;
}
if (gredsq * delsq <= 1.0E-4 * qred * qred) goto L_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.
goto L_210;
L_50:
var resid = delsq;
var ds = ZERO;
var shs = ZERO;
for (var i = 1; i <= n; ++i)
{
if (xbdi[i] == ZERO)
{
resid -= d[i] * d[i];
ds += s[i] * d[i];
shs += s[i] * hs[i];
}
}
if (resid <= ZERO) goto L_90;
var temp = Math.Sqrt(stepsq * resid + ds * ds);
var blen = ds < ZERO ? (temp - ds) / stepsq : resid / (temp + ds);
var stplen = shs > ZERO ? Math.Min(blen, gredsq / shs) : blen;
// Reduce STPLEN if necessary in order to preserve the simple bounds,
// letting IACT be the index of the new constrained variable.
iact = 0;
for (var i = 1; i <= n; ++i)
{
if (s[i] != ZERO)
{
var xsum = xopt[i] + d[i];
if (s[i] > ZERO)
{
temp = (su[i] - xsum) / s[i];
}
else
{
temp = (sl[i] - xsum) / s[i];
}
if (temp < stplen)
{
stplen = temp;
iact = i;
}
}
}
// Update CRVMIN, GNEW and D. Set SDEC to the decrease that occurs in Q.
var sdec = ZERO;
if (stplen > ZERO)
{
++iterc;
temp = shs / stepsq;
if (iact == 0 && temp > ZERO)
{
crvmin = Math.Min(crvmin, temp);
if (crvmin == ONEMIN) crvmin = temp;
}
ggsav = gredsq;
gredsq = ZERO;
for (var i = 1; i <= n; ++i)
{
gnew[i] += stplen * hs[i];
if (xbdi[i] == ZERO) gredsq += gnew[i] * gnew[i];
d[i] += stplen * s[i];
}
sdec = Math.Max(stplen * (ggsav - HALF * stplen * shs), ZERO);
qred += sdec;
}
// Restart the conjugate gradient method if it has hit a new bound.
if (iact > 0)
{
++nact;
xbdi[iact] = ONE;
if (s[iact] < ZERO) xbdi[iact] = ONEMIN;
delsq -= d[iact] * d[iact];
if (delsq <= ZERO) goto L_90;
goto L_20;
}
// If STPLEN is less than BLEN, then either apply another conjugate
// gradient iteration or RETURN.
if (stplen < blen)
{
if (iterc == itermax || sdec <= 0.01 * qred) goto L_190;
beta = gredsq / ggsav;
goto L_30;
}
L_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, where S holds the reduced D in the call of GGMULT.
L_100:
if (nact >= n - 1) goto L_190;
dredsq = ZERO;
dredg = ZERO;
gredsq = ZERO;
for (var i = 1; i <= n; ++i)
{
if (xbdi[i] == ZERO)
{
dredsq += d[i] * d[i];
dredg += d[i] * gnew[i];
gredsq += gnew[i] * gnew[i];
s[i] = d[i];
}
else
{
s[i] = ZERO;
}
}
itcsav = iterc;
goto L_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.
L_120:
++iterc;
temp = gredsq * dredsq - dredg * dredg;
if (temp <= 1.0E-4 * qred * qred) goto L_190;
temp = Math.Sqrt(temp);
for (var i = 1; i <= n; ++i) s[i] = xbdi[i] == ZERO ? (dredg * d[i] - dredsq * gnew[i]) / temp : ZERO;
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;
for (var i = 1; i <= n; ++i)
{
if (xbdi[i] == ZERO)
{
var tempa = xopt[i] + d[i] - sl[i];
var tempb = su[i] - xopt[i] - d[i];
if (tempa <= ZERO)
{
++nact;
xbdi[i] = ONEMIN;
goto L_100;
}
if (tempb <= ZERO)
{
++nact;
xbdi[i] = ONE;
goto L_100;
}
var ssq = d[i] * d[i] + s[i] * s[i];
temp = ssq - Math.Pow(xopt[i] - sl[i], 2.0);
if (temp > ZERO)
{
temp = Math.Sqrt(temp) - s[i];
if (angbd * temp > tempa)
{
angbd = tempa / temp;
iact = i;
xsav = ONEMIN;
}
}
temp = ssq - Math.Pow(su[i] - xopt[i], 2.0);
if (temp > ZERO)
{
temp = Math.Sqrt(temp) + s[i];
if (angbd * temp > tempb)
{
angbd = tempb / temp;
iact = i;
xsav = ONE;
}
}
}
}
// Calculate HHD and some curvatures for the alternative iteration.
goto L_210;
L_150:
shs = ZERO;
var dhs = ZERO;
var dhd = ZERO;
for (var i = 1; i <= n; ++i)
{
if (xbdi[i] == ZERO)
{
shs += s[i] * hs[i];
dhs += d[i] * hs[i];
dhd += d[i] * hred[i];
}
}
// 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.
