diff --git a/matlab/lmmcp/lmmcp.m b/matlab/lmmcp/lmmcp.m
index 90a8f41952ffe3c7cdd8b08a3474d0dfbd500d0f..f6c0864b0a95e006efa594607c9720306f79e9ac 100644
--- a/matlab/lmmcp/lmmcp.m
+++ b/matlab/lmmcp/lmmcp.m
@@ -22,9 +22,10 @@ function [x,FVAL,EXITFLAG,OUTPUT,JACOB] = lmmcp(FUN,x,lb,ub,options,varargin)
% preprocess : activate preprocessor for phase I (default = 1)
% presteps : number of iterations in phase I (default = 20)
% Termination parameters
-% epsilon2 : termination value of the merit function (default = 1E-16)
-% MaxIter : maximum number of iterations (default = 500)
+% MaxIter : Maximum number of iterations (default = 500)
% tmin : safeguard stepsize (default = 1E-12)
+% TolFun : Termination tolerance on the function value, a positive
+% scalar (default = sqrt(eps))
% Stepsize parameters
% m : number of previous function values to use in the nonmonotone
% line search rule (default = 10)
@@ -85,7 +86,7 @@ function [x,FVAL,EXITFLAG,OUTPUT,JACOB] = lmmcp(FUN,x,lb,ub,options,varargin)
% e-mail: kanzow@mathematik.uni-wuerzburg.de
% petra@mathematik.uni-wuerzburg.de
-% ------------Initialization----------------
+%% Initialization
defaultopt = struct(...
'beta', 0.55,...
'Big', 1e10,...
@@ -93,7 +94,6 @@ defaultopt = struct(...
'deltamin', 1,...
'Display', 'none',...
'epsilon1', 1e-6,...
- 'epsilon2', 1e-16,...
'eta', 0.95,...
'kwatch', 20,...
'lambda1', 0.1,...
@@ -106,6 +106,7 @@ defaultopt = struct(...
'sigma1', 0.5,...
'sigma2', 2,...
'tmin', 1e-12,...
+ 'TolFun', sqrt(eps),...
'watchdog', 1);
if nargin < 4
@@ -122,21 +123,22 @@ else
options = catstruct(defaultopt,options);
end
+warning('off','MATLAB:rankDeficientMatrix')
switch options.Display
- case {'off','none'}
- verbosity = 0;
- case {'iter','iter-detailed'}
- verbosity = 2;
- case {'final','final-detailed'}
- verbosity = 1;
- otherwise
- verbosity = 0;
+ case {'off','none'}
+ verbosity = 0;
+ case {'iter','iter-detailed'}
+ verbosity = 2;
+ case {'final','final-detailed'}
+ verbosity = 1;
+ otherwise
+ verbosity = 0;
end
% parameter settings
eps1 = options.epsilon1;
-eps2 = options.epsilon2;
+eps2 = 0.5*options.TolFun^2;
null = options.null;
Big = options.Big;
@@ -212,14 +214,14 @@ aux(1) = Psix;
MaxPsi = Psix;
if watchdog==1
- kbest = k;
- xbest = x;
- Phibest = Phix;
- Psibest = Psix;
- DPhibest = DPhix;
- DPsibest = DPsix;
- normDPsibest = normDPsix;
-end;
+ kbest = k;
+ xbest = x;
+ Phibest = Phix;
+ Psibest = Psix;
+ DPhibest = DPhix;
+ DPsibest = DPsix;
+ normDPsibest = normDPsix;
+end
% initial output
if verbosity > 1
@@ -229,76 +231,74 @@ if verbosity > 1
fprintf('%4.0f %24.5e %24.5e\n',k,Psix,normDPsix);
end
-%
-% Preprocessor using local method
-%
+%% Preprocessor using local method
if preprocess==1
if verbosity > 1
disp('************************** Preprocessor ****************************')
end
-
- normpLM=1;
- while (k < presteps) && (Psix > eps2) && (normpLM>null)
- k=k+1;
-
- % choice of Levenberg-Marquardt parameter, note that we do not use
- % the condition estimator for large-scale problems, although this
- % may cause numerical problems in some examples
-
- i=0;
- mu=0;
- if n<100
- i=1;
- mu=1e-16;
- if condest(DPhix'*DPhix)>1e25
- mu=1e-6/(k+1);
- end
- end
- if i==1
- pLM= [ DPhix ; sqrt(mu)*speye(n)]\[-Phix;sparse(n,1)];
- else
- pLM=-DPhix\Phix;
- end
- normpLM=norm(pLM);
-
- % compute the projected Levenberg-Marquard step onto box Xk
- lbnew=max(min(lb-x,0),-delta);
- ubnew=min(max(ub-x,0),delta);
- d=max(lbnew,min(pLM,ubnew));
- xnew=x+d;
-
- % function evaluations etc.
