qpsub.m

工具箱 (分枝定界法工具箱) matlab 中使用

               projSD = pinv(projH)*(-Zgf);
            else % LLS
               projH = HZ'*HZ; 
               Zgf = Z'*gf;
               projSD = pinv(projH)*(-Zgf);
            end
            
            % Check if compatible
            if norm(projH*projSD+Zgf) > 10*eps*(norm(projH) + norm(Zgf))
               % system is incompatible --> it's a "chute": use SD from compdir
               % unbounded in SD direction
               if norm(SD) > errnorm
                  if normalize < 0
                     STEPMIN=abs((X(numberOfVariables)+1e-5)/(SD(numberOfVariables)+eps));
                  else 
                     STEPMIN = 1e16;
                  end
                  X=X+STEPMIN*SD;
                  how='unbounded'; 
                  % was exitflag = 5;
                  exitflag = -1;
               else % norm(SD) <= errnorm
                  how = 'ill posed';
                  %was exitflag = 6;
                  exitflag = -1;
               end
               if verbosity > 0
                  if norm(SD) > errnorm
                     disp('Exiting: The solution is unbounded and at infinity;')
                     disp('         the constraints are not restrictive enough.') 
                  else
                     disp('Exiting: The search direction is close to zero; ')
                     disp('      the problem is ill-posed.')
                     disp('      The gradient of the objective function may be zero')
                     disp('         or the problem may be badly conditioned.')
                  end
               end % if verbosity > 0
               output.iterations = iterations;
               return
            else % Convex -- move to the minimum (compatible system)
               SD = Z*projSD;
               dirType = 'singular';
               % First check if constraint is violated.
               GSD=A*SD;
               indf = find((GSD > errnorm * norm(SD))  &  ~aix);
               if isempty(indf) % No constraints to hit
                  STEPMIN=1;
                  delete_constr = 1;
                  dist=[]; ind2=[]; ind=[];
               else % Find distance to the nearest constraint
                  dist = abs(cstr(indf)./GSD(indf));
                  [STEPMIN,ind2] =  min(dist);
                  ind2 = find(dist == STEPMIN);
                  % Bland's rule for anti-cycling: if there is more than one 
                  % blocking constraint then add the one with the smallest index.
                  ind=indf(min(ind2));
               end
               if STEPMIN > 1  % Overstepped minimum; reset STEPMIN
                  STEPMIN = 1;
                  delete_constr = 1;
               end
               X = X + STEPMIN*SD; 
            end
         end % if ~is_qp | smallRealEig < -eps
      end % if strcmp(dirType, NewtonStep)
   end % if ~isempty(indf)& isfinite(STEPMIN) % Hit a constraint
   
   if delete_constr
      % Note: only reach here if a minimum in the current subspace found
      if ACTCNT>0
         if ACTCNT>=numberOfVariables-1, 
            % Avoid case when CIND is greater than ACTCNT
            if CIND <= ACTCNT
               ACTSET(CIND,:)=[];
               ACTIND(CIND)=[]; 
            end
         end
         if is_qp
            rlambda = -R\(Q'*(H*X+f));
         elseif LLS
            rlambda = -R\(Q'*(H'*(H*X-f)));
            % else: lp does not reach this point
         end
         actlambda = rlambda;
         actlambda(eqix) = abs(rlambda(eqix));
         indlam = find(actlambda < 0);
         if (~length(indlam)) 
            lambda(indepInd(ACTIND)) = normf * (rlambda./normA(ACTIND));
            output.iterations = iterations;
            return
         end
         % Remove constraint
         lind = find(ACTIND == min(ACTIND(indlam)));
         lind=lind(1);
         ACTSET(lind,:) = [];
         aix(ACTIND(lind)) = 0;
         [Q,R]=qrdelete(Q,R,lind);
         ACTIND(lind) = [];
         ACTCNT = ACTCNT - 2;
         simplex_iter = 0;
         ind = 0;
      else % ACTCNT == 0
         output.iterations = iterations;
         return
      end
      delete_constr = 0;
   end
   
   % Calculate gradient w.r.t objective at this point
   if is_qp
      gf=H*X+f;
   elseif LLS % LLS
      gf=H'*(H*X-f);
      % else gf=f still true.
   end
   
