[37] | 1 | function varargout = norm(varargin) |
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| 2 | %NORM (overloaded) |
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| 3 | % |
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| 4 | % t = NORM(x,P) |
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| 5 | % |
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| 6 | % The variable t can only be used in convexity preserving |
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| 7 | % operations such as t<0, max(t,y)<1, minimize t etc. |
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| 8 | % |
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| 9 | % For matrices... |
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| 10 | % NORM(X) models the largest singular value of X, max(svd(X)). |
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| 11 | % NORM(X,2) is the same as NORM(X). |
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| 12 | % NORM(X,1) models the 1-norm of X, the largest column sum, max(sum(abs(X))). |
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| 13 | % NORM(X,inf) models the infinity norm of X, the largest row sum, max(sum(abs(X'))). |
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| 14 | % NORM(X,'fro') models the Frobenius norm, sqrt(sum(diag(X'*X))). |
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| 15 | % For vectors... |
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| 16 | % NORM(V) = norm(V,2) = standard Euclidean norm. |
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| 17 | % NORM(V,inf) = max(abs(V)). |
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| 18 | % NORM(V,1) = sum(abs(V)) |
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| 19 | % |
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| 20 | % SEE ALSO SUMK, SUMABSK |
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| 21 | |
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| 22 | % Author Johan Löfberg |
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| 23 | % $Id: norm.m,v 1.1 2006/08/10 18:00:21 joloef Exp $ |
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| 24 | |
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| 25 | |
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| 26 | %% *************************************************** |
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| 27 | % This file defines a nonlinear operator for YALMIP |
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| 28 | % |
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| 29 | % It can take three different inputs |
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| 30 | % For DOUBLE inputs, it returns standard double values |
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| 31 | % For SDPVAR inputs, it generates an internal variable |
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| 32 | % |
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| 33 | % When first input is 'model' it returns the graph |
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| 34 | % in the first output and structure describing some |
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| 35 | % properties of the operator. |
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| 36 | |
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| 37 | %% *************************************************** |
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| 38 | switch class(varargin{1}) |
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| 39 | |
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| 40 | case 'double' % What is the numerical value of this argument (needed for displays etc) |
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| 41 | % SHOULD NEVER HAPPEN, THIS SHOULD BE CAUGHT BY BUILT-IN |
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| 42 | error('Overloaded SDPVAR/NORM CALLED WITH DOUBLE. Report error') |
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| 43 | |
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| 44 | case 'sdpvar' % Overloaded operator for SDPVAR objects. Pass on args and save them. |
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| 45 | if nargin == 1 |
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| 46 | varargout{1} = yalmip('addextendedvariable',mfilename,varargin{1},2); |
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| 47 | else |
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| 48 | switch varargin{2} |
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| 49 | case {1,2,inf,'inf','fro'} |
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| 50 | varargout{1} = yalmip('addextendedvariable',mfilename,varargin{:}); |
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| 51 | otherwise |
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| 52 | error('norm(x,P) only supported for P = 1, 2, inf and ''fro'''); |
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| 53 | end |
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| 54 | end |
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| 55 | |
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| 56 | case 'char' % YALMIP sends 'model' when it wants the epigraph or hypograph |
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| 57 | switch varargin{1} |
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| 58 | case 'graph' |
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| 59 | t = varargin{2}; |
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| 60 | X = varargin{3}; |
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| 61 | p = varargin{4}; |
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| 62 | |
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| 63 | % Code below complicated by two things |
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| 64 | % 1: Absolute value for complex data -> cone constraints on |
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| 65 | % elements |
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| 66 | % 2: SUBSREF does not call SDPVAR subsref -> use extsubsref.m |
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| 67 | |
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| 68 | % FIX : Exploit symmetry to create smaller problem |
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| 69 | switch p |
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| 70 | case 1 |
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| 71 | z = sdpvar(size(X,1),size(X,2),'full'); |
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| 72 | if min(size(X))>1 |
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| 73 | if isreal(X) |
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| 74 | F = set(-z < X < z); |
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| 75 | else |
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| 76 | F = set([]); |
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| 77 | for i = 1:size(X,1) |
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| 78 | for j = 1:size(X,2) |
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| 79 | xi = extsubsref(X,i,j); |
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| 80 | zi = extsubsref(z,i,j); |
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| 81 | F = F + set(cone([real(xi);imag(xi)],zi)); |
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| 82 | end |
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| 83 | end |
