inline void
TrsvLT
( Orientation orientation, UnitOrNonUnit diag,
const DistMatrix<F>& L, DistMatrix<F>& x )
{
#ifndef RELEASE
PushCallStack("internal::TrsvLT");
if( L.Grid() != x.Grid() )
throw std::logic_error("{L,x} must be distributed over the same grid");
if( orientation == NORMAL )
throw std::logic_error("TrsvLT expects a (conjugate-)transpose option");
if( L.Height() != L.Width() )
throw std::logic_error("L must be square");
if( x.Width() != 1 && x.Height() != 1 )
throw std::logic_error("x must be a vector");
const int xLength = ( x.Width() == 1 ? x.Height() : x.Width() );
if( L.Width() != xLength )
throw std::logic_error("Nonconformal TrsvLT");
#endif
const Grid& g = L.Grid();
if( x.Width() == 1 )
{
// Matrix views
DistMatrix<F> L10(g), L11(g);
DistMatrix<F>
xT(g), x0(g),
xB(g), x1(g),
x2(g);
// Temporary distributions
DistMatrix<F,STAR,STAR> L11_STAR_STAR(g);
DistMatrix<F,STAR,STAR> x1_STAR_STAR(g);
DistMatrix<F,MC, STAR> x1_MC_STAR(g);
DistMatrix<F,MC, MR > z1(g);
DistMatrix<F,MR, MC > z1_MR_MC(g);
DistMatrix<F,MR, STAR> z_MR_STAR(g);
// Views of z[MR,* ]
DistMatrix<F,MR,STAR> z0_MR_STAR(g),
z1_MR_STAR(g);
z_MR_STAR.AlignWith( L );
Zeros( x.Height(), 1, z_MR_STAR );
// Start the algorithm
PartitionUp
( x, xT,
xB, 0 );
while( xT.Height() > 0 )
{
RepartitionUp
( xT, x0,
x1,
/**/ /**/
xB, x2 );
const int n0 = x0.Height();
const int n1 = x1.Height();
L10.LockedView( L, n0, 0, n1, n0 );
L11.LockedView( L, n0, n0, n1, n1 );
z0_MR_STAR.View( z_MR_STAR, 0, 0, n0, 1 );
z1_MR_STAR.View( z_MR_STAR, n0, 0, n1, 1 );
x1_MC_STAR.AlignWith( L10 );
z1.AlignWith( x1 );
//----------------------------------------------------------------//
if( x2.Height() != 0 )
{
z1_MR_MC.SumScatterFrom( z1_MR_STAR );
z1 = z1_MR_MC;
Axpy( F(1), z1, x1 );
}
x1_STAR_STAR = x1;
L11_STAR_STAR = L11;
Trsv
( LOWER, orientation, diag,
L11_STAR_STAR.LockedLocalMatrix(),
x1_STAR_STAR.LocalMatrix() );
x1 = x1_STAR_STAR;
x1_MC_STAR = x1_STAR_STAR;
Gemv
( orientation, F(-1),
L10.LockedLocalMatrix(),
x1_MC_STAR.LockedLocalMatrix(),
F(1), z0_MR_STAR.LocalMatrix() );
//----------------------------------------------------------------//
x1_MC_STAR.FreeAlignments();
z1.FreeAlignments();
SlidePartitionUp
( xT, x0,
/**/ /**/
x1,
xB, x2 );
}
}
else
{
// Matrix views
DistMatrix<F> L10(g), L11(g);
DistMatrix<F>
xL(g), xR(g),
x0(g), x1(g), x2(g);
// Temporary distributions
DistMatrix<F,STAR,STAR> L11_STAR_STAR(g);
DistMatrix<F,STAR,STAR> x1_STAR_STAR(g);
DistMatrix<F,STAR,MC > x1_STAR_MC(g);
DistMatrix<F,STAR,MR > z_STAR_MR(g);
// Views of z[* ,MR], which will store updates to x
DistMatrix<F,STAR,MR> z0_STAR_MR(g),
z1_STAR_MR(g);
z_STAR_MR.AlignWith( L );
Zeros( 1, x.Width(), z_STAR_MR );
// Start the algorithm
PartitionLeft( x, xL, xR, 0 );
while( xL.Width() > 0 )
{
RepartitionLeft
( xL, /**/ xR,
x0, x1, /**/ x2 );
const int n0 = x0.Width();
const int n1 = x1.Width();
L10.LockedView( L, n0, 0, n1, n0 );
L11.LockedView( L, n0, n0, n1, n1 );
z0_STAR_MR.View( z_STAR_MR, 0, 0, 1, n0 );
z1_STAR_MR.View( z_STAR_MR, 0, n0, 1, n1 );
x1_STAR_MC.AlignWith( L10 );
//----------------------------------------------------------------//
if( x2.Width() != 0 )
x1.SumScatterUpdate( F(1), z1_STAR_MR );
x1_STAR_STAR = x1;
L11_STAR_STAR = L11;
Trsv
( LOWER, orientation, diag,
L11_STAR_STAR.LockedLocalMatrix(),
x1_STAR_STAR.LocalMatrix() );
x1 = x1_STAR_STAR;
x1_STAR_MC = x1_STAR_STAR;
Gemv
( orientation, F(-1),
L10.LockedLocalMatrix(),
x1_STAR_MC.LockedLocalMatrix(),
F(1), z0_STAR_MR.LocalMatrix() );
//----------------------------------------------------------------//
x1_STAR_MC.FreeAlignments();
SlidePartitionLeft
( xL, /**/ xR,
x0, /**/ x1, x2 );
}
}
#ifndef RELEASE
PopCallStack();
#endif
}
Int ADMM
( const Matrix<Real>& A,
const Matrix<Real>& b, const Matrix<Real>& c,
Matrix<Real>& z,
const ADMMCtrl<Real>& ctrl )
{
DEBUG_ONLY(CSE cse("lp::direct::ADMM"))
// Cache a custom partially-pivoted LU factorization of
// | rho*I A^H | = | B11 B12 |
// | A 0 | | B21 B22 |
// by (justifiably) avoiding pivoting in the first n steps of
// the factorization, so that
// [I,rho*I] = lu(rho*I).
