void RLS
( const ElementalMatrix<Real>& APre,
const ElementalMatrix<Real>& bPre,
Real rho,
ElementalMatrix<Real>& xPre,
const socp::affine::Ctrl<Real>& ctrl )
{
DEBUG_CSE
DistMatrixReadProxy<Real,Real,MC,MR>
AProx( APre ),
bProx( bPre );
DistMatrixWriteProxy<Real,Real,MC,MR>
xProx( xPre );
auto& A = AProx.GetLocked();
auto& b = bProx.GetLocked();
auto& x = xProx.Get();
const Int m = A.Height();
const Int n = A.Width();
const Grid& g = A.Grid();
DistMatrix<Int,VC,STAR> orders(g), firstInds(g);
Zeros( orders, m+n+3, 1 );
Zeros( firstInds, m+n+3, 1 );
{
const Int localHeight = orders.LocalHeight();
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
{
const Int i = orders.GlobalRow(iLoc);
if( i < m+1 )
{
orders.SetLocal( iLoc, 0, m+1 );
firstInds.SetLocal( iLoc, 0, 0 );
}
else
{
orders.SetLocal( iLoc, 0, n+2 );
firstInds.SetLocal( iLoc, 0, m+1 );
}
}
}
// G := | -1 0 0 |
// | 0 0 A |
// | 0 -1 0 |
// | 0 0 -I |
// | 0 0 0 |
DistMatrix<Real> G(g);
{
Zeros( G, m+n+3, n+2 );
G.Set( 0, 0, -1 );
auto GA = G( IR(1,m+1), IR(2,n+2) );
GA = A;
G.Set( m+1, 1, -1 );
auto GI = G( IR(m+2,m+n+2), IR(2,n+2) );
Identity( GI, n, n );
GI *= -1;
}
// h := | 0 |
// | b |
// | 0 |
// | 0 |
// | 1 |
DistMatrix<Real> h(g);
Zeros( h, m+n+3, 1 );
auto hb = h( IR(1,m+1), ALL );
hb = b;
h.Set( END, 0, 1 );
// c := [1; rho; 0]
DistMatrix<Real> c(g);
Zeros( c, n+2, 1 );
c.Set( 0, 0, 1 );
c.Set( 1, 0, rho );
DistMatrix<Real> AHat(g), bHat(g);
Zeros( AHat, 0, n+2 );
Zeros( bHat, 0, 1 );
DistMatrix<Real> xHat(g), y(g), z(g), s(g);
SOCP( AHat, G, bHat, c, h, orders, firstInds, xHat, y, z, s, ctrl );
x = xHat( IR(2,END), ALL );
}
Int ADMM
( const AbstractDistMatrix<Real>& APre,
const AbstractDistMatrix<Real>& bPre,
const AbstractDistMatrix<Real>& cPre,
AbstractDistMatrix<Real>& zPre,
const ADMMCtrl<Real>& ctrl )
{
EL_DEBUG_CSE
DistMatrixReadProxy<Real,Real,MC,MR>
AProx( APre ),
bProx( bPre ),
cProx( cPre );
DistMatrixWriteProxy<Real,Real,MC,MR>
zProx( zPre );
auto& A = AProx.GetLocked();
auto& b = bProx.GetLocked();
auto& c = cProx.GetLocked();
auto& z = zProx.Get();
// 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.
const Int m = A.Height();
const Int n = A.Width();
const Grid& grid = A.Grid();
DistMatrix<Real> U12(grid), L21(grid), B22(grid), bPiv(grid);
U12.Align( 0, n%U12.RowStride() );
L21.Align( n%L21.ColStride(), 0 );
B22.Align( n%B22.ColStride(), n%B22.RowStride() );
Adjoint( A, U12 );
L21 = A;
L21 *= 1/ctrl.rho;
Herk( LOWER, NORMAL, -1/ctrl.rho, A, B22 );
MakeHermitian( LOWER, B22 );
DistPermutation P2(grid);
LU( B22, P2 );
P2.PermuteRows( L21 );
bPiv = b;
P2.PermuteRows( bPiv );
// Possibly form the inverse of L22 U22
DistMatrix<Real> X22(grid);
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;
DistMatrix<Real> g(grid), xTmp(grid), y(grid), t(grid);
Zeros( g, m+n, 1 );
PartitionDown( g, xTmp, y, n );
DistMatrix<Real> x(grid), u(grid), zOld(grid), xHat(grid);
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 );
if( grid.Rank() == 0 )
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 && grid.Rank() == 0 )
cout << "ADMM failed to converge" << endl;
x = xTmp;
return numIter;
}
