1 | function [Fdual,objdual,X,t,err] = dualize(F,obj,auto,extlp,extend) |
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2 | % DUALIZE Create the dual of an SDP given in primal form |
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3 | % |
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4 | % [Fd,objd,X,t,err] = dualize(F,obj,auto) |
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5 | % |
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6 | % Input |
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7 | % F : Primal constraint in form AX=b+dt, X>0, t free. |
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8 | % obj : Primal cost CX+ct |
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9 | % auto : If set to 0, YALMIP will not automatically handle variables |
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10 | % and update variable values when the dual problem is solved. |
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11 | % extlp: If set to 0, YALMIP will not try to perform variables changes in |
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12 | % order to convert simple translated LP cones (as in x>1) to |
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13 | % standard unit cone constraints (x>0) |
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14 | % |
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15 | % Output |
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16 | % Fd : Dual constraints in form C-A'y>0, c-dy==0 |
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17 | % obj : Dual cost b'y (to be MAXIMIZED!) |
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18 | % X : The detected primal cone variables |
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19 | % t : The detected primal free variables |
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20 | % err : Error status (returns 0 if no problems) |
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21 | % |
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22 | % Example |
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23 | % See the HTML help. |
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24 | % |
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25 | % See also DUAL, SOLVESDP, PRIMALIZE |
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26 | |
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27 | % Author Johan Löfberg |
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28 | % $Id: dualize.m,v 1.51 2006/10/18 07:59:09 joloef Exp $ |
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29 | |
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30 | % Check for unsupported problems |
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31 | |
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32 | if isempty(F) |
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33 | F = set([]); |
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34 | end |
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35 | |
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36 | if nargin < 2 |
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37 | obj = []; |
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38 | end |
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39 | |
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40 | err = 0; |
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41 | p1 = ~(isreal(F) & isreal(obj)); |
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42 | p2 = ~(islinear(F) & islinear(obj)); |
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43 | p3 = any(is(F,'integer')) | any(is(F,'binary')); |
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44 | if p1 | p2 | p3 |
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45 | if nargout == 5 |
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46 | Fdual = set([]);objdual = [];y = []; X = []; t = []; err = 1; |
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47 | else |
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48 | problems = {'Cannot dualize complex-valued problems','Cannot dualize nonlinear problems','Cannot dualize discrete problems'}; |
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49 | error(problems{min(find([p1 p2 p3]))}); |
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50 | end |
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51 | end |
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52 | |
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53 | if nargin<5 |
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54 | extend = 1; |
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55 | end |
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56 | |
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57 | if extend |
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58 | options.dualize = 1; |
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59 | options.allowmilp = 0; |
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60 | options.solver = ''; |
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61 | [F,failure,cause] = expandmodel(F,obj,options); |
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62 | if failure |
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63 | error('Failed during convexity propagation. Avoid nonlinear operators when applying dualization.'); |
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64 | end |
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65 | end |
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66 | |
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67 | if nargin<3 |
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68 | auto = 1; |
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69 | end |
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70 | |
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71 | if nargin<4 |
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72 | extlp = 1; |
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73 | end |
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74 | |
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75 | % Cones and equalities |
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76 | F_AXb = set([]); |
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77 | |
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78 | |
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79 | % Shiftmatrix is a bit messy at the moment. |
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80 | % We want to be able to allow cones X>shift |
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81 | % by using a new variable X-shift = Z |
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82 | shiftMatrix = {}; |
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83 | X={}; |
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84 | |
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85 | % First, get variables in initial SDP cones |
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86 | % We need this to avoid adding the same variable twice |
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87 | % when we add simple LP constraints (as in P>0, P(1,3)>0) |
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88 | varSDP = []; |
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89 | SDPset = zeros(length(F),1); |
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90 | isSDP = is(F,'sdp'); |
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91 | for i = 1:length(F) |