- [Fxnew,DFxnew] = feval(FUN,xnew,varargin{:});
- Phixnew=Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
- Psixnew=0.5*(Phixnew'*Phixnew);
- normPhixnew=norm(Phixnew);
-
- % update of delta
- if normPhixnew<=eta*normPhix
- delta=max(deltamin,sigma2*delta);
- elseif normPhixnew>5*eta*normPhix
- delta=max(deltamin,sigma1*delta);
- end
-
- % update
- x=xnew;
- Fx=Fxnew;
- DFx=DFxnew;
- Phix=Phixnew;
- Psix=Psixnew;
- normPhix=normPhixnew;
- DPhix=DPhi(x,Fx,DFx,lb,ub,lambda1,lambda2,n,Indexset);
- DPsix=DPhix'*Phix;
- normDPsix=norm(DPsix,inf);
-
- % output at each iteration
- t=1;
- if verbosity > 1
- fprintf('%4.0f %24.5e %24.5e %11.7g\n',k,Psix,normDPsix,t);
+
+ normpLM=1;
+ while (k < presteps) && (Psix > eps2) && (normpLM>null)
+ k = k+1;
+
+ % choice of Levenberg-Marquardt parameter, note that we do not use
+ % the condition estimator for large-scale problems, although this
+ % may cause numerical problems in some examples
+
+ i = false;
+ mu = 0;
+ if n<100
+ i = true;
+ mu = 1e-16;
+ if condest(DPhix'*DPhix)>1e25
+ mu = 1e-6/(k+1);
end
- end
+ end
+ if i
+ pLM = [DPhix; sqrt(mu)*speye(n)]\[-Phix; zeros(n,1)];
+ else
+ pLM = -DPhix\Phix;
+ end
+ normpLM = norm(pLM);
+
+ % compute the projected Levenberg-Marquard step onto box Xk
+ lbnew = max(min(lb-x,0),-delta);
+ ubnew = min(max(ub-x,0),delta);
+ d = max(lbnew,min(pLM,ubnew));
+ xnew = x+d;
+
+ % function evaluations etc.
