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Pathfollowv2.m
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function [x,y,result,itn,finaleta] = Pathfollowv2(problem,p1,p2,x0,y0,pickmul0,rate0)
% Generates a point for approximating the solution to the parametric NLP
% min f(x,p)
% subject to cL <= c(x,p) <= cU
%
% prob(p) defines the problem. the dependence of f and c on p is assumed to
% be linear
% x0 is the solution to prob(p1), or an estimate thereof.
% If none is given, prob(p1) is solved as a
% standalone optimization problem.
if nargin==7
rate = rate0;
else
rate = 1.4;
end
if nargin==6
pickmul = pickmul0;
else
pickmul = 1;
end
options = optimset('Display','off','Algorithm','active-set');
%options = optimset('Display','off','Algorithm','interior-point-convex');
%optionslin = optimset('Display','off','Algorithm', 'active-set');
%optionslin = optimset('Display','off','Algorithm', 'dual-simplex');
optionslin = optimoptions('linprog','Algorithm','dual-simplex','Display','off');
result=0;
p=p1;
param = parampath;
clear prob;
sym pp;
theprob = @(pp)problem(pp);
prob = theprob(p);
m = prob.m;
me = prob.me;
eqs = 1:me;
ineqs = me+1:m;
n = prob.n ;
cL = prob.cl; cU = prob.cu;
if nargin == 3
% Solve the base problem
% Do this later
end
[f,g] = prob.obj(x0,p);
[c,J] = prob.cons( x0,p );
[etaold,~] = calceta(x0,y0,prob,f,g,c,J);
if etaold>.01
%get a more accurate multiplier
Act = setdiff(find(c<.01),eqs);
[yact,~,exitflag,~,~] = quadprog([J(Act,:);J(eqs,:)]*[J(Act,:);J(eqs,:)]',(-2*g'*[J(Act,:);J(eqs,:)]')',[],[],[],[],[zeros(length(Act),1);-Inf*ones(length(eqs),1)],[Inf*ones(length(Act)+length(eqs),1)], y0(union(Act,eqs)),options);
y0(Act) = yact(1:length(Act));
y0(eqs) = yact(length(Act)+1:end);
[etaold,~] = calceta(x0,y0,prob,f,g,c,J);
end
epsA = find(abs(y0) > etaold^param.gamma);
epsA = setdiff(epsA,eqs);
epsF = setdiff(ineqs,epsA);
Act = union(epsA,union(eqs,find(c< etaold^(param.gamma))));
converged = 0;
itn = 0;
%%4g %8.1e %s
fprintf('Solving problem %s with p=%3.1e using estimate at p=%3.1e\n',prob.name,p2(1),p1(1));
eqs = eqs';
t = 0;
deltat = param.deltat0;
fprintf('iteration p t eta(x,y,p) deltat sizeepsA Success\n');
fprintf('%6g %3.1e %3.1e %5.2e %3.1e %4g \n',itn,p(1),t,etaold,deltat,length(epsA));
x=x0;
y=y0;
%H = prob.hess(x,y);
etanew = etaold;
success = 0;
fail = 0;
etamax = etaold;
alphahat = 1;
ts = [0];
ys =[x(1)];
testrepeat = 0;
xs = [x'];
ys = [y'];
ts = [0];
Deltats = [deltat];
Etats = [etanew];
%make y satisfy the necessary conditions
dcdp = prob.dcdp;
while ~converged
itn = itn+1;
success = 0;
pred = 0;
%ts = [ts;t+deltat];
p0 = p2*(t)+p1*(1-t);
clear p;
p = p2*(t+deltat)+p1*(1-t-deltat);
clear prob;
prob = theprob(p0);
m = prob.m;
n = prob.n ;
cL = prob.cl; cU = prob.cu;