   
   % Update X and calculate constraints
   cstr = A*X-B;
   cstr(eqix) = abs(cstr(eqix));
   % Check no constraint is violated
   if normalize < 0 
      if X(numberOfVariables,1) < eps
         output.iterations = iterations;
         return;
      end
   end
   
   if max(cstr) > 1e5 * errnorm
      if max(cstr) > norm(X) * errnorm 
         if ( verbosity > 0 ) & ( exitflag == 1 )
            disp('Note: The problem is badly conditioned;')
            disp('         the solution may not be reliable') 
            % verbosity = 0;
         end
         how='unreliable'; 
         % exitflag = 2;
         exitflag = -1;
         if 0
            X=X-STEPMIN*SD;
            output.iterations = iterations;
            return
         end
      end
   end
   
   if ind % Hit a constraint
      aix(ind)=1;
      ACTSET(CIND,:)=A(ind,:);
      ACTIND(CIND)=ind;
      [m,n]=size(ACTSET);
      [Q,R] = qrinsert(Q,R,CIND,A(ind,:)');
   end
   if oldind 
      aix(oldind) = 0; 
   end
   if ~simplex_iter
      % Z = null(ACTSET);
      [m,n]=size(ACTSET);
      Z = Q(:,m+1:n);
      ACTCNT=ACTCNT+1;
      if ACTCNT == numberOfVariables - 1, simplex_iter = 1; end
      CIND=ACTCNT+1;
      oldind = 0; 
   else
      rlambda = -R\(Q'*gf);
      
      if isinf(rlambda(1)) & rlambda(1) < 0 
         fprintf('         Working set is singular; results may still be reliable.\n');
         [m,n] = size(ACTSET);
         rlambda = -(ACTSET + sqrt(eps)*randn(m,n))'\gf;
      end
      actlambda = rlambda;
      actlambda(eqix)=abs(actlambda(eqix));
      indlam = find(actlambda<0);
      if length(indlam)
         if STEPMIN > errnorm
            % If there is no chance of cycling then pick the constraint 
            % which causes the biggest reduction in the cost function. 
            % i.e the constraint with the most negative Lagrangian 
            % multiplier. Since the constraints are normalized this may 
            % result in less iterations.
            [minl,CIND] = min(actlambda);
         else
            % Bland's rule for anti-cycling: if there is more than one 
            % negative Lagrangian multiplier then delete the constraint
            % with the smallest index in the active set.
            CIND = find(ACTIND == min(ACTIND(indlam)));
         end
         
         [Q,R]=qrdelete(Q,R,CIND);
         Z = Q(:,numberOfVariables);
         oldind = ACTIND(CIND);
      else
         lambda(indepInd(ACTIND))= normf * (rlambda./normA(ACTIND));
         output.iterations = iterations;
         return
      end
   end %if ACTCNT<numberOfVariables
      
   if (is_qp)
      Zgf = Z'*gf; 
      if ~isempty(Zgf) & (norm(Zgf) < 1e-15) 
         SD = zeros(numberOfVariables,1); 
      else
         [SD, dirType] = compdir(Z,H,gf,numberOfVariables,f);
      end
   elseif (LLS)
      Zgf = Z'*gf;
      HZ = H*Z;
      if (norm(Zgf) < 1e-15)
         SD = zeros(numberOfVariables,1);
      else
         HXf=H*X-f;
         gf=H'*(HXf);
         [mm,nn]=size(HZ);
         if mm >= nn
            [QHZ, RHZ] =  qr(HZ);
            Pd = QHZ'*HXf;
            % SD = - Z*(RHZ(1:nn, 1:nn) \ Pd(1:nn,:));
            % Now need to check which is dependent
            if min(size(RHZ))==1 % Make sure RHZ isn't a vector
               depInd = find( abs(RHZ(1,1)) < tolDep);
            else
               depInd = find( abs(diag(RHZ)) < tolDep );
            end  
         end
         if mm >= nn & isempty(depInd) % Newton step
            SD = - Z*(RHZ(1:nn, 1:nn) \ Pd(1:nn,:));
            dirType = NewtonStep;
         else % steepest descent direction
            SD = -Z*(Z'*gf);
            dirType = SteepDescent;
         end
      end
   else % LP
      if ~simplex_iter
         SD = -Z*(Z'*gf);
         gradsd = norm(SD);
      else
         gradsd = Z'*gf;
         if  gradsd > 0
            SD = -Z;
         else
            SD = Z;
         end
      end
      if abs(gradsd) < 1e-10  % Search direction null
         % Check whether any constraints can be deleted from active set.
         % rlambda = -ACTSET'\gf;
         if ~oldind
            rlambda = -R\(Q'*gf);
         end
         actlambda = rlambda;
         actlambda(1:neqcstr) = abs(actlambda(1:neqcstr));
         indlam = find(actlambda < errnorm);
         lambda(indepInd(ACTIND)) = normf * (rlambda./normA(ACTIND));
         if ~length(indlam)
            output.iterations = iterations;
            return
         end
         cindmax = length(indlam);
         cindcnt = 0;
         newactcnt = 0;
         while (abs(gradsd) < 1e-10) & (cindcnt < cindmax)
            cindcnt = cindcnt + 1;
            if oldind
               % Put back constraint which we deleted
               [Q,R] = qrinsert(Q,R,CIND,A(oldind,:)');
            else
               simplex_iter = 0;
               if ~newactcnt
                  newactcnt = ACTCNT - 1;
               end
            end
            CIND = indlam(cindcnt);
            oldind = ACTIND(CIND);
            