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92 | if isSDP(i); |
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93 | Fi = sdpvar(F(i)); |
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94 | if is(Fi,'shiftsdpcone') |
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95 | vars = getvariables(Fi); |
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96 | if isempty(findrows(varSDP,[vars(1) vars(end)])) |
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97 | SDPset(i) = 1; |
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98 | varSDP = [varSDP;vars(1) vars(end)]; |
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99 | shiftMatrix{end+1} = getbasematrix(Fi,0); |
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100 | X{end+1}=Fi; |
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101 | end |
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102 | end |
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103 | end |
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104 | end |
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105 | F_SDP = F(find(SDPset)); |
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106 | F = F(find(~SDPset)); |
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107 | |
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108 | % Same thing for second order cones |
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109 | % However, we must not add any SOC cones |
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110 | % that we already defined as SDP cones |
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111 | varSOC = []; |
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112 | SOCset = zeros(length(F),1); |
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113 | isSOCP = is(F,'socp'); |
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114 | for i = 1:length(F) |
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115 | if isSOCP(i);%is(F(i),'socp') |
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116 | Fi = sdpvar(F(i)); |
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117 | if is(Fi,'socone') |
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118 | vars = getvariables(Fi); |
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119 | % Make sure these variables are not SDP cone variables |
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120 | % This can actually only happen for set(X>0) + set(Xcone((2:end,1),X(1))) |
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121 | if ~isempty(varSDP) |
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122 | inSDP = any(varSDP(:,1)<=vars(1)& vars(1) <=varSDP(:,2)) | any(varSDP(:,1)<=vars(end)& vars(end) <=varSDP(:,2)); |
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123 | else |
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124 | inSDP = 0; |
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125 | end |
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126 | if ~inSDP |
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127 | SOCset(i) = 1; |
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128 | vars = getvariables(Fi); |
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129 | varSOC = [varSOC;vars(1) vars(end)]; |
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130 | shiftMatrix{end+1} = getbasematrix(Fi,0); |
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131 | X{end+1}=Fi; |
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132 | end |
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133 | end |
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134 | end |
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135 | end |
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136 | F_SOC = F(find(SOCset)); |
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137 | F = F(find(~SOCset)); |
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138 | |
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139 | % Merge SDP and SOC data |
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140 | varCONE = [varSDP;varSOC]; |
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141 | F_CONE = F_SDP + F_SOC; |
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142 | |
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143 | % Find all LP constraints, add slacks and extract simple cones |
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144 | % to speed up things, we treat LP cone somewhat different |
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145 | % compared to the conceptually similiar SOCP/SDP cones |
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146 | % This code is pretty messy, since there are a lot off odd |
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147 | % cases to take care of (x>0 and x>1 etc etc) |
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148 | elementwise = is(F,'element-wise'); |
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149 | elementwise_index = find(elementwise); |
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150 | if ~isempty(elementwise_index) |
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151 | |
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152 | % Find element-wise inequalities |
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153 | Flp = F(elementwise_index); |
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154 | F = F(find(~elementwise)); % remove these LPs |
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155 | % Find LP cones |
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156 | lpconstraint = []; |
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157 | for i = 1:length(Flp) |
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158 | temp = sdpvar(Flp(i)); |
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159 | if min(size(temp))>1 |
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160 | temp = temp(:); |
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161 | end |
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162 | lpconstraint = [lpconstraint reshape(temp,1,length(temp))]; |
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163 | end |
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164 | |
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165 | % Find all constraints of type a_i+x_i >0 and extract the unique and |
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166 | % most constraining inequalities (i.e. remove redundant lower bounds) |
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167 | base = getbase(lpconstraint); |
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168 | candidates = find(sum(base(:,2:end)~=0,2)==1); |
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169 | if length(candidates)>0 |
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170 | % The other ones... |
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171 | alwayskeep = find(sum(base(:,2:end)~=0,2)~=1); |
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172 | w1 = lpconstraint(alwayskeep); |
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173 | w2 = lpconstraint(candidates); |
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174 | |
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175 | % Find unique rows |
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176 | base = getbase(w2); |
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177 | [i,uniquerows,k] = unique(base(:,2:end)*randn(size(base,2)-1,1)); |
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178 | aUniqueRow=k(:)'; |