+ [Fxnew,DFxnew] = feval(FUN,xnew,varargin{:});
+ Phixnew = Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
+ Psixnew = 0.5*(Phixnew'*Phixnew);
+ normPhixnew = norm(Phixnew);
+
+ % update of delta
+ if normPhixnew<=eta*normPhix
+ delta = max(deltamin,sigma2*delta);
+ elseif normPhixnew>5*eta*normPhix
+ delta = max(deltamin,sigma1*delta);
+ end
+
+ % update
+ x = xnew;
+ Fx = Fxnew;
+ DFx = DFxnew;
+ Phix = Phixnew;
+ Psix = Psixnew;
+ normPhix = normPhixnew;
+ DPhix = DPhi(x,Fx,DFx,lb,ub,lambda1,lambda2,n,Indexset);
+ DPsix = DPhix'*Phix;
+ normDPsix = norm(DPsix,inf);
+
+ % output at each iteration
+ t=1;
+ if verbosity > 1
+ fprintf('%4.0f %24.5e %24.5e %11.7g\n',k,Psix,normDPsix,t);
+ end
+ end
end
% terminate program or redefine current iterate as original initial point
@@ -323,12 +323,10 @@ elseif preprocess==1 && Psix>=eps2
if verbosity > 1
disp('******************** Restart with initial point ********************')
fprintf('%4.0f %24.5e %24.5e\n',k_main,Psix0,normDPsix0);
- end
+ end
end
-%
-% Main algorithm
-%
+%% Main algorithm
if verbosity > 1
disp('************************** Main program ****************************')
@@ -336,96 +334,96 @@ end
while (k < kmax) && (Psix > eps2)
- % choice of Levenberg-Marquardt parameter, note that we do not use
- % the condition estimator for large-scale problems, although this
- % may cause numerical problems in some examples
+ % choice of Levenberg-Marquardt parameter, note that we do not use
+ % the condition estimator for large-scale problems, although this
+ % may cause numerical problems in some examples
+
+ i = false;
+ if n<100
+ i = true;
+ mu = 1e-16;
+ if condest(DPhix'*DPhix)>1e25
+ mu = 1e-1/(k+1);
+ end
+ end
+
+ % compute a Levenberg-Marquard direction
+
+ if i
+ d = [DPhix; sqrt(mu)*speye(n)]\[-Phix; zeros(n,1)];
+ else
+ d = -DPhix\Phix;
+ end
- i=0;
- if n<100
- i=1;
- mu=1e-16;
- if condest(DPhix'*DPhix)>1e25
- mu=1e-1/(k+1);
- end
- end
-
- % compute a Levenberg-Marquard direction
-
- if i==1
- d= [ DPhix ; sqrt(mu)*eye(n)]\[-Phix;zeros(n,1)];
- else
- d=-DPhix\Phix;
- end
-
- % computation of steplength t using the nonmonotone Armijo-rule
- % starting with the 6-th iteration
-
- % computation of steplength t using the monotone Armijo-rule if
- % d is a 'good' descent direction or k<=5
-
- t = 1;
- xnew = x+d;
- Fxnew = feval(FUN,xnew,varargin{:});
- Phixnew = Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
- Psixnew = 0.5*(Phixnew'*Phixnew);
- const = sigma*DPsix'*d;
-
- while (Psixnew > MaxPsi + const*t) && (t > tmin)
- t = t*beta;
- xnew = x+t*d;
- Fxnew = feval(FUN,xnew,varargin{:});
- Phixnew = Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
- Psixnew = 0.5*(Phixnew'*Phixnew);
- end
-
- % updatings
- x=xnew;
- Fx=Fxnew;
- Phix=Phixnew;
- Psix=Psixnew;
- [~,DFx]=feval(FUN,x,varargin{:});
- DPhix=DPhi(x,Fx,DFx,lb,ub,lambda1,lambda2,n,Indexset);
- DPsix=DPhix'*Phix;
- normDPsix=norm(DPsix);
- k=k+1;
- k_main=k_main+1;
-
- if k_main<=5
- aux(mod(k_main,m)+1)=Psix;
+ % computation of steplength t using the nonmonotone Armijo-rule
+ % starting with the 6-th iteration
+
