/usr/include/vigra/quadprog.hxx is in libvigraimpex-dev 1.10.0+git20160211.167be93+dfsg-2+b5.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 | /************************************************************************/
/* */
/* Copyright 2008 by Ullrich Koethe */
/* */
/* This file is part of the VIGRA computer vision library. */
/* The VIGRA Website is */
/* http://hci.iwr.uni-heidelberg.de/vigra/ */
/* Please direct questions, bug reports, and contributions to */
/* ullrich.koethe@iwr.uni-heidelberg.de or */
/* vigra@informatik.uni-hamburg.de */
/* */
/* 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 activeSet 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. */
/* */
/************************************************************************/
#ifndef VIGRA_QUADPROG_HXX
#define VIGRA_QUADPROG_HXX
#include <limits>
#include "mathutil.hxx"
#include "matrix.hxx"
#include "linear_solve.hxx"
#include "numerictraits.hxx"
#include "array_vector.hxx"
namespace vigra {
namespace detail {
template <class T, class C1, class C2, class C3>
bool quadprogAddConstraint(MultiArrayView<2, T, C1> & R, MultiArrayView<2, T, C2> & J, MultiArrayView<2, T, C3> & d,
int activeConstraintCount, double& R_norm)
{
typedef typename MultiArrayShape<2>::type Shape;
int n=columnCount(J);
linalg::detail::qrGivensStepImpl(0, subVector(d, activeConstraintCount, n),
J.subarray(Shape(activeConstraintCount,0), Shape(n,n)));
if (abs(d(activeConstraintCount,0)) <= NumericTraits<T>::epsilon() * R_norm) // problem degenerate
return false;
R_norm = std::max<T>(R_norm, abs(d(activeConstraintCount,0)));
++activeConstraintCount;
// add d as a new column to R
columnVector(R, Shape(0, activeConstraintCount - 1), activeConstraintCount) = subVector(d, 0, activeConstraintCount);
return true;
}
template <class T, class C1, class C2, class C3>
void quadprogDeleteConstraint(MultiArrayView<2, T, C1> & R, MultiArrayView<2, T, C2> & J, MultiArrayView<2, T, C3> & u,
int activeConstraintCount, int constraintToBeRemoved)
{
typedef typename MultiArrayShape<2>::type Shape;
int newActiveConstraintCount = activeConstraintCount - 1;
if(constraintToBeRemoved == newActiveConstraintCount)
return;
std::swap(u(constraintToBeRemoved,0), u(newActiveConstraintCount,0));
columnVector(R, constraintToBeRemoved).swapData(columnVector(R, newActiveConstraintCount));
linalg::detail::qrGivensStepImpl(0, R.subarray(Shape(constraintToBeRemoved, constraintToBeRemoved),
Shape(newActiveConstraintCount,newActiveConstraintCount)),
J.subarray(Shape(constraintToBeRemoved, 0),
Shape(newActiveConstraintCount,newActiveConstraintCount)));
}
} // namespace detail
/** \addtogroup Optimization Optimization and Regression
*/
//@{
/** Solve Quadratic Programming Problem.
The quadraticProgramming() function implements the algorithm described in
D. Goldfarb, A. Idnani: <i>"A numerically stable dual method for solving
strictly convex quadratic programs"</i>, Mathematical Programming 27:1-33, 1983.
for the solution of (convex) quadratic programming problems by means of a primal-dual method.
<b>\#include</b> \<vigra/quadprog.hxx\> <br/>
Namespaces: vigra
<b>Declaration:</b>
\code
namespace vigra {
template <class T, class C1, class C2, class C3, class C4, class C5, class C6, class C7>
T
quadraticProgramming(MultiArrayView<2, T, C1> const & GG, MultiArrayView<2, T, C2> const & g,
MultiArrayView<2, T, C3> const & CE, MultiArrayView<2, T, C4> const & ce,
MultiArrayView<2, T, C5> const & CI, MultiArrayView<2, T, C6> const & ci,
MultiArrayView<2, T, C7> & x);
}
\endcode
The problem must be specified in the form:
\f{eqnarray*}
\mbox{minimize } &\,& \frac{1}{2} \mbox{\bf x}'\,\mbox{\bf G}\, \mbox{\bf x} + \mbox{\bf g}'\,\mbox{\bf x} \\
\mbox{subject to} &\,& \mbox{\bf C}_E\, \mbox{\bf x} = \mbox{\bf c}_e \\
&\,& \mbox{\bf C}_I\,\mbox{\bf x} \ge \mbox{\bf c}_i
\f}
Matrix <b>G</b> G must be symmetric positive definite, and matrix <b>C</b><sub>E</sub> must have full row rank.
Matrix and vector dimensions must be as follows:
<ul>
<li> <b>G</b>: [n * n], <b>g</b>: [n * 1]
<li> <b>C</b><sub>E</sub>: [me * n], <b>c</b><sub>e</sub>: [me * 1]
<li> <b>C</b><sub>I</sub>: [mi * n], <b>c</b><sub>i</sub>: [mi * 1]
<li> <b>x</b>: [n * 1]
</ul>
The function writes the optimal solution into the vector \a x and returns the cost of this solution.