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179 | keep = []; |
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180 | rhsLP = base(:,1); |
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181 | rr = histc(k,[1:max(k)]); |
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182 | if all(rr==1) |
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183 | lpconstraint = [w1 w2]; |
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184 | else |
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185 | for i=1:length(k) |
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186 | sameRow=find(k==k(i)); |
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187 | if length(sameRow)==1 |
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188 | keep = [keep sameRow]; |
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189 | else |
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190 | rhs=base(sameRow,1); |
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191 | [val,pos]=min(rhsLP(sameRow)); |
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192 | keep = [keep sameRow(pos)]; |
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193 | end |
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194 | end |
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195 | lpconstraint = [w1 w2(unique(keep))]; |
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196 | end |
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197 | |
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198 | end |
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199 | |
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200 | % LP cone will be saved in a vector for speed |
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201 | x = []; |
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202 | |
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203 | % Pure cones of the type x>0 |
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204 | base = getbase(lpconstraint); |
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205 | purelpcones = (base(:,1)==0) & (sum(base(:,2:end),2)==1) & (sum(base(:,2:end)==1,2)==1); |
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206 | if ~isempty(purelpcones) |
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207 | x = [x lpconstraint(purelpcones)]; |
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208 | lpconstraint = lpconstraint(find(~purelpcones)); |
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209 | end |
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210 | |
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211 | |
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212 | % Translated cones x>k, k positive |
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213 | % User does not want to make variable changes based on k |
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214 | % But if k>=0, we can at-least mark x as a simple LP cone variable and |
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215 | % thus avoid a free variable. |
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216 | if ~extlp & ~isempty(lpconstraint) |
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217 | base = getbase(lpconstraint); |
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218 | lpcones = (base(:,1)<0) & (sum(base(:,2:end),2)==1) & (sum(base(:,2:end)==1,2)==1); |
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219 | if ~isempty(find(lpcones)) |
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220 | s = recover(getvariables(lpconstraint(find(lpcones)))); |
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221 | x = [x reshape(s,1,length(s))]; |
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222 | end |
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223 | end |
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224 | |
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225 | % Translated cones x>k |
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226 | % Extract these and perform the associated variable change y=x-k |
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227 | if ~isempty(lpconstraint)%Avoid warning in 5.3.1 |
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228 | base = getbase(lpconstraint); |
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229 | lpcones = (sum(base(:,2:end),2)==1) & (sum(base(:,2:end)==1,2)==1); |
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230 | if ~isempty(lpcones) & extlp |
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231 | x = [x lpconstraint(find(lpcones))]; |
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232 | nlp = lpconstraint(find(~lpcones)); |
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233 | if ~isempty(nlp) |
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234 | s = sdpvar(1,length(nlp)); |
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235 | F_AXb = F_AXb + set(nlp-s==0); |
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236 | x = [x s]; |
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237 | end |
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238 | elseif length(lpconstraint) > 0 |
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239 | s = sdpvar(1,length(lpconstraint)); |
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240 | x = [x s]; % New LP cones |
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241 | F_AXb = F_AXb + set(lpconstraint-s==0); |
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242 | end |
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243 | end |
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244 | |
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245 | |
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246 | % Sort asccording to variable index |
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247 | % (Code below assumes x is sorted in increasing variables indicies) |
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248 | base = getbase(x);base = base(:,2:end);[i,j,k] = find(base); |
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249 | x = x(i); |
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250 | xv = getvariables(x); |
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251 | |
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252 | % For mixed LP/SDP problems, we must ensure that LP cone variables are |
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253 | % not actually an element in a SDP cone variable |
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254 | if ~isempty(varCONE) |
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255 | keep = zeros(length(x),1); |
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256 | for i = 1:length(xv) |
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257 | if any(varCONE(:,1)<=xv(i) & xv(i) <=varCONE(:,2)) |
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258 | else |
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259 | keep(i) = 1; |
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260 | end |
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261 | end |
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262 | if ~all(keep) |
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263 | % We need to add some explicit constraints on some elements and |
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264 | % remove the x variables since they are already in a cone |
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265 | % variable |