[f,g] = prob.obj(x,p0);
[c,J] = prob.cons(x,p0);
keepgoingrank = 1;
yold=y;
if itn == 1243 || itn == 1265 || itn == 290
1
end
while keepgoingrank
nAct = length(eqs)+length(epsA);
rk = rank(J([eqs;epsA],:));
if rk == nAct
keepgoingrank = 0;
break;
end
NullJ = null(J([eqs;epsA],:)');
wnJ = NullJ(:,1);
wnJ(find(wnJ>0)) = -eps;
alphahat = min(min(-(y([epsA])./wnJ((length(eqs)+1):end))),1);
y([eqs;epsA]) = y([eqs;epsA])+alphahat*wnJ;
epsA = setdiff(find(y>max(param.etamin,etaold^param.gamma)),eqs);
end
%norm(y-yold)
HC = prob.chess(x);
H = prob.hess(x,y,p);
[etaold,~] = calceta(x,y,prob,f,g,c,J);
xhat = x;
yhat = y;
if (etaold >=1)
%fprintf = 'Initial eta too large, decreasing delta t\n');
deltat = deltat*param.gamma1;
success = 0;
else
%options = optimset('Display','off','TolFun',eps*etaold,'TolX',eps*etaold);
delc = -[H -J([eqs;epsA],:)'; J([eqs;epsA],:) zeros(length(epsA)+length(eqs))]\[g-J'*y;c([eqs;epsA])];
delxc = delc(1:n);
delyc = delc(n+1:end);
delycf = zeros(m,1);
delycf([eqs;epsA]) = delyc;
Jnew = J;
for i=1:m
Jnew(i,:) = Jnew(i,:)+(squeeze(HC(:,:,i))*delxc)';
end
[deltax,~,exitflag,~,lambda] = quadprog(H,zeros(n,1),-Jnew(epsF,:),c(epsF)+Jnew(epsF,:)*delxc+dcdp(epsF,:)*(deltat*(p2-p1)),[Jnew(eqs,:);Jnew(epsA,:)],[-dcdp(eqs,:)*(deltat*(p2-p1));-dcdp(epsA,:)*(deltat*(p2-p1))],[],[], x,options);
%if exitflag < 0
% keyboard;
%end
dynew = zeros(m,1);
dynew(epsF) = (lambda.ineqlin(1:(length(epsF))));
dynew([eqs;epsA]) = -lambda.eqlin;
prob = theprob(p);
[f,g] = prob.obj(xhat+delxc+deltax,p);
[c,J] = prob.cons(xhat +delxc+deltax,p);
ynew = y+dynew+delycf;
[etanew,z] = calceta(xhat+delxc+deltax,ynew,prob,f,g,c,J);
if (etanew<=param.isopt) || (etanew <= etaold^rate && etanew< max(0.01,etamax)) || (etanew < etamax) %|| (fail>=param.maxfail && etanew< (etaold+etamax)/2)%^(1+param.gamma)
success = 1;
xold = x;
yold = y;
%ynew
y = ynew;
t = t+deltat;
x = x+deltax+delxc;
if (etanew <= etaold^1.6 || etanew<=param.isopt) && ~testrepeat
deltat = min(1-t,deltat/param.gamma1);
else
deltat = min(1-t,deltat);
end
testrepeat = 0;
etaold = etanew;
epsA = find(abs(y) > max(param.etamin,etaold^param.gamma));
epsA = setdiff(epsA,eqs);
epsF = setdiff(ineqs,epsA);
Act = [eqs;epsA;setdiff(setdiff(find(c< max(param.etamin,etaold^(param.gamma))),eqs),epsA)];
% union(epsA,union(eqs,find(c< etaold^(param.gamma))));
%epsA
fail = 0;
H = prob.hess(x,y,p);
%get new multiplier
%x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)
lb = zeros(length(Act),1);
lb(1:me) = -Inf;
ub = Inf*ones(length(Act),1);
if pickmul
[ydel,~,exitflag] = linprog(prob.dcdp(Act,:)*deltat*(p2-p1), [J(Act,:)';-J(Act,:)'],[g+abs(z);abs(z)-g],[],[],lb,ub,y(Act),optionslin);
% if exitflag~=1
% x = xold;
% y = yold;
% deltat = deltat*param.gamma1;
% continue;
