pde_surrogate/mmasub.m

%-------------------------------------------------------
%    This is the file mmasub.m
%
function [xmma,ymma,zmma,lam,xsi,eta,mu,zet,s,low,upp] = ...
mmasub(m,n,iter,xval,xmin,xmax,xold1,xold2, ...
f0val,df0dx,fval,dfdx,low,upp,a0,a,c,d);
%
%    Version September 2007 (and a small change August 2008)
%
%    Krister Svanberg <krille@math.kth.se>
%    Department of Mathematics, SE-10044 Stockholm, Sweden.
%
%    This function mmasub performs one MMA-iteration, aimed at
%    solving the nonlinear programming problem:
%         
%      Minimize  f_0(x) + a_0*z + sum( c_i*y_i + 0.5*d_i*(y_i)^2 )
%    subject to  f_i(x) - a_i*z - y_i <= 0,  i = 1,...,m
%                xmin_j <= x_j <= xmax_j,    j = 1,...,n
%                z >= 0,   y_i >= 0,         i = 1,...,m
%*** INPUT:
%
%   m    = The number of general constraints.
%   n    = The number of variables x_j.
%  iter  = Current iteration number ( =1 the first time mmasub is called).
%  xval  = Column vector with the current values of the variables x_j.
%  xmin  = Column vector with the lower bounds for the variables x_j.
%  xmax  = Column vector with the upper bounds for the variables x_j.
%  xold1 = xval, one iteration ago (provided that iter>1).
%  xold2 = xval, two iterations ago (provided that iter>2).
%  f0val = The value of the objective function f_0 at xval.
%  df0dx = Column vector with the derivatives of the objective function
%          f_0 with respect to the variables x_j, calculated at xval.
%  fval  = Column vector with the values of the constraint functions f_i,
%          calculated at xval.
%  dfdx  = (m x n)-matrix with the derivatives of the constraint functions
%          f_i with respect to the variables x_j, calculated at xval.
%          dfdx(i,j) = the derivative of f_i with respect to x_j.
%  low   = Column vector with the lower asymptotes from the previous
%          iteration (provided that iter>1).
%  upp   = Column vector with the upper asymptotes from the previous
%          iteration (provided that iter>1).
%  a0    = The constants a_0 in the term a_0*z.
%  a     = Column vector with the constants a_i in the terms a_i*z.
%  c     = Column vector with the constants c_i in the terms c_i*y_i.
%  d     = Column vector with the constants d_i in the terms 0.5*d_i*(y_i)^2.
%     
%*** OUTPUT:
%
%  xmma  = Column vector with the optimal values of the variables x_j
%          in the current MMA subproblem.
%  ymma  = Column vector with the optimal values of the variables y_i
%          in the current MMA subproblem.
%  zmma  = Scalar with the optimal value of the variable z
%          in the current MMA subproblem.
%  lam   = Lagrange multipliers for the m general MMA constraints.
%  xsi   = Lagrange multipliers for the n constraints alfa_j - x_j <= 0.
%  eta   = Lagrange multipliers for the n constraints x_j - beta_j <= 0.
%   mu   = Lagrange multipliers for the m constraints -y_i <= 0.
%  zet   = Lagrange multiplier for the single constraint -z <= 0.
%   s    = Slack variables for the m general MMA constraints.
%  low   = Column vector with the lower asymptotes, calculated and used
%          in the current MMA subproblem.
%  upp   = Column vector with the upper asymptotes, calculated and used
%          in the current MMA subproblem.
%
%epsimin = sqrt(m+n)*10^(-9);
epsimin = 10^(-7);
raa0 = 0.00001;
move = 1.0;
albefa = 0.1;
asyinit = 0.5;
asyincr = 1.2;
asydecr = 0.7;
eeen = ones(n,1);
eeem = ones(m,1);
zeron = zeros(n,1);

% Calculation of the asymptotes low and upp :
if iter < 2.5
  low = xval - asyinit*(xmax-xmin);
  upp = xval + asyinit*(xmax-xmin);
else
  zzz = (xval-xold1).*(xold1-xold2);
  factor = eeen;
  factor(find(zzz > 0)) = asyincr;
  factor(find(zzz < 0)) = asydecr;
  low = xval - factor.*(xold1 - low);
  upp = xval + factor.*(upp - xold1);
  lowmin = xval - 10*(xmax-xmin);
  lowmax = xval - 0.01*(xmax-xmin);
  uppmin = xval + 0.01*(xmax-xmin);
  uppmax = xval + 10*(xmax-xmin);
  low = max(low,lowmin);
  low = min(low,lowmax);
  upp = min(upp,uppmax);
  upp = max(upp,uppmin);
end

% Calculation of the bounds alfa and beta :

zzz1 = low + albefa*(xval-low);
zzz2 = xval - move*(xmax-xmin);
zzz  = max(zzz1,zzz2);
alfa = max(zzz,xmin);
zzz1 = upp - albefa*(upp-xval);
zzz2 = xval + move*(xmax-xmin);
zzz  = min(zzz1,zzz2);
beta = min(zzz,xmax);

% Calculations of p0, q0, P, Q and b.

xmami = xmax-xmin;
xmamieps = 0.00001*eeen;
xmami = max(xmami,xmamieps);
xmamiinv = eeen./xmami;
ux1 = upp-xval;
ux2 = ux1.*ux1;
xl1 = xval-low;
xl2 = xl1.*xl1;
uxinv = eeen./ux1;
xlinv = eeen./xl1;
%
p0 = zeron;
q0 = zeron;
p0 = max(df0dx,0);
q0 = max(-df0dx,0);
%p0(find(df0dx > 0)) = df0dx(find(df0dx > 0));
%q0(find(df0dx < 0)) = -df0dx(find(df0dx < 0));
pq0 = 0.001*(p0 + q0) + raa0*xmamiinv;
p0 = p0 + pq0;
q0 = q0 + pq0;
p0 = p0.*ux2;
q0 = q0.*xl2;
%
P = sparse(m,n);
Q = sparse(m,n);
P = max(dfdx,0);
Q = max(-dfdx,0);
%P(find(dfdx > 0)) = dfdx(find(dfdx > 0));
%Q(find(dfdx < 0)) = -dfdx(find(dfdx < 0));
PQ = 0.001*(P + Q) + raa0*eeem*xmamiinv';
P = P + PQ;
Q = Q + PQ;
P = P * spdiags(ux2,0,n,n);
Q = Q * spdiags(xl2,0,n,n);
b = P*uxinv + Q*xlinv - fval ;
%
%%% Solving the subproblem by a primal-dual Newton method
[xmma,ymma,zmma,lam,xsi,eta,mu,zet,s] = ...
subsolv(m,n,epsimin,low,upp,alfa,beta,p0,q0,P,Q,a0,a,b,c,d);