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266 | xcone = x(find(~keep)); |
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267 | s = sdpvar(1,length(xcone)); |
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268 | F_AXb = F_AXb + set(xcone-s==0); |
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269 | x = x(find(keep)); |
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270 | x = [x s]; |
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271 | end |
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272 | end |
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273 | else |
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274 | x = []; |
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275 | end |
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276 | |
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277 | % Check for mixed cones, ie terms C-A'y > 0. |
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278 | keep = ones(length(F),1); |
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279 | isSDP = is(F,'sdp'); |
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280 | isSOCP = is(F,'socp'); |
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281 | |
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282 | % Pre-allocate all SDP slacks in one call |
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283 | % This is a lot faster |
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284 | if nnz(isSDP) > 0 |
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285 | SDPindicies = find(isSDP)'; |
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286 | for i = 1:length(SDPindicies)%find(isSDP)' |
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287 | Fi = sdpvar(F(SDPindicies(i))); |
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288 | ns(i) = size(Fi,1); |
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289 | ms(i) = ns(i); |
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290 | end |
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291 | Slacks = sdpvar(ns,ms); |
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292 | if ~isa(Slacks,'cell') |
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293 | Slacks = {Slacks}; |
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294 | end |
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295 | end |
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296 | prei = 1; |
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297 | for i = 1:length(F) |
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298 | if isSDP(i) |
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299 | % Semidefinite dual-form cone |
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300 | Fi = sdpvar(F(i)); |
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301 | n = size(Fi,1); |
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302 | % S = sdpvar(n,n); |
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303 | S = Slacks{prei};prei = prei + 1; |
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304 | slack = Fi-S; |
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305 | ind = find(triu(reshape(1:n^2,n,n))); |
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306 | F_AXb = F_AXb + set(slack(ind)==0); |
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307 | F_CONE = F_CONE + lmi(S,[],[],[],1); |
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308 | shiftMatrix{end+1} = spalloc(n,n,0); |
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309 | X{end+1}=S; |
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310 | keep(i)=0; |
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311 | elseif isSOCP(i) |
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312 | % SOC dual-form cone |
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313 | Fi = sdpvar(F(i)); |
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314 | n = size(Fi,1); |
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315 | S = sdpvar(n,1); |
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316 | % S = Slacks{i}; |
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317 | slack = Fi-S; |
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318 | F_AXb = F_AXb + set(slack==0); |
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319 | F_CONE = F_CONE + set(cone(S(2:end),S(1))); |
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320 | shiftMatrix{end+1} = spalloc(n,1,0); |
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321 | X{end+1}=S; |
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322 | keep(i)=0; |
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323 | end |
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324 | end |
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325 | |
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326 | % Now, remove all mixed cones... |
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327 | F = F(find(keep)); |
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328 | |
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329 | % Get the equalities |
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330 | AXbset = is(F,'equality'); |
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331 | if any(AXbset) |
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332 | % Get the constraints |
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333 | F_AXb = F_AXb + F(find(AXbset)); |
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334 | F = F(find(~AXbset)); |
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335 | end |
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336 | |
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337 | % Is thee something we missed in our tests? |
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338 | if length(F)>0 |
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339 | error('DUALIZE can only treat standard SDPs (and LPs) at the moment.') |
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340 | end |
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341 | |
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342 | |
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343 | % Sort the SDP cone variables X according to YALMIP |
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344 | % This is just to simplify some indexing later |
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345 | ns = []; |
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346 | first_var = []; |
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347 | for i = 1:length(F_CONE) |
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348 | ns = [ns length(X{i})]; |
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349 | first_var = [first_var min(getvariables(X{i}))]; |
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350 | end |
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351 | [sorted,index] = sort(first_var); |
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352 | X={X{index}}; |
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353 | shiftMatrix={shiftMatrix{index}}; |
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354 | |
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355 | shift = []; |
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356 | for i = 1:length(F_CONE) |
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357 | ns(i) = length(X{i}); |
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358 | if size(X{i},2)==1 |
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359 | % SOCP |