% end
linobj = prob.dcdp(Act,:)*deltat*(p2-p1);
if exitflag==1 && (linobj'*ydel < linobj'*y(Act)-sqrt(eps))
y(Act) = ydel;
epsA = find(abs(y) > max(param.etamin,etaold^param.gamma));
epsA = setdiff(epsA,eqs);
epsF = setdiff(ineqs,epsA);
Act = union(epsA,union(eqs,find(c< max(param.etamin^0.5,etaold^(param.gamma)))));
%y
%epsA
end
end
xs = [xs;x'];
ts = [ts;t];
ys = [ys;y'];
Deltats = [Deltats;deltat];
Etats = [Etats;etanew];
if etanew > etamax
etamax = etanew;
end
else
deltat = deltat*param.gamma1;
if fail == 0
testrepeat = 1;
end
fail = fail+1;
end
end
if deltat <1/(100000*param.itnmax) && t<1
result = 0;
return
end
sizeact = length(epsA);
% Print iteration, exit if necessary
if mod(itn,10) == 0
fprintf('iteration p t eta(x,y,p) deltat sizeepsA Success\n');
end
fprintf('%6g %3.1e %3.1e %5.2e %3.1e %4g %4g\n',itn,p(1),t,etaold,deltat,sizeact,success);
% ys = [ys;x(1)];
% ts = [ts;t];
if t >= 1 && success% && etanew < param.isopt
fprintf('Successful approximate solution to %s with p=%3.1e\n',prob.name,p2(1));
result = 1;
finaleta = etanew;
break;
end
if itn>=param.itnmax
fprintf('Maximum iterations reached\n');
result = 0;
break;
end
end
g
J
y
% figure(1);
% for i=1:n
% h(i) = subplot(n+m+1,1,i);
% plot(ts,xs(:,i));
% ylabel(strcat('x_',num2str(i)));
% set(h(i),'ylim',[min(xs(:,i))-sqrt(eps),max(xs(:,i))+sqrt(eps)]);
% end
% for j=1:m
% h(n+j) = subplot(n+m+1,1,j+n);
% plot(ts,ys(:,j));
% ylabel(strcat('y_',num2str(j)));
% set(h(n+j),'ylim',[min(ys(:,j))-sqrt(eps),max(ys(:,j))+sqrt(eps)]);
% end
% xlabel('t')
% % h(n+m+1) = subplot(n+m+1,1,m+n+1);
% % plot(ts,Deltats);
% % ylabel('\Delta t');
% % %set(h(n+j),'ylim',[min(ys(:,j))-sqrt(eps),max(ys(:,j))+sqrt(eps)]);
figure(1);
title('dual-degenerate PC-QP');
for i=1:n
h(i) = subplot(n+m,1,i);
plot(ts,xs(:,i));
ylabel(strcat('x(',num2str(i),')'));
%set(h(i),'ylim',[min(xs(:,i)),max(xs(:,i))]);
if (min(xs(:,i)) ~= max(xs(:,i)))
set(h(i),'ylim',[min(xs(:,i)),max(xs(:,i))]);
end
end
for j=1:m
h(n+j) = subplot(n+m,1,j+n);
plot(ts,ys(:,j));
ylabel(strcat('y(',num2str(j),')'));
if (min(ys(:,j)) ~= max(ys(:,j)))
set(h(n+j),'ylim',[min(ys(:,j)),max(ys(:,j))]);
end
end
xlabel('t');
figure(2);
plot(ts,Etats);
ylabel('\eta');
xlabel('t');
title('dual-degenerate PC-QP');
%linkaxes(h')
%plot(ts,ys,'-.o')
end
function [eta,z] = calceta(x,y,prob,f,g,c,J)
vl = c(prob.me+1:end);
z = g-J'*y;
cmp = max(0,y(prob.me+1:end).*min(vl,1));
vioL = min(0,vl);
vioE = c(1:prob.me);
maxComp = norm( cmp ,Inf);
maxViol = norm( [ vioL; vioE ] ,Inf);
eta = .1*max(maxComp,max(maxViol,norm(z,Inf)));
end
function param = parampath
param = struct('gamma', 0.7, ...
'gamma1', 0.6, ...
'itnmax',50000, ...%10000
'deltat0',.05, ...%.05
'maxfail',3, ...
'isopt',0.01,... %1e-6
'etamin',10^(-6));
end