278 lines
12 KiB
Mathematica
278 lines
12 KiB
Mathematica
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function varargout = norm(varargin)
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%NORM (overloaded)
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%
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% t = NORM(x,P)
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%
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% The variable t can only be used in convexity preserving
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% operations such as t<=1, max(t,y)<=1, minimize t etc.
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%
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% For matrices...
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% NORM(X) models the largest singular value of X, max(svd(X)).
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% NORM(X,2) is the same as NORM(X).
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% NORM(X,1) models the 1-norm of X, the largest column sum, max(sum(abs(X))).
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% NORM(X,inf) models the infinity norm of X, the largest row sum, max(sum(abs(X'))).
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% NORM(X,'inf') same as above
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% NORM(X,'fro') models the Frobenius norm, sqrt(sum(diag(X'*X))).
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% NORM(X,'nuc') models the Nuclear norm, sum of singular values.
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% NORM(X,'*') same as above
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% NORM(X,'tv') models the (isotropic) total variation semi-norm
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% For vectors...
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% NORM(V) = norm(V,2) = standard Euclidean norm.
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% NORM(V,inf) = max(abs(V)).
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% NORM(V,1) = sum(abs(V))
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%
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% SEE ALSO SUMK, SUMABSK
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%% ***************************************************
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% This file defines a nonlinear operator for YALMIP
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%
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% It can take three different inputs
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% For DOUBLE inputs, it returns standard double values
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% For SDPVAR inputs, it generates an internal variable
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%
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% When first input is 'model' it returns the graph
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% in the first output and structure describing some
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% properties of the operator.
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%% ***************************************************
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switch class(varargin{1})
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case 'sdpvar' % Overloaded operator for SDPVAR objects. Pass on args and save them.
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if nargin == 1
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varargout{1} = yalmip('define',mfilename,varargin{1},2);
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else
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switch varargin{2}
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case {1,2,inf,'inf','fro'}
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varargout{1} = yalmip('define',mfilename,varargin{:});
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case 'tv'
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if ~isreal(varargin{1})
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error('Total variation norm not yet implemented for complex arguments');
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end
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if min(varargin{1}.dim)==1
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varargout{1} = norm(diff(varargin{1}),1);
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return
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end
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varargout{1} = yalmip('define','norm_tv',varargin{:});
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case {'nuclear','*'}
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if min(size(varargin{1}))==1
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varargout{1} = norm(varargin{1},1);
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else
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varargout{1} = yalmip('define','norm_nuclear',varargin{:});
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end
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otherwise
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if isreal(varargin{1}) & min(size(varargin{1}))==1 & isnumeric(varargin{2})
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varargout{1} = pnorm(varargin{:});
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else
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error('norm(x,P) only supported for P = 1, 2, inf, ''fro'' and ''nuclear''');
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end
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end
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end
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case 'char' % YALMIP sends 'model' when it wants the epigraph or hypograph
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switch varargin{1}
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case 'graph'
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t = varargin{2};
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X = varargin{3};
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p = varargin{4};
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% Code below complicated by two things
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% 1: Absolute value for complex data -> cone constraints on
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% elements
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% 2: SUBSREF does not call SDPVAR subsref -> use extsubsref.m
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switch p
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case 1
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if issymmetric(X)
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Z = sdpvar(size(X,1),size(X,2));
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else
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Z = sdpvar(size(X,1),size(X,2),'full');
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end
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if min(size(X))>1
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if isreal(X)
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z = reshape(Z,[],1);
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x = reshape(X,[],1);
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F = (-z <= x <= z);
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else
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F = ([]);
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zvec = reshape(Z,1,[]);
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xrevec=reshape(real(X),1,[]);
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ximvec=reshape(imag(X),1,[]);
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F = [F,cone([zvec;xrevec;ximvec])];
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end
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F = F + (sum(Z,1) <= t);
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else
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if isreal(X)
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% Standard definition
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% F = (-t <= X <= t);
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X = reshape(X,[],1);
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Z = reshape(Z,[],1);
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Xbase = getbase(X);
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Constant = find(~any(Xbase(:,2:end),2));
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if ~isempty(Constant)
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% Exploit elements without any
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% decision variables
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r1 = ones(length(Z),1);
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r2 = zeros(length(Z),1);
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r1(Constant) = 0;
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r2(Constant) = abs(Xbase(Constant,1));
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Z = Z.*r1 + r2;
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end
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F = (-Z <= X <= Z) + (sum(Z) <= t);
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else
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F = (cone([reshape(Z,1,[]);real(reshape(X,1,[]));imag(reshape(X,1,[]))]));
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F = F + (sum(Z) <= t);
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end
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end
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case 2
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if min(size(X))>1
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F = ([t*eye(size(X,1)) X;X' t*eye(size(X,2))])>=0;
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else
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F = cone(X(:),t);
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end
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case {inf,'inf'}
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if min(size(X))>1