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360 | shift = [shift;shiftMatrix{i}(:)]; |
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361 | else |
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362 | % SDP |
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363 | ind = find(tril(reshape(1:ns(i)^2,ns(i),ns(i)))); |
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364 | shift = [shift;shiftMatrix{i}(ind)]; |
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365 | end |
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366 | end |
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367 | |
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368 | % Free variables (here called t) is everything except the cone variables |
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369 | X_variables = getvariables(F_CONE); |
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370 | x_variables = getvariables(x); |
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371 | Xx_variables = [X_variables x_variables]; |
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372 | |
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373 | other_variables = [getvariables(obj) getvariables(F_AXb)]; |
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374 | % For quadratic case |
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375 | %other_variables = [depends(obj) getvariables(F_AXb)]; |
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376 | |
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377 | all_variables = uniquestripped([other_variables Xx_variables]); |
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378 | |
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379 | % Avoid set-diff |
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380 | if isequal(all_variables,Xx_variables) |
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381 | t_variables = []; |
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382 | t_in_all = []; |
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383 | t = []; |
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384 | else |
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385 | t_variables = setdiff(all_variables,Xx_variables); |
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386 | ind = ismembc(all_variables,t_variables); |
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387 | t_in_all = find(ind); |
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388 | t = recover(t_variables); |
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389 | end |
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390 | |
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391 | ind = ismembc(all_variables,x_variables); |
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392 | x_in_all = find(ind); |
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393 | ind = ismembc(all_variables,X_variables); |
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394 | X_in_all = find(ind); |
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395 | |
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396 | vecF1 = []; |
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397 | nvars = length(all_variables); |
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398 | for i = 1:length(F_AXb) |
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399 | AXb = sdpvar(F_AXb(i)); |
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400 | mapper = find(ismembc(all_variables,getvariables(AXb))); |
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401 | |
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402 | [n,m] = size(AXb); |
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403 | data = getbase(AXb); |
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404 | [iF,jF,sF] = find(data); |
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405 | if 1 % New |
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406 | smapper = [1 1+mapper(:)']; |
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407 | F_structemp = sparse(iF,smapper(jF),sF,n*m,1+nvars); |
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408 | else |
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409 | F_structemp = spalloc(n*m,1+nvars,nnz(data)); |
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410 | F_structemp(:,[1 1+mapper(:)'])= data; |
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411 | end |
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412 | vecF1 = [vecF1;F_structemp]; |
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413 | end |
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414 | |
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415 | %Remove trivially redundant constraints |
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416 | h = 1+rand(size(vecF1,2),1); |
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417 | h = vecF1*h; |
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418 | [dummy,uniquerows] = uniquesafe(h); |
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419 | if length(uniquerows) < length(h) |
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420 | % Sort to ensure run-to-run consistency |
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421 | vecF1 = vecF1((sort(uniquerows)),:); |
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422 | end |
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423 | |
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424 | if isempty(obj) |
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425 | vecF1(end+1,1) = 0; |
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426 | else |
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427 | if is(obj,'linear') |
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428 | mapper = find(ismembc(all_variables,getvariables(obj))); |
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429 | [n,m] = size(obj); |
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430 | data = getbase(obj); |
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431 | [iF,jF,sF] = find(data); |
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432 | if 1 |
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433 | smapper = [1 1+mapper(:)']; |
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434 | F_structemp = sparse(iF,smapper(jF),sF,n*m,1+nvars); |
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435 | else |
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436 | F_structemp = spalloc(n*m,1+nvars,nnz(data)); |
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437 | F_structemp(:,[1 1+mapper(:)'])= data; |
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438 | end |
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439 | vecF1 = [vecF1;F_structemp]; |
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440 | else |
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441 | % FIX: Generalize to QP duality |
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442 | % min c'x+0.5x'Qx, Ax==b, x>=0 |
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443 | % max b'y-0.5x'Qx, c-A'y+Qx >=0 |
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444 | [Q,c,xreally,info] = quaddecomp(obj,recover(all_variables)) |
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445 | mapper = find(ismembc(all_variables,getvariables(c'*xreally))); |
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446 | [n,m] = size(c'*xreally); |
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447 | data = getbase(c'*xreally); |