            [Q,R]=qrdelete(Q,R,CIND);
            [m,n]=size(ACTSET);
            Z = Q(:,m:n);
            
            if m ~= numberOfVariables
               SD = -Z*Z'*gf;
               gradsd = norm(SD);
            else
               gradsd = Z'*gf;
               if  gradsd > 0
                  SD = -Z;
               else
                  SD = Z;
               end
            end
         end
         if abs(gradsd) < 1e-10  % Search direction still null
            output.iterations = iterations;
            return;
         end
         lambda = zeros(ncstr,1);
         if newactcnt 
            ACTCNT = newactcnt;
         end
      end
   end
   
   if simplex_iter & oldind
      % Avoid case when CIND is greater than ACTCNT
      if CIND <= ACTCNT
         ACTIND(CIND)=[];
         ACTSET(CIND,:)=[];
         CIND = numberOfVariables;
      end
   end 
end % while 1
if iterations > maxiter
   exitflag = 0;
   how = 'ill-conditioned';   
end

output.iterations = iterations;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[Q,R,A,B,CIND,X,Z,actlambda,how,...
      ACTSET,ACTIND,ACTCNT,aix,eqix,neqcstr,ncstr,remove,exitflag]= ...
   eqnsolv(A,B,eqix,neqcstr,ncstr,numberOfVariables,LLS,H,X,f,normf,normA,verbosity, ...
   aix,how,exitflag)
% EQNSOLV Helper function for QPSUB.
%    Finds a feasible point with respect to the equality constraints.
%    If the equalities are dependent but not consistent, warning
%    messages are given. If the equalities are dependent but consistent, 
%    the redundant constraints are removed and the corresponding variables 
%    adjusted.

% set tolerances
tolDep = 100*numberOfVariables*eps;      
tolCons = 1e-10;

actlambda = [];
aix(eqix)=ones(neqcstr,1);
ACTSET=A(eqix,:);
ACTIND=eqix;
ACTCNT=neqcstr;
CIND=neqcstr+1;
Z=[]; Anew=[]; Bnew=[]; remove =[];

% See if the equalities form a consistent system:
%   QR factorization of A
[Qa,Ra,Ea]=qr(A(eqix,:));
% Now need to check which is dependent
if min(size(Ra))==1 % Make sure Ra isn't a vector
   depInd = find( abs(Ra(1,1)) < tolDep);
else
   depInd = find( abs(diag(Ra)) < tolDep );
end
if neqcstr > numberOfVariables
   depInd = [depInd; ((numberOfVariables+1):neqcstr)'];
end      

if ~isempty(depInd)
   if verbosity > 0
      disp('The equality constraints are dependent.')
   end
   how='dependent';
   exitflag = 1;
   bdepInd =  abs(Qa(:,depInd)'*B(eqix)) >= tolDep ;
   
   if any( bdepInd ) % Not consistent
      how='infeasible';   
      exitflag = 9;exitflag = -1;
      if verbosity > 0
         disp('The system of equality constraints is not consistent.');
         if ncstr > neqcstr
            disp('The inequality constraints may or may not be satisfied.');
         end
         disp('  There is no feasible solution.')
      end
   else % the equality constraints are consistent
      numDepend = nnz(depInd);
      % delete the redundant constraints:
      % By QR factoring the transpose, we see which columns of A'
      %   (rows of A) move to the end
      [Qat,Rat,Eat]=qr(ACTSET');        
      [i,j] = find(Eat); % Eat permutes the columns of A' (rows of A)
      remove = i(depInd);
      if verbosity > 0
         disp('The system of equality constraints is consistent. Removing');
         disp('the following dependent constraints before continuing:');
         disp(remove)
      end
      A(eqix(remove),:)=[];
      B(eqix(remove))=[];
      neqcstr = neqcstr - nnz(remove);
      ncstr = ncstr - nnz(remove);
      eqix = 1:neqcstr;
      aix=[ones(neqcstr,1); zeros(ncstr-neqcstr,1)];
      ACTIND = eqix;
      ACTSET=A(eqix,:);
      
      CIND = neqcstr+1;
      ACTCNT = neqcstr;
   end % consistency check
end % dependency check

if ~strcmp(how,'infeasible')
   % Find a feasible point
   if max(abs(A(eqix,:)*X-B(eqix))) > tolCons
      X = A(eqix,:)\B(eqix);  
   end
end

[Q,R]=qr(ACTSET');
Z = Q(:,neqcstr+1:numberOfVariables);

% End of eqnsolv.m