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| 84 | end |
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| 85 | F = F + set(sum(z,1) < t); |
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| 86 | else |
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| 87 | if isreal(X) |
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| 88 | F = set(-z < X < z) + set(sum(z) < t); |
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| 89 | [M,m] = derivebounds(X); |
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| 90 | bounds(z,0,max(abs([M -m]),[],2)); |
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| 91 | bounds(t,0,sum(max(abs([M -m]),[],2))); |
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| 92 | else |
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| 93 | F = set([]); |
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| 94 | for i = 1:length(X) |
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| 95 | xi = extsubsref(X,i); |
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| 96 | zi = extsubsref(z,i); |
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| 97 | F = F + set(cone([real(xi);imag(xi)],zi)); |
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| 98 | end |
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| 99 | F = F + set(sum(z) < t); |
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| 100 | end |
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| 101 | end |
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| 102 | case 2 |
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| 103 | z = sdpvar(size(X,1),size(X,2)); |
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| 104 | if min(size(X))>1 |
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| 105 | F = set([t*eye(size(X,1)) X;X' t*eye(size(X,2))]); |
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| 106 | else |
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| 107 | F = set(cone(X(:),t)); |
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| 108 | end |
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| 109 | case {inf,'inf'} |
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| 110 | if min(size(X))>1 |
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| 111 | z = sdpvar(size(X,1),size(X,2),'full'); |
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| 112 | if isreal(X) |
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| 113 | F = set(-z < X < z); |
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| 114 | else |
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| 115 | F = set([]); |
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| 116 | for i = 1:size(X,1) |
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| 117 | for j = 1:size(X,2) |
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| 118 | xi = extsubsref(X,i,j); |
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| 119 | zi = extsubsref(z,i,j); |
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| 120 | F = F + set(cone([real(xi);imag(xi)],zi)); |
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| 121 | end |
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| 122 | end |
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| 123 | end |
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| 124 | F = F + set(sum(z,2) < t); |
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| 125 | else |
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| 126 | if isreal(X) |
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| 127 | F = set(-t < X < t); |
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| 128 | [M,m,infbound] = derivebounds(X); |
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| 129 | if ~infbound |
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| 130 | F = F + set(0<t<max(M)); |
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| 131 | end |
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| 132 | else |
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| 133 | F = set([]); |
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| 134 | for i = 1:length(X) |
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| 135 | xi = extsubsref(X,i); |
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| 136 | F = F + set(cone([real(xi);imag(xi)],t)); |
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| 137 | end |
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| 138 | end |
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| 139 | end |
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| 140 | case 'fro' |
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| 141 | X.dim(1)=X.dim(1)*X.dim(2); |
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| 142 | X.dim(2)=1; |
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| 143 | F = set(cone(X,t)); |
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| 144 | otherwise |
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| 145 | end |
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| 146 | varargout{1} = F; |
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| 147 | varargout{2} = struct('convexity','convex','monotonicity','none','definiteness','positive'); |
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| 148 | varargout{3} = X; |
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| 149 | case 'milp' |
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| 150 | |
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| 151 | t = varargin{2}; |
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| 152 | X = varargin{3}; |
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| 153 | p = varargin{4}; |
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| 154 | 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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| 155 | varargout{1} = []; |
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| 156 | varargout{2} = []; |
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| 157 | varargout{3} = []; |
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| 158 | else |
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| 159 | if p==1 |
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| 160 | X = reshape(X,length(X),1); |
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| 161 | absX = sdpvar(length(X),1); |
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| 162 | d = binvar(length(X),1); |
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| 163 | [M,m] = derivebounds(X); |
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| 164 | F = set([]); |
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| 165 | positive = find(m >= 0); |
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| 166 | negative = find(M <= 0); |
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| 167 | |
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| 168 | % d(find(positive)) = 1; |
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| 169 | % d(find(negative)) = 0; |
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| 170 | if ~isempty(positive) |
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| 171 | d = subsasgn(d,struct('type','()','subs',{{positive}}),1); |
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| 172 | end |
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| 173 | if ~isempty(negative) |
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| 174 | d = subsasgn(d,struct('type','()','subs',{{negative}}),0); |
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| 175 | end |
