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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