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/* ========================================================================== */
/* === Source/Mongoose_QPNapUp.cpp ========================================== */
/* ========================================================================== */
/* -----------------------------------------------------------------------------
* Mongoose Graph Partitioning Library Copyright (C) 2017-2018,
* Scott P. Kolodziej, Nuri S. Yeralan, Timothy A. Davis, William W. Hager
* Mongoose is licensed under Version 3 of the GNU General Public License.
* Mongoose is also available under other licenses; contact authors for details.
* -------------------------------------------------------------------------- */
/* ========================================================================== */
/* === QPNapUp ============================================================== */
/* ========================================================================== */
/* Find x that minimizes ||x-y|| while satisfying the constraints
0 <= x <= 1, a'x = b.
The algorithm is described in the napsack comments.
It is assumed that the starting guess lambda for the dual multiplier is <=
the correct multiplier. Hence, lambda will be increased. The slope of
the dual function, neglecting b, starts out larger than b. We stop
when we reach b. We assume that a >= 0, so that as lambda increases,
x_i (lambda) decreases. Hence, the only bound variables that can become
free are those with x_i (lambda) >= 1 */
#include "Mongoose_QPNapUp.hpp"
#include "Mongoose_Debug.hpp"
#include "Mongoose_Internal.hpp"
#include "Mongoose_Logger.hpp"
#include "Mongoose_QPMinHeap.hpp"
namespace Mongoose
{
double QPNapUp /* return lambda */
(const double *x, /* holds y on input, not modified */
const Int n, /* size of x */
double lambda, /* initial guess for the shift */
const double *a, /* input constraint vector */
double b, /* input constraint scalar */
double *breakpts, /* break points */
Int *bound_heap, /* work array */
Int *free_heap /* work array */
)
{
Int i, k, e, maxsteps, n_bound, n_free;
double ai, asum, a2sum, minbound, minfree, t;
minbound = INFINITY;
minfree = INFINITY;
/* -------------------------------------------------------------- */
/* construct the heaps */
/* -------------------------------------------------------------- */
n_bound = 0;
n_free = 0;
asum = 0.;
a2sum = 0.;
for (i = 0; i < n; i++)
{
ai = (a) ? a[i] : 1;
double xi = x[i] - ai * lambda;
if (xi > 1.)
{
n_bound++;
bound_heap[n_bound] = i;
asum += ai;
t = (x[i] - 1.) / ai;
minbound = std::min(minbound, t);
breakpts[i] = t;
}
else if (xi > 0.)
{
n_free++;
free_heap[n_free] = i;
asum += x[i] * ai;
a2sum += ai * ai;
t = x[i] / ai;
minfree = std::min(minfree, t);
breakpts[i] = t;
}
}
maxsteps = 2 * n + 1;
for (k = 1; k <= maxsteps; k++)
{
/*------------------------------------------------------------------- */
/* check to see if zero slope achieved without changing the free set */
/* remember that the slope must always be adjusted by b */
/*------------------------------------------------------------------- */
double new_break = std::min(minfree, minbound);
double s = asum - new_break * a2sum;
if ((s <= b) || (new_break == INFINITY)) /* done */
{
if (a2sum != 0.)
{
lambda = (asum - b) / a2sum;
}
return (lambda);
}
lambda = new_break;
if (k == 1)
{
QPMinHeap_build(free_heap, n_free, breakpts);
QPMinHeap_build(bound_heap, n_bound, breakpts);
}
/* -------------------------------------------------------------- */
/* update the heaps */
/* -------------------------------------------------------------- */
if (n_free > 0)
{
while (breakpts[e = free_heap[1]] <= lambda)
{
ai = (a) ? a[e] : 1;
a2sum -= ai * ai;
asum -= ai * x[e];
n_free = QPMinHeap_delete(free_heap, n_free, breakpts);
if (n_free == 0)
{
a2sum = 0.;
break;
}
}
}
if (n_bound > 0)
{
while (breakpts[e = bound_heap[1]] <= lambda)
{
n_bound = QPMinHeap_delete(bound_heap, n_bound, breakpts);
ai = (a) ? a[e] : 1;
a2sum += ai * ai;
asum += ai * (x[e] - 1.);
t = x[e] / ai;
breakpts[e] = t;
n_free = QPMinHeap_add(e, free_heap, breakpts, n_free);
if (n_bound == 0)
break;
}
}
/*------------------------------------------------------------------- */
/* get the biggest entry in each heap */
/*------------------------------------------------------------------- */
minfree = (n_free > 0 ? breakpts[free_heap[1]] : INFINITY);
minbound = (n_bound > 0 ? breakpts[bound_heap[1]] : INFINITY);
}
/* this should not happen */
ASSERT(false);
lambda = 0.;
return lambda;
}
} // end namespace Mongoose
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