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subsolv.m
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subsolv.m
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%-------------------------------------------------------------
% This is the file subsolv.m
%
% Version Dec 2006.
% Krister Svanberg <krille@math.kth.se>
% Department of Mathematics, KTH,
% SE-10044 Stockholm, Sweden.
%
function [xmma,ymma,zmma,lamma,xsimma,etamma,mumma,zetmma,smma] = ...
subsolv(m,n,epsimin,low,upp,alfa,beta,p0,q0,P,Q,a0,a,b,c,d)
% This function subsolv solves the MMA subproblem:
%
% minimize SUM[ p0j/(uppj-xj) + q0j/(xj-lowj) ] + a0*z +
% + SUM[ ci*yi + 0.5*di*(yi)^2 ],
%
% subject to SUM[ pij/(uppj-xj) + qij/(xj-lowj) ] - ai*z - yi <= bi,
% alfaj <= xj <= betaj, yi >= 0, z >= 0.
%
% Input: m, n, low, upp, alfa, beta, p0, q0, P, Q, a0, a, b, c, d.
% Output: xmma,ymma,zmma, slack variables and Lagrange multiplers.
%
een = ones(n,1);
eem = ones(m,1);
epsi = 1;
epsvecn = epsi*een;
epsvecm = epsi*eem;
x = 0.5*(alfa+beta);
y = eem;
z = 1;
lam = eem;
xsi = een./(x-alfa);
xsi = max(xsi,een);
eta = een./(beta-x);
eta = max(eta,een);
mu = max(eem,0.5*c);
zet = 1;
s = eem;
itera = 0;
while epsi > epsimin
epsvecn = epsi*een;
epsvecm = epsi*eem;
ux1 = upp-x;
xl1 = x-low;
ux2 = ux1.*ux1;
xl2 = xl1.*xl1;
uxinv1 = een./ux1;
xlinv1 = een./xl1;
plam = p0 + P'*lam ;
qlam = q0 + Q'*lam ;
gvec = P*uxinv1 + Q*xlinv1;
dpsidx = plam./ux2 - qlam./xl2 ;
rex = dpsidx - xsi + eta;
rey = c + d.*y - mu - lam;
rez = a0 - zet - a'*lam;
relam = gvec - a*z - y + s - b;
rexsi = xsi.*(x-alfa) - epsvecn;
reeta = eta.*(beta-x) - epsvecn;
remu = mu.*y - epsvecm;
rezet = zet*z - epsi;
res = lam.*s - epsvecm;
residu1 = [rex' rey' rez]';
residu2 = [relam' rexsi' reeta' remu' rezet res']';
residu = [residu1' residu2']';
residunorm = sqrt(residu'*residu);
residumax = max(abs(residu));
ittt = 0;
while residumax > 0.9*epsi & ittt < 200
ittt=ittt + 1;
itera=itera + 1;
ux1 = upp-x;
xl1 = x-low;
ux2 = ux1.*ux1;
xl2 = xl1.*xl1;
ux3 = ux1.*ux2;
xl3 = xl1.*xl2;
uxinv1 = een./ux1;
xlinv1 = een./xl1;
uxinv2 = een./ux2;
xlinv2 = een./xl2;
plam = p0 + P'*lam ;
qlam = q0 + Q'*lam ;
gvec = P*uxinv1 + Q*xlinv1;
GG = P*spdiags(uxinv2,0,n,n) - Q*spdiags(xlinv2,0,n,n);
dpsidx = plam./ux2 - qlam./xl2 ;
delx = dpsidx - epsvecn./(x-alfa) + epsvecn./(beta-x);
dely = c + d.*y - lam - epsvecm./y;
delz = a0 - a'*lam - epsi/z;
dellam = gvec - a*z - y - b + epsvecm./lam;
diagx = plam./ux3 + qlam./xl3;
diagx = 2*diagx + xsi./(x-alfa) + eta./(beta-x);
diagxinv = een./diagx;
diagy = d + mu./y;
diagyinv = eem./diagy;
diaglam = s./lam;
diaglamyi = diaglam+diagyinv;
