void dcauchy(int n, double *x, double *xl, double *xu, double *A, double *g, double delta, double *alpha, double *s)
{
/*
c **********
c
c Subroutine dcauchy
c
c This subroutine computes a Cauchy step that satisfies a trust
c region constraint and a sufficient decrease condition.
c
c The Cauchy step is computed for the quadratic
c
c q(s) = 0.5*s'*A*s + g'*s,
c
c where A is a symmetric matrix , and g is a vector. Given a
c parameter alpha, the Cauchy step is
c
c s[alpha] = P[x - alpha*g] - x,
c
c with P the projection onto the n-dimensional interval [xl,xu].
c The Cauchy step satisfies the trust region constraint and the
c sufficient decrease condition
c
c || s || <= delta, q(s) <= mu_0*(g'*s),
c
c where mu_0 is a constant in (0,1).
c
c parameters:
c
c n is an integer variable.
c On entry n is the number of variables.
c On exit n is unchanged.
c
c x is a double precision array of dimension n.
c On entry x specifies the vector x.
c On exit x is unchanged.
c
c xl is a double precision array of dimension n.
c On entry xl is the vector of lower bounds.
c On exit xl is unchanged.
c
c xu is a double precision array of dimension n.
c On entry xu is the vector of upper bounds.
c On exit xu is unchanged.
c
c A is a double precision array of dimension n*n.
c On entry A specifies the matrix A.
c On exit A is unchanged.
c
c g is a double precision array of dimension n.
c On entry g specifies the gradient g.
c On exit g is unchanged.
c
c delta is a double precision variable.
c On entry delta is the trust region size.
c On exit delta is unchanged.
c
c alpha is a double precision variable.
c On entry alpha is the current estimate of the step.
c On exit alpha defines the Cauchy step s[alpha].
c
c s is a double precision array of dimension n.
c On entry s need not be specified.
c On exit s is the Cauchy step s[alpha].
c
c **********
*/
double one = 1, zero = 0;
/* Constant that defines sufficient decrease.
Interpolation and extrapolation factors. */
double mu0 = 0.01, interpf = 0.1, extrapf = 10;
int search, interp, nbrpt, nsteps = 1, i, inc = 1;
double alphas, brptmax, brptmin, gts, q;
double *wa = (double *) xmalloc(sizeof(double)*n);
/* Find the minimal and maximal break-point on x - alpha*g. */
for (i=0;i<n;i++)
wa[i] = -g[i];
dbreakpt(n, x, xl, xu, wa, &nbrpt, &brptmin, &brptmax);
/* Evaluate the initial alpha and decide if the algorithm
must interpolate or extrapolate. */
dgpstep(n, x, xl, xu, -(*alpha), g, s);
if (dnrm2_(&n, s, &inc) > delta)
interp = 1;
else
{
dsymv_("U", &n, &one, A, &n, s, &inc, &zero, wa, &inc);
gts = ddot_(&n, g, &inc, s, &inc);
q = 0.5*ddot_(&n, s, &inc, wa, &inc) + gts;
interp = q >= mu0*gts ? 1 : 0;
}
/* Either interpolate or extrapolate to find a successful step. */
if (interp)
{
/* Reduce alpha until a successful step is found. */
search = 1;
while (search)
{
/* This is a crude interpolation procedure that
will be replaced in future versions of the code. */
nsteps++;
(*alpha) *= interpf;
dgpstep(n, x, xl, xu, -(*alpha), g, s);
if (dnrm2_(&n, s, &inc) <= delta)
{
dsymv_("U", &n, &one, A, &n, s, &inc, &zero, wa, &inc);
gts = ddot_(&n, g, &inc, s, &inc);
q = 0.5*ddot_(&n, s, &inc, wa, &inc) + gts;
search = q > mu0*gts ? 1 : 0;
}
}
}
else
{
search = 1;
alphas = *alpha;
/* Increase alpha until a successful step is found. */
while (search && (*alpha) <= brptmax)
{
/* This is a crude extrapolation procedure that
will be replaced in future versions of the code. */
nsteps++;
alphas = *alpha;
(*alpha) *= extrapf;
dgpstep(n, x, xl, xu, -(*alpha), g, s);
if (dnrm2_(&n, s, &inc) <= delta)
{
dsymv_("U", &n, &one, A, &n, s, &inc, &zero, wa, &inc);
gts = ddot_(&n, g, &inc, s, &inc);
q = 0.5*ddot_(&n, s, &inc, wa, &inc) + gts;
search = q < mu0*gts ? 1 : 0;
}
else
search = 0;
}
*alpha = alphas;
dgpstep(n, x, xl, xu, -(*alpha), g, s);
}
free(wa);
}
void dprsrch(int n, double *x, double *xl, double *xu, double *A, double *g, double *w)
{
/*
c **********
c
c Subroutine dprsrch
c
c This subroutine uses a projected search to compute a step
c that satisfies a sufficient decrease condition for the quadratic
c
c q(s) = 0.5*s'*A*s + g'*s,
c
c where A is a symmetric matrix and g is a vector. Given the
c parameter alpha, the step is
c
c s[alpha] = P[x + alpha*w] - x,
c
c where w is the search direction and P the projection onto the
c n-dimensional interval [xl,xu]. The final step s = s[alpha]
c satisfies the sufficient decrease condition
c
c q(s) <= mu_0*(g'*s),
c
c where mu_0 is a constant in (0,1).
