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/* */
/* Copyright 2008 by Ullrich Koethe */
/* */
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#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
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