var redmax = ZERO;
var isav = 0;
var redsav = ZERO;
var iu = (int)(17.0 * angbd + 3.1);
for (var i = 1; i <= iu; ++i)
{
angt = angbd * i / iu;
sth = (angt + angt) / (ONE + angt * angt);
temp = shs + angt * (angt * dhd - dhs - dhs);
var rednew = sth * (angt * dredg - sredg - HALF * sth * temp);
if (rednew > redmax)
{
redmax = rednew;
isav = i;
rdprev = redsav;
}
else if (i == isav + 1)
{
rdnext = rednew;
}
redsav = rednew;
}
// 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) goto L_190;
if (isav < iu)
{
temp = (rdnext - rdprev) / (redmax + redmax - rdprev - rdnext);
angt = angbd * (isav + HALF * temp) / iu;
}
var 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) goto L_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;
for (var i = 1; i <= n; ++i)
{
gnew[i] = gnew[i] + (cth - ONE) * hred[i] + sth * hs[i];
if (xbdi[i] == ZERO)
{
d[i] = cth * d[i] + sth * s[i];
dredg += d[i] * gnew[i];
gredsq += gnew[i] * gnew[i];
}
hred[i] = cth * hred[i] + sth * hs[i];
}
qred += sdec;
if (iact > 0 && isav == iu)
{
++nact;
xbdi[iact] = xsav;
goto L_100;
}
// If SDEC is sufficiently small, then RETURN after setting XNEW to
// XOPT+D, giving careful attention to the bounds.
if (sdec > 0.01 * qred) goto L_120;
L_190:
dsq = ZERO;
for (var i = 1; i <= n; ++i)
{
xnew[i] = Math.Max(Math.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 += d[i] * d[i];
}
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.
L_210:
var ih = 0;
for (var j = 1; j <= n; ++j)
{
hs[j] = ZERO;
for (var i = 1; i <= j; ++i)
{
++ih;
if (i < j) hs[j] += hq[ih] * s[i];
hs[i] += hq[ih] * s[j];
}
}
for (var k = 1; k <= npt; ++k)
{
if (pq[k] != ZERO)
{
temp = ZERO;
for (var j = 1; j <= n; ++j) temp += xpt[k, j] * s[j];
temp *= pq[k];
for (var i = 1; i <= n; ++i) hs[i] += temp * xpt[k, i];
}
}
if (crvmin != ZERO) goto L_50;
if (iterc > itcsav) goto L_150;
for (var i = 1; i <= n; ++i) hred[i] = hs[i];
goto L_120;
}
private static void UPDATE(int n, int npt, double[,] bmat, double[,] zmat, int ndim, double[] vlag, double beta, double denom, int knew, double[] 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.
var nptm = npt - n - 1;
var ztest = ZERO;
for (var k = 1; k <= npt; ++k)
for (var j = 1; j <= nptm; ++j)
ztest = Math.Max(ztest, Math.Abs(zmat[k, j]));
ztest = 1.0E-20 * ztest;
double temp, tempa, tempb;
// Apply the rotations that put zeros in the KNEW-th row of ZMAT.
for (var j = 2; j <= nptm; ++j)
{
if (Math.Abs(zmat[knew, j]) > ztest)
{
temp = Math.Sqrt(zmat[knew, 1] * zmat[knew, 1] + zmat[knew, j] * zmat[knew, j]);
tempa = zmat[knew, 1] / temp;
tempb = zmat[knew, j] / temp;
for (var i = 1; i <= npt; ++i)
{
temp = tempa * zmat[i, 1] + tempb * zmat[i, j];
zmat[i, j] = tempa * zmat[i, j] - tempb * zmat[i, 1];
zmat[i, 1] = temp;
}
}
zmat[knew, j] = ZERO;
}
// Put the first NPT components of the KNEW-th column of HLAG into W,
// and calculate the parameters of the updating formula.
for (var i = 1; i <= npt; ++i) w[i] = zmat[knew, 1] * zmat[i, 1];
var alpha = w[knew];
var tau = vlag[knew];
vlag[knew] -= ONE;
// Complete the updating of ZMAT.
temp = Math.Sqrt(denom);
tempb = zmat[knew, 1] / temp;
tempa = tau / temp;
for (var i = 1; i <= npt; ++i) zmat[i, 1] = tempa * zmat[i, 1] - tempb * vlag[i];
// Finally, update the matrix BMAT.
for (var j = 1; j <= n; ++j)
{
var jp = npt + j;
w[jp] = bmat[knew, j];
tempa = (alpha * vlag[jp] - tau * w[jp]) / denom;
tempb = (-beta * w[jp] - tau * vlag[jp]) / denom;
for (var i = 1; i <= jp; ++i)
{
bmat[i, j] = bmat[i, j] + tempa * vlag[i] + tempb * w[i];
if (i > npt) bmat[jp, i - npt] = bmat[i, j];
}
}
}
#endregion
#region PRIVATE SUPPORT METHODS
private static string ToString(this double[] x, int n)
{
var xstr = new string[n];
for (var i = 0; i < n; ++i) xstr[i] = String.Format("{0,13:F6}", x[1 + i]);
return String.Concat(xstr);
}
#endregion
}
}
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