c
c The search direction w must be a descent direction for the
c quadratic q at x such that the quadratic is decreasing
c in the ray x + alpha*w for 0 <= alpha <= 1.
c
c parameters:
c
c n is an integer variable.
c On entry n is the number of variables.
c On exit n is unchanged.
c
c x is a double precision array of dimension n.
c On entry x specifies the vector x.
c On exit x is set to the final point P[x + alpha*w].
c
c xl is a double precision array of dimension n.
c On entry xl is the vector of lower bounds.
c On exit xl is unchanged.
c
c xu is a double precision array of dimension n.
c On entry xu is the vector of upper bounds.
c On exit xu is unchanged.
c
c A is a double precision array of dimension n*n.
c On entry A specifies the matrix A
c On exit A is unchanged.
c
c g is a double precision array of dimension n.
c On entry g specifies the vector g.
c On exit g is unchanged.
c
c w is a double prevision array of dimension n.
c On entry w specifies the search direction.
c On exit w is the step s[alpha].
c
c **********
*/
double one = 1, zero = 0;
/* Constant that defines sufficient decrease. */
/* Interpolation factor. */
double mu0 = 0.01, interpf = 0.5;
double *wa1 = (double *) xmalloc(sizeof(double)*n);
double *wa2 = (double *) xmalloc(sizeof(double)*n);
/* Set the initial alpha = 1 because the quadratic function is
decreasing in the ray x + alpha*w for 0 <= alpha <= 1 */
double alpha = 1, brptmin, brptmax, gts, q;
int search = 1, nbrpt, nsteps = 0, i, inc = 1;
/* Find the smallest break-point on the ray x + alpha*w. */
dbreakpt(n, x, xl, xu, w, &nbrpt, &brptmin, &brptmax);
/* Reduce alpha until the sufficient decrease condition is
satisfied or x + alpha*w is feasible. */
while (search && alpha > brptmin)
{
/* Calculate P[x + alpha*w] - x and check the sufficient
decrease condition. */
nsteps++;
dgpstep(n, x, xl, xu, alpha, w, wa1);
F77_CALL(dsymv)("U", &n, &one, A, &n, wa1, &inc, &zero, wa2, &inc);
gts = F77_CALL(ddot)(&n, g, &inc, wa1, &inc);
q = 0.5*F77_CALL(ddot)(&n, wa1, &inc, wa2, &inc) + gts;
if (q <= mu0*gts)
search = 0;
else
/* This is a crude interpolation procedure that
will be replaced in future versions of the code. */
alpha *= interpf;
}
/* Force at least one more constraint to be added to the active
set if alpha < brptmin and the full step is not successful.
There is sufficient decrease because the quadratic function
is decreasing in the ray x + alpha*w for 0 <= alpha <= 1. */
if (alpha < 1 && alpha < brptmin)
alpha = brptmin;
/* Compute the final iterate and step. */
dgpstep(n, x, xl, xu, alpha, w, wa1);
F77_CALL(daxpy)(&n, &alpha, w, &inc, x, &inc);
for (i=0;i<n;i++)
x[i] = mymax(xl[i], mymin(x[i], xu[i]));
memcpy(w, wa1, sizeof(double)*n);
free(wa1);
free(wa2);
}