+ % computation of steplength t using the monotone Armijo-rule if
+ % d is a 'good' descent direction or k<=5
+
+ t = 1;
+ xnew = x+d;
+ Fxnew = feval(FUN,xnew,varargin{:});
+ Phixnew = Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
+ Psixnew = 0.5*(Phixnew'*Phixnew);
+ const = sigma*DPsix'*d;
+
+ while (Psixnew > MaxPsi + const*t) && (t > tmin)
+ t = t*beta;
+ xnew = x+t*d;
+ Fxnew = feval(FUN,xnew,varargin{:});
+ Phixnew = Phi(xnew,Fxnew,lb,ub,lambda1,lambda2,n,Indexset);
+ Psixnew = 0.5*(Phixnew'*Phixnew);
+ end
+
+ % updatings
+ x = xnew;
+ Fx = Fxnew;
+ Phix = Phixnew;
+ Psix = Psixnew;
+ [~,DFx] = feval(FUN,x,varargin{:});
+ DPhix = DPhi(x,Fx,DFx,lb,ub,lambda1,lambda2,n,Indexset);
+ DPsix = DPhix'*Phix;
+ normDPsix = norm(DPsix);
+ k = k+1;
+ k_main = k_main+1;
+
+ if k_main<=5
+ aux(mod(k_main,m)+1) = Psix;
+ MaxPsi = Psix;
+ else
+ aux(mod(k_main,m)+1) = Psix;
+ MaxPsi = max(aux);
+ end
+
+ % updatings for the watchdog strategy
+ if watchdog ==1
+ if Psix<Psibest
+ kbest = k;
+ xbest = x;
+ Phibest = Phix;
+ Psibest = Psix;
+ DPhibest = DPhix;
+ DPsibest = DPsix;
+ normDPsibest = normDPsix;
+ elseif k-kbest>kwatch
+ x=xbest;
+ Phix=Phibest;
+ Psix=Psibest;
+ DPhix=DPhibest;
+ DPsix=DPsibest;
+ normDPsix=normDPsibest;
MaxPsi=Psix;
- else
- aux(mod(k_main,m)+1) = Psix;
- MaxPsi = max(aux);
- end;
-
- % updatings for the watchdog strategy
- if watchdog ==1
- if Psix<Psibest
- kbest = k;
- xbest = x;
- Phibest = Phix;
- Psibest = Psix;
- DPhibest = DPhix;
- DPsibest = DPsix;
- normDPsibest = normDPsix;
- elseif k-kbest>kwatch
- x=xbest;
- Phix=Phibest;
- Psix=Psibest;
- DPhix=DPhibest;
- DPsix=DPsibest;
- normDPsix=normDPsibest;
- MaxPsi=Psix;
- end;
- end;
-
- if verbosity > 1
- % output at each iteration
- fprintf('%4.0f %24.5e %24.5e %11.7g\n',k,Psix,normDPsix,t);
- end
-end;
-
-% final output
+ end
+ end
+
+ if verbosity > 1
+ % output at each iteration
+ fprintf('%4.0f %24.5e %24.5e %11.7g\n',k,Psix,normDPsix,t);
+ end
+end
+
+%% Final output
if Psix<=eps2
EXITFLAG = 1;
if verbosity > 0, disp('Approximate solution found.'); end
@@ -446,9 +444,10 @@ OUTPUT.Psix = Psix;
OUTPUT.normDPsix = normDPsix;
JACOB = DFx;
-% Subfunctions
+%% Subfunctions
function y = Phi(x,Fx,lb,ub,lambda1,lambda2,n,Indexset)
+%% PHI
y = zeros(2*n,1);
phi_u = sqrt((ub-x).^2+Fx.^2)-ub+x+Fx;
@@ -472,7 +471,7 @@ y([LZ; I3]) = lambda2*(max(0,x(I3)-lb(I3)).*max(0,Fx(I3))+max(0,ub(I3)-x(I3)).*m
function H = DPhi(x,Fx,DFx,lb,ub,lambda1,lambda2,n,Indexset)
-% DPHI evaluates an element of the C-subdifferential of operator Phi
+%% DPHI evaluates an element of the C-subdifferential of operator Phi
null = 1e-8;
beta_l = zeros(n,1);
@@ -521,14 +520,14 @@ Da(I1b) = (x(I1b)-lb(I1b))./denom1(I1b)-1;
Db(I1b) = Fx(I1b)./denom1(I1b)-1;
I1b = Indexset==1 & beta_l~=0;
if any(I1b)
- Da(I1b) = z(I1b)./denom2(I1b)-1;
- Db(I1b) = (DFx(I1b,:)*z)./denom2(I1b)-1;
+ Da(I1b) = z(I1b)./denom2(I1b)-1;
+ Db(I1b) = (DFx(I1b,:)*z)./denom2(I1b)-1;
end
-I1a = I(Indexset==1 & alpha_l==1);
+I1a = I(Indexset==1 & alpha_l==1);
if any(I1a)
- H2(I1a,:) = repmat(x(I1a)-lb(I1a),1,n).*DFx(I1a,:)+sparse(1:length(I1a),I1a, ...