// The factorization would then proceed with
// B21 := B21 U11^{-1} = A (rho*I)^{-1} = A/rho
// B12 := L11^{-1} B12 = I A^H = A^H.
// The Schur complement would then be
// B22 := B22 - B21 B12 = 0 - (A*A^H)/rho.
// We then factor said matrix with LU with partial pivoting and
// swap the necessary rows of B21 in order to implicitly commute
// the row pivots with the Gauss transforms in the manner standard
// for GEPP. Unless A A' is singular, pivoting should not be needed,
// as Cholesky factorization of the negative matrix should be valid.
//
// The result is the factorization
// | I 0 | | rho*I A^H | = | I 0 | | rho*I U12 |,
// | 0 P22 | | A 0 | | L21 L22 | | 0 U22 |
// where [L22,U22] are stored within B22.
Matrix<Real> U12, L21, B22, bPiv;
Adjoint( A, U12 );
L21 = A;
L21 *= 1/ctrl.rho;
Herk( LOWER, NORMAL, -1/ctrl.rho, A, B22 );
MakeHermitian( LOWER, B22 );
// TODO: Replace with sparse-direct Cholesky version?
Matrix<Int> rowPiv2;
LU( B22, rowPiv2 );
ApplyRowPivots( L21, rowPiv2 );
bPiv = b;
ApplyRowPivots( bPiv, rowPiv2 );
// Possibly form the inverse of L22 U22
Matrix<Real> X22;
if( ctrl.inv )
{
X22 = B22;
MakeTrapezoidal( LOWER, X22 );
FillDiagonal( X22, Real(1) );
TriangularInverse( LOWER, UNIT, X22 );
Trsm( LEFT, UPPER, NORMAL, NON_UNIT, Real(1), B22, X22 );
}
Int numIter=0;
const Int m = A.Height();
const Int n = A.Width();
Matrix<Real> g, xTmp, y, t;
Zeros( g, m+n, 1 );
PartitionDown( g, xTmp, y, n );
Matrix<Real> x, u, zOld, xHat;
Zeros( z, n, 1 );
Zeros( u, n, 1 );
Zeros( t, n, 1 );
while( numIter < ctrl.maxIter )
{
zOld = z;
// Find x from
// | rho*I A^H | | x | = | rho*(z-u)-c |
// | A 0 | | y | | b |
// via our cached custom factorization:
//
// |x| = inv(U) inv(L) P' |rho*(z-u)-c|
// |y| |b |
// = |rho*I U12|^{-1} |I 0 | |I 0 | |rho*(z-u)-c|
// = |0 U22| |L21 L22| |0 P22'| |b |
// = " " |rho*(z-u)-c|
// | P22' b |
xTmp = z;
xTmp -= u;
xTmp *= ctrl.rho;
xTmp -= c;
y = bPiv;
Gemv( NORMAL, Real(-1), L21, xTmp, Real(1), y );
if( ctrl.inv )
{
Gemv( NORMAL, Real(1), X22, y, t );
y = t;
}
else
{
Trsv( LOWER, NORMAL, UNIT, B22, y );
Trsv( UPPER, NORMAL, NON_UNIT, B22, y );
}
Gemv( NORMAL, Real(-1), U12, y, Real(1), xTmp );
xTmp *= 1/ctrl.rho;
// xHat := alpha*x + (1-alpha)*zOld
xHat = xTmp;
xHat *= ctrl.alpha;
Axpy( 1-ctrl.alpha, zOld, xHat );
// z := pos(xHat+u)
z = xHat;
z += u;
LowerClip( z, Real(0) );
// u := u + (xHat-z)
u += xHat;
u -= z;
const Real objective = Dot( c, xTmp );
// rNorm := || x - z ||_2
t = xTmp;
t -= z;
const Real rNorm = FrobeniusNorm( t );
// sNorm := |rho| || z - zOld ||_2
t = z;
t -= zOld;
const Real sNorm = Abs(ctrl.rho)*FrobeniusNorm( t );
const Real epsPri = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Max(FrobeniusNorm(xTmp),FrobeniusNorm(z));
const Real epsDual = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Abs(ctrl.rho)*FrobeniusNorm(u);
if( ctrl.print )
{
t = xTmp;
LowerClip( t, Real(0) );
t -= xTmp;
const Real clipDist = FrobeniusNorm( t );
cout << numIter << ": "
<< "||x-z||_2=" << rNorm << ", "
<< "epsPri=" << epsPri << ", "
<< "|rho| ||z-zOld||_2=" << sNorm << ", "
<< "epsDual=" << epsDual << ", "
<< "||x-Pos(x)||_2=" << clipDist << ", "
<< "c'x=" << objective << endl;
}
if( rNorm < epsPri && sNorm < epsDual )
break;
++numIter;
}
if( ctrl.maxIter == numIter )
cout << "ADMM failed to converge" << endl;
x = xTmp;
return numIter;
}