If the problem is infeasible, <tt>std::numeric_limits::infinity()</tt> is returned. In this case
the value of vector \a x is undefined.
<b>Usage:</b>
Minimize <tt> f = 0.5 * x'*G*x + g'*x </tt> subject to <tt> -1 <= x <= 1</tt>.
The solution is <tt> x' = [1.0, 0.5, -1.0] </tt> with <tt> f = -22.625</tt>.
\code
double Gdata[] = {13.0, 12.0, -2.0,
12.0, 17.0, 6.0,
-2.0, 6.0, 12.0};
double gdata[] = {-22.0, -14.5, 13.0};
double CIdata[] = { 1.0, 0.0, 0.0,
0.0, 1.0, 0.0,
0.0, 0.0, 1.0,
-1.0, 0.0, 0.0,
0.0, -1.0, 0.0,
0.0, 0.0, -1.0};
double cidata[] = {-1.0, -1.0, -1.0, -1.0, -1.0, -1.0};
Matrix<double> G(3,3, Gdata),
g(3,1, gdata),
CE, // empty since there are no equality constraints
ce, // likewise
CI(7,3, CIdata),
ci(7,1, cidata),
x(3,1);
double f = quadraticProgramming(G, g, CE, ce, CI, ci, x);
\endcode
This algorithm can also be used to solve non-negative least squares problems
(see \ref nonnegativeLeastSquares() for an alternative algorithm). Consider the
problem to minimize <tt> f = (A*x - b)' * (A*x - b) </tt> subject to <tt> x >= 0</tt>.
Expanding the product in the objective gives <tt>f = x'*A'*A*x - 2*b'*A*x + b'*b</tt>.
This is equivalent to the problem of minimizing <tt>fn = 0.5 * x'*G*x + g'*x</tt> with
<tt>G = A'*A</tt> and <tt>g = -A'*b</tt> (the constant term <tt>b'*b</tt> has no
effect on the optimal solution and can be dropped). The following code computes the
solution <tt>x' = [18.4493, 0, 4.50725]</tt>:
\code
double A_data[] = {
1, -3, 2,
-3, 10, -5,
2, -5, 6
};
double b_data[] = {
27,
-78,
64
};
Matrix<double> A(3, 3, A_data),
b(3, 1, b_data),
G = transpose(A)*A,
g = -(transpose(A)*b),
CE, // empty since there are no equality constraints
ce, // likewise
CI = identityMatrix<double>(3), // constrain all elements of x
ci(3, 1, 0.0), // ... to be non-negative
x(3, 1);
quadraticProgramming(G, g, CE, ce, CI, ci, x);
\endcode
*/
doxygen_overloaded_function(template <...> unsigned int quadraticProgramming)
template <class T, class C1, class C2, class C3, class C4, class C5, class C6, class C7>
T
quadraticProgramming(MultiArrayView<2, T, C1> const & G, MultiArrayView<2, T, C2> const & g,
MultiArrayView<2, T, C3> const & CE, MultiArrayView<2, T, C4> const & ce,
MultiArrayView<2, T, C5> const & CI, MultiArrayView<2, T, C6> const & ci,
MultiArrayView<2, T, C7> & x)
{
using namespace linalg;
typedef typename MultiArrayShape<2>::type Shape;
int n = rowCount(g),
me = rowCount(ce),
mi = rowCount(ci),
constraintCount = me + mi;
vigra_precondition(columnCount(G) == n && rowCount(G) == n,
"quadraticProgramming(): Matrix shape mismatch between G and g.");
vigra_precondition(rowCount(x) == n,
"quadraticProgramming(): Output vector x has illegal shape.");
vigra_precondition((me > 0 && columnCount(CE) == n && rowCount(CE) == me) ||
(me == 0 && columnCount(CE) == 0),
"quadraticProgramming(): Matrix CE has illegal shape.");
vigra_precondition((mi > 0 && columnCount(CI) == n && rowCount(CI) == mi) ||
(mi == 0 && columnCount(CI) == 0),
"quadraticProgramming(): Matrix CI has illegal shape.");
Matrix<T> J = identityMatrix<T>(n);
{
Matrix<T> L(G.shape());
choleskyDecomposition(G, L);
// find unconstrained minimizer of the quadratic form 0.5 * x G x + g' x
choleskySolve(L, -g, x);
// compute the inverse of the factorized matrix G^-1, this is the initial value for J
linearSolveLowerTriangular(L, J, J);
}
// current solution value
T f_value = 0.5 * dot(g, x);
T epsilonZ = NumericTraits<T>::epsilon() * sq(J.norm(0)),
inf = std::numeric_limits<T>::infinity();
Matrix<T> R(n, n), r(constraintCount, 1), u(constraintCount,1);
T R_norm = NumericTraits<T>::one();
// incorporate equality constraints
for (int i=0; i < me; ++i)
{
MultiArrayView<2, T, C3> np = rowVector(CE, i);
Matrix<T> d = J*transpose(np);
Matrix<T> z = transpose(J).subarray(Shape(0, i), Shape(n,n))*subVector(d, i, n);
linearSolveUpperTriangular(R.subarray(Shape(0, 0), Shape(i,i)),