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Z = sdpvar(size(X,1),size(X,2),'full');
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if isreal(X)
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F = (-Z <= X <= Z);
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else
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F = ([]);
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for i = 1:size(X,1)
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for j = 1:size(X,2)
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xi = extsubsref(X,i,j);
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zi = extsubsref(Z,i,j);
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F = F + (cone([real(xi);imag(xi)],zi));
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end
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end
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end
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F = F + (sum(Z,2) <= t);
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else
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if isreal(X)
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F = (-t <= X <= t);
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[M,m,infbound] = derivebounds(X);
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if ~infbound
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F = F + (0 <= t <= max(max(abs([m M]))));
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end
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else
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F = ([]);
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for i = 1:length(X)
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xi = extsubsref(X,i);
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F = F + (cone([real(xi);imag(xi)],t));
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end
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end
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end
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case 'fro'
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X.dim(1)=X.dim(1)*X.dim(2);
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X.dim(2)=1;
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F = (cone(X,t));
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case 'nuclear'
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U = sdpvar(X.dim(2));
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V = sdpvar(X.dim(1));
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F = [trace(U)+trace(V) <= 2*t, [U X';X V]>=0];
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case 'tv'
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Dx = [diff(X,1,1);zeros(1,X.dim(2))];
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Dy = [diff(X,1,2) zeros(X.dim(1),1)];
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T = sdpvar(X.dim(1),X.dim(2),'full');
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F = cone([reshape(T,1,[]);reshape(Dx,1,[]);reshape(Dy,1,[])]);
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F = [F, sum(sum(T)) <= t];
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otherwise
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end
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varargout{1} = F;
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varargout{2} = struct('convexity','convex','monotonicity','none','definiteness','positive','model','graph');
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varargout{3} = X;
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case 'exact'
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t = varargin{2};
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X = varargin{3};
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p = varargin{4};
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if ~isreal(X) | isequal(p,2) | isequal(p,'fro') | min(size(X))>1 % Complex valued data, matrices and 2-norm not supported
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varargout{1} = [];
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varargout{2} = [];
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varargout{3} = [];
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else
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if p==1
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X = reshape(X,length(X),1);
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absX = sdpvar(length(X),1);
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d = binvar(length(X),1);
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[M,m] = derivebounds(X);
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if all(abs(sign(m)-sign(M))<=1)
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% Silly convex case. Some coding to care of the
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% case sign(0)=0...
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d = ones(length(X),1);
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d(m<0)=-1;
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F = (t - sum(absX) == 0) + (absX == d.*X);
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else
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F = ([]);
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% Some fixes to remove trivial constraints
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% which caused problems in a user m
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positive = find(m >= 0);
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negative = find(M <= 0);
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fixed = find(m==M);
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if ~isempty(fixed)
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positive = setdiff(positive,fixed);
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negative = setdiff(negative,fixed);
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end
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if ~isempty(positive)
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d = subsasgn(d,struct('type','()','subs',{{positive}}),1);
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end
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if ~isempty(negative)
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d = subsasgn(d,struct('type','()','subs',{{negative}}),0);
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end
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if ~isempty(fixed)
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notfixed = setdiff(1:length(m),fixed);
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addsum = sum(abs(m(fixed)));
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m = m(notfixed);
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M = M(notfixed);
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X = extsubsref(X,notfixed);
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absX = extsubsref(absX,notfixed);
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d = extsubsref(d,notfixed);
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else
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addsum = 0;
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end
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maxABSX = max([abs(m) abs(M)],[],2);
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% d==0 ---> X<0 and absX = -X
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F = F + (X <= M.*d) + (0 <= absX+X <= 2*maxABSX.*d);
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% d==1 ---> X>0 and absX = X
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F = F + (X >= m.*(1-d)) + (0 <= absX-X <= 2*maxABSX.*(1-d));
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F = F + (t - sum(absX)-addsum == 0);
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end
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else
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F = max_integer_model([X;-X],t);
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end
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varargout{1} = F;
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varargout{2} = struct('convexity','convex','monotonicity','milp','definiteness','positive','model','integer');
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varargout{3} = X;
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end
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otherwise
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error('SDPVAR/NORM called with CHAR argument?');
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end
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otherwise
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error('Strange type on first argument in SDPVAR/NORM');
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end
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function F = findmax(F,M,m,X,t)
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n = length(X);
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d = binvar(n,1);
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F = F + (sum(d)==1);
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F = F + (-(max(M)-min(m))*(1-d) <= t-X <= (max(M)-min(m))*(1-d));
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kk = [];
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ii = [];
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for i = 1:n
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k = [1:1:i-1 i+1:1:n]';
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ii = [ii;repmat(i,n-1,1)];
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kk = [kk;k];
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Mm = M(k)-m(i);
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end
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xii = extsubsref(X,ii);
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dii = extsubsref(d,ii);
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xkk = extsubsref(X,kk);
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F = F + (xkk <= xii+(M(kk)-m(ii)).*(1-dii));
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