- Fx(I1a),length(I1a),n,length(I1a));
+ H2(I1a,:) = bsxfun(@times,x(I1a)-lb(I1a),DFx(I1a,:))+...
+ sparse(1:length(I1a),I1a,Fx(I1a),length(I1a),n,length(I1a));
end
I2 = Indexset==2;
@@ -544,14 +543,14 @@ Da(I2b) = (ub(I2b)-x(I2b))./denom1(I2b)-1;
Db(I2b) = -Fx(I2b)./denom1(I2b)-1;
I2b = Indexset==2 & beta_u~=0;
if any(I2b)
- Da(I2b) = -z(I2b)./denom2(I2b)-1;
- Db(I2b) = -(DFx(I2b,:)*z)./denom2(I2b)-1;
+ Da(I2b) = -z(I2b)./denom2(I2b)-1;
+ Db(I2b) = -(DFx(I2b,:)*z)./denom2(I2b)-1;
end
-I2a = I(Indexset==2 & alpha_u==1);
+I2a = I(Indexset==2 & alpha_u==1);
if any(I2a)
- H2(I2a,:) = repmat(x(I2a)-ub(I2a),1,n).*DFx(I2a,:)+sparse(1:length(I2a),I2a, ...
- Fx(I2a),length(I2a),n,length(I2a));
+ H2(I2a,:) = bsxfun(@times,x(I2a)-ub(I2a),DFx(I2a,:))+...
+ sparse(1:length(I2a),I2a,Fx(I2a),length(I2a),n,length(I2a));
end
I3 = Indexset==3;
@@ -593,23 +592,23 @@ end
Da(I3) = ai(I3)+bi(I3).*ci(I3);
Db(I3) = bi(I3).*di(I3);
-I3a = I(Indexset==3 & alpha_l==1 & alpha_u==1);
+I3a = I(Indexset==3 & alpha_l==1 & alpha_u==1);
if any(I3a)
- H2(I3a,:) = repmat(-lb(I3a)-ub(I3a)+2*x(I3a),1,n).*DFx(I3a,:)+...
- 2*sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
+ H2(I3a,:) = bsxfun(@times,-lb(I3a)-ub(I3a)+2*x(I3a),DFx(I3a,:))+...
+ 2*sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
end
-I3a = I(Indexset==3 & alpha_l==1 & alpha_u~=1);
+I3a = I(Indexset==3 & alpha_l==1 & alpha_u~=1);
if any(I3a)
- H2(I3a,:) = repmat(x(I3a)-lb(I3a),1,n).*DFx(I3a,:)+...
- sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
+ H2(I3a,:) = bsxfun(@times,x(I3a)-lb(I3a),DFx(I3a,:))+...
+ sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
end
-I3a = I(Indexset==3 & alpha_l~=1 & alpha_u==1);
+I3a = I(Indexset==3 & alpha_l~=1 & alpha_u==1);
if any(I3a)
- H2(I3a,:) = repmat(x(I3a)-ub(I3a),1,n).*DFx(I3a,:)+...
- sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
+ H2(I3a,:) = bsxfun(@times,x(I3a)-ub(I3a),DFx(I3a,:))+...
+ sparse(1:length(I3a),I3a,Fx(I3a),length(I3a),n,length(I3a));
end
-%H1 = sparse(1:n,1:n,Da,n,n,n)+Db(:,ones(n,1)).*DFx;
-H1 = bsxfun(@times,Db,DFx);
-H1 = spdiags(diag(H1)+Da,0,H1);
-H = [lambda1*H1; lambda2*H2];
+H1 = bsxfun(@times,Db,DFx);
+H1 = spdiags(diag(H1)+Da,0,H1);
+
+H = [lambda1*H1; lambda2*H2];