if m < n
blam = dellam + dely./diagy - GG*(delx./diagx);
bb = [blam' delz]';
Alam = spdiags(diaglamyi,0,m,m) + GG*spdiags(diagxinv,0,n,n)*GG';
AA = [Alam a
a' -zet/z ];
solut = AA\bb;
dlam = solut(1:m);
dz = solut(m+1);
dx = -delx./diagx - (GG'*dlam)./diagx;
else
diaglamyiinv = eem./diaglamyi;
dellamyi = dellam + dely./diagy;
Axx = spdiags(diagx,0,n,n) + GG'*spdiags(diaglamyiinv,0,m,m)*GG;
azz = zet/z + a'*(a./diaglamyi);
axz = -GG'*(a./diaglamyi);
bx = delx + GG'*(dellamyi./diaglamyi);
bz = delz - a'*(dellamyi./diaglamyi);
AA = [Axx axz
axz' azz ];
bb = [-bx' -bz]';
solut = AA\bb;
dx = solut(1:n);
dz = solut(n+1);
dlam = (GG*dx)./diaglamyi - dz*(a./diaglamyi) + dellamyi./diaglamyi;
end
%
dy = -dely./diagy + dlam./diagy;
dxsi = -xsi + epsvecn./(x-alfa) - (xsi.*dx)./(x-alfa);
deta = -eta + epsvecn./(beta-x) + (eta.*dx)./(beta-x);
dmu = -mu + epsvecm./y - (mu.*dy)./y;
dzet = -zet + epsi/z - zet*dz/z;
ds = -s + epsvecm./lam - (s.*dlam)./lam;
xx = [ y' z lam' xsi' eta' mu' zet s']';
dxx = [dy' dz dlam' dxsi' deta' dmu' dzet ds']';
%
stepxx = -1.01*dxx./xx;
stmxx = max(stepxx);
stepalfa = -1.01*dx./(x-alfa);
stmalfa = max(stepalfa);
stepbeta = 1.01*dx./(beta-x);
stmbeta = max(stepbeta);
stmalbe = max(stmalfa,stmbeta);
stmalbexx = max(stmalbe,stmxx);
stminv = max(stmalbexx,1);
steg = 1/stminv;
%
xold = x;
yold = y;
zold = z;
lamold = lam;
xsiold = xsi;
etaold = eta;
muold = mu;
zetold = zet;
sold = s;
%
itto = 0;
resinew = 2*residunorm;
while resinew > residunorm & itto < 50
itto = itto+1;
x = xold + steg*dx;
y = yold + steg*dy;
z = zold + steg*dz;
lam = lamold + steg*dlam;
xsi = xsiold + steg*dxsi;
eta = etaold + steg*deta;
mu = muold + steg*dmu;
zet = zetold + steg*dzet;
s = sold + steg*ds;
ux1 = upp-x;
xl1 = x-low;
ux2 = ux1.*ux1;
xl2 = xl1.*xl1;
uxinv1 = een./ux1;
xlinv1 = een./xl1;
plam = p0 + P'*lam ;
qlam = q0 + Q'*lam ;
gvec = P*uxinv1 + Q*xlinv1;
dpsidx = plam./ux2 - qlam./xl2 ;
rex = dpsidx - xsi + eta;
rey = c + d.*y - mu - lam;
rez = a0 - zet - a'*lam;
relam = gvec - a*z - y + s - b;
rexsi = xsi.*(x-alfa) - epsvecn;
reeta = eta.*(beta-x) - epsvecn;
remu = mu.*y - epsvecm;
rezet = zet*z - epsi;
res = lam.*s - epsvecm;
residu1 = [rex' rey' rez]';
residu2 = [relam' rexsi' reeta' remu' rezet res']';
residu = [residu1' residu2']';
resinew = sqrt(residu'*residu);
steg = steg/2;
end
residunorm=resinew;
residumax = max(abs(residu));
steg = 2*steg;
end
if ittt > 198
epsi
ittt
end
epsi = 0.1*epsi;
end
xmma = x;
ymma = y;
zmma = z;
lamma = lam;
xsimma = xsi;
etamma = eta;
mumma = mu;
zetmma = zet;
smma = s;
%-------------------------------------------------------------