subVector(d, 0, i),
subVector(r, 0, i));
// compute step in primal space so that the constraint becomes satisfied
T step = (squaredNorm(z) <= epsilonZ) // i.e. z == 0
? 0.0
: (-dot(np, x) + ce(i,0)) / dot(z, np);
x += step * z;
u(i,0) = step;
subVector(u, 0, i) -= step * subVector(r, 0, i);
f_value += 0.5 * sq(step) * dot(z, np);
vigra_precondition(vigra::detail::quadprogAddConstraint(R, J, d, i, R_norm),
"quadraticProgramming(): Equality constraints are linearly dependent.");
}
int activeConstraintCount = me;
// determine optimum solution and corresponding active inequality constraints
ArrayVector<int> activeSet(mi);
for (int i = 0; i < mi; ++i)
activeSet[i] = i;
int constraintToBeAdded = 0;
T ss = 0.0;
for (int i = activeConstraintCount-me; i < mi; ++i)
{
T s = dot(rowVector(CI, activeSet[i]), x) - ci(activeSet[i], 0);
if (s < ss)
{
ss = s;
constraintToBeAdded = i;
}
}
int iter = 0, maxIter = 10*mi;
while(iter++ < maxIter)
{
if (ss >= 0.0) // all constraints are satisfied
return f_value; // => solved!
// determine step direction in the primal space (through J, see the paper)
MultiArrayView<2, T, C5> np = rowVector(CI, activeSet[constraintToBeAdded]);
Matrix<T> d = J*transpose(np);
Matrix<T> z = transpose(J).subarray(Shape(0, activeConstraintCount), Shape(n,n))*subVector(d, activeConstraintCount, n);
// compute negative of the step direction in the dual space
linearSolveUpperTriangular(R.subarray(Shape(0, 0), Shape(activeConstraintCount,activeConstraintCount)),
subVector(d, 0, activeConstraintCount),
subVector(r, 0, activeConstraintCount));
// determine minimum step length in primal space such that activeSet[constraintToBeAdded] becomes feasible
T primalStep = (squaredNorm(z) <= epsilonZ) // i.e. z == 0
? inf
: -ss / dot(z, np);
// determine maximum step length in dual space that doesn't violate dual feasibility
// and the corresponding index
T dualStep = inf;
int constraintToBeRemoved = 0;
for (int k = me; k < activeConstraintCount; ++k)
{
if (r(k,0) > 0.0)
{
if (u(k,0) / r(k,0) < dualStep)
{
dualStep = u(k,0) / r(k,0);
constraintToBeRemoved = k;
}
}
}
// the step is chosen as the minimum of dualStep and primalStep
T step = std::min(dualStep, primalStep);
// take step and update matrices
if (step == inf)
{
// case (i): no step in primal or dual space possible
return inf; // QPP is infeasible
}
if (primalStep == inf)
{
// case (ii): step in dual space
subVector(u, 0, activeConstraintCount) -= step * subVector(r, 0, activeConstraintCount);
vigra::detail::quadprogDeleteConstraint(R, J, u, activeConstraintCount, constraintToBeRemoved);
--activeConstraintCount;
std::swap(activeSet[constraintToBeRemoved-me], activeSet[activeConstraintCount-me]);
continue;
}
// case (iii): step in primal and dual space
x += step * z;
// update the solution value
f_value += 0.5 * sq(step) * dot(z, np);
// u = [u 1]' + step * [-r 1]
subVector(u, 0, activeConstraintCount) -= step * subVector(r, 0, activeConstraintCount);
u(activeConstraintCount,0) = step;
if (step == primalStep)
{
// add constraintToBeAdded to the active set
vigra::detail::quadprogAddConstraint(R, J, d, activeConstraintCount, R_norm);
std::swap(activeSet[constraintToBeAdded], activeSet[activeConstraintCount-me]);
++activeConstraintCount;
}
else
{
// drop constraintToBeRemoved from the active set
vigra::detail::quadprogDeleteConstraint(R, J, u, activeConstraintCount, constraintToBeRemoved);
--activeConstraintCount;
std::swap(activeSet[constraintToBeRemoved-me], activeSet[activeConstraintCount-me]);
}
// update values of inactive inequality constraints
ss = 0.0;
for (int i = activeConstraintCount-me; i < mi; ++i)
{
// compute CI*x - ci with appropriate row permutation
T s = dot(rowVector(CI, activeSet[i]), x) - ci(activeSet[i], 0);
if (s < ss)
{
ss = s;
constraintToBeAdded = i;
}
}
}
return inf; // too many iterations
}
//@}
} // namespace vigra
#endif
|