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| 176 | |
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| 177 | F = F + set(X <= M.*d) + set(2*m.*d <= absX+X <= 2*M.*d); |
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| 178 | F = F + set(X >= m.*(1-d)) + set(2*m.*(1-d) <= absX-X <= 2*M.*(1-d)); |
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| 179 | F = F + set(t - sum(absX) == 0); |
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| 180 | |
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| 181 | else |
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| 182 | |
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| 183 | if 0 |
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| 184 | |
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| 185 | %2^n cases |
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| 186 | %e.g in 2d, |
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| 187 | %norm([x;y],inf) = y, y>0, y>x.y>-x |
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| 188 | % = x, x>0, x>y.x>-y |
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| 189 | |
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| 190 | n = length(X); |
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| 191 | X = reshape(X,n,1); |
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| 192 | absX = sdpvar(n,1); |
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| 193 | d = binvar(n,1); |
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| 194 | [M,m] = derivebounds(X); |
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| 195 | |
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| 196 | F = set(sum(d)==0); |
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| 197 | |
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| 198 | top = 1; |
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| 199 | for i = 1:n |
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| 200 | xi = extsubsref(X,i); |
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| 201 | y = extsubsref(X,setdiff(1:n,i)); |
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| 202 | for sign_abs_largest_variable = -1:2:1 |
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| 203 | di = extsubsref(d,top); |
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| 204 | for j = setdiff(1:n,i) |
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| 205 | y = extsubsref(X,j); |
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| 206 | for sign_other = -1:2:1 |
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| 207 | F = F + set(xi*sign_abs_largest_variable >= sign_other*y); |
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| 208 | F = F + set(-M*100*(1-di) <= xi*sign_abs_largest_variable-t <= t+M*100*(1-di)); |
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| 209 | end |
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| 210 | end |
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| 211 | top = top + 1; |
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| 212 | end |
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| 213 | end |
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| 214 | |
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| 215 | |
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| 216 | |
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| 217 | |
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| 218 | else |
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| 219 | % OLD |
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| 220 | n = length(X); |
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| 221 | X = reshape(X,n,1); |
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| 222 | absX = sdpvar(n,1); |
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| 223 | d = binvar(n,1); |
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| 224 | [M,m] = derivebounds(X); |
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| 225 | F = set([]); |
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| 226 | F = F + set(X <= M.*d) + set(2*m.*d <= absX+X <= 2*M.*d); |
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| 227 | F = F + set(X >= m.*(1-d)) + set(2*m.*(1-d) <= absX-X <= 2*M.*(1-d)); |
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| 228 | M = max(M,-m); |
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| 229 | d = binvar(n,1); |
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| 230 | F = F + set(sum(d)==1); |
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| 231 | F = F + set(absX <= t <= absX + M.*(1-d)); |
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| 232 | |
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| 233 | kk = []; |
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| 234 | ii = []; |
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| 235 | for i = 1:n |
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| 236 | k = [1:1:i-1 i+1:1:n]'; |
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| 237 | ii = [ii;repmat(i,n-1,1)]; |
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| 238 | kk = [kk;k]; |
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| 239 | Mm = M(k); |
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| 240 | end |
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| 241 | xii = extsubsref(absX,ii); |
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| 242 | dii = extsubsref(d,ii); |
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| 243 | xkk = extsubsref(absX,kk); |
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| 244 | F = F + set(xkk <= xii+(M(kk)-m(ii)).*(1-dii)); |
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| 245 | end |
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| 246 | |
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| 247 | % for i = 1:n |
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| 248 | % xi = extsubsref(absX,i); |
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| 249 | % di = extsubsref(d,i); |
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| 250 | % for k = [1:1:i-1 i+1:1:n] |
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| 251 | % xk = extsubsref(absX,k); |
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| 252 | % F = F + set(xk <= xi+M(k)*(1-di)); |
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| 253 | % end |
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| 254 | % end |
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| 255 | |
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| 256 | end |
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| 257 | varargout{1} = F; |
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| 258 | varargout{2} = struct('convexity','milp','monotonicity','milp','definiteness','positive'); |
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| 259 | varargout{3} = X; |
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| 260 | end |
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| 261 | otherwise |
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| 262 | error('SDPVAR/NORM called with CHAR argument?'); |
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| 263 | end |
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| 264 | otherwise |
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| 265 | error('Strange type on first argument in SDPVAR/NORM'); |
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| 266 | end |
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