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448 | F_structemp = spalloc(n*m,1+nvars,nnz(data)); |
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449 | F_structemp(:,[1 1+mapper(:)'])= data; |
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450 | vecF1 = [vecF1;F_structemp] |
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451 | end |
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452 | |
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453 | end |
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454 | |
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455 | vecF1(end+1,1) = 0; |
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456 | Fbase = vecF1; |
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457 | |
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458 | %Fbase = unique(Fbase','rows')'; |
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459 | |
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460 | b = Fbase(1:end-2,1); |
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461 | |
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462 | Fbase = -Fbase(1:end-1,2:end); |
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463 | vecA = []; |
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464 | Fbase_t = Fbase(:,t_in_all); |
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465 | Fbase_x = Fbase(:,x_in_all); |
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466 | Fbase_X = Fbase; |
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467 | %Fbase_X(:,unionstripped(t_in_all,x_in_all)) = []; |
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468 | if 1 |
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469 | removethese = unique([t_in_all x_in_all]); |
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470 | if length(removethese) > 0.5*size(Fbase_X,2) |
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471 | Fbase_X = Fbase_X(:,setdiff(1:size(Fbase_X,2),removethese)); |
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472 | else |
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473 | Fbase_X(:,[t_in_all x_in_all]) = []; |
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474 | end |
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475 | else |
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476 | removecols = uniquestripped([t_in_all x_in_all]); |
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477 | if ~isempty(removecols) |
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478 | [i,j,k] = find(Fbase_X); |
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479 | keep = find(~ismember(j,removecols)); |
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480 | i = i(keep); |
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481 | k = k(keep); |
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482 | j = j(keep); |
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483 | map = find(1:length(unique(j)),j); |
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484 | |
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485 | end |
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486 | end |
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487 | |
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488 | % Shift due to translated dual cones X = Z+shift |
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489 | if length(shift > 0) |
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490 | b = b + Fbase_X(1:end-1,:)*shift; |
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491 | end |
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492 | if length(x)>0 |
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493 | % Arrgh |
---|
494 | base = getbase(x); |
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495 | constant = base(:,1); |
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496 | base = base(:,2:end);[i,j,k] = find(base); |
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497 | b = b + Fbase_x(1:end-1,:)*constant(i); |
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498 | end |
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499 | |
---|
500 | start = 0; |
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501 | n_cones = length(ns); |
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502 | % All LPs in one cone |
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503 | if length(x)>0 |
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504 | n_lp = 1; |
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505 | else |
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506 | n_lp = 0; |
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507 | end |
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508 | n_free = length(t_variables); |
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509 | |
---|
510 | % SDP cones |
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511 | for j = 1:1:n_cones |
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512 | |
---|
513 | if size(X{j},1)==size(X{j},2) |
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514 | % SDP cone |
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515 | ind = reshape(1:ns(j)^2,ns(j),ns(j)); |
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516 | ind = find(tril(ind)); |
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517 | |
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518 | % Get non-symmetric constraint AiX=b |
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519 | Fi = Fbase_X(1:length(b),start+(1:length(ind)))'/2; |
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520 | |
---|
521 | if 1 |
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522 | [iF,jF,sF] = find(Fi); |
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523 | iA = ind(iF); |
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524 | jA = jF; |
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525 | sA = sF; |
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526 | the_col = 1+floor((iA-1)/ns(j)); |
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527 | the_row = iA-(the_col-1)*ns(j); |
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528 | these_must_be_transposed = find(the_row > the_col); |
---|
529 | if ~isempty(these_must_be_transposed) |
---|
530 | new_rowcols = the_col(these_must_be_transposed) + (the_row(these_must_be_transposed)-1)*ns(j); |
---|
531 | iA = [iA;new_rowcols]; |
---|
532 | jA = [jA;jA(these_must_be_transposed)]; |
---|
533 | sA = [sA;sA(these_must_be_transposed)]; |
---|
534 | end |
---|
535 | % Fix diagonal term |
---|
536 | diags = find(diag(1:ns(j))); |
---|
537 | id = find(ismembc(iA,diags)); |
---|
538 | sA(id) = 2*sA(id); |
---|
539 | Ai = sparse(iA,jA,sA,ns(j)^2,length(b)); |
---|
540 | |
---|
541 | else % Old slooooooow version |
---|
542 | Ai = spalloc(ns(j)^2,length(b),nnz(Fi)); |
---|
543 | Ai(ind,:) = Fi; |
---|
544 | % Symmetrize |
---|
545 | [rowcols,varindicies,vals]=find(Ai); |
---|
546 | the_col = 1+floor((rowcols-1)/ns(j)); |
---|
547 | the_row = rowcols-(the_col-1)*ns(j); |
---|
548 | these_must_be_transposed = find(the_row > the_col); |
---|
549 | if ~isempty(these_must_be_transposed) |
---|
550 | new_rowcols = the_col(these_must_be_transposed) + (the_row(these_must_be_transposed)-1)*ns(j); |
---|
551 | Ai(sub2ind(size(Ai),new_rowcols,varindicies(these_must_be_transposed))) = vals(these_must_be_transposed); |
---|
552 | end |
---|
553 | |
---|
554 | % Fix diagonal term |
---|
555 | diags = find(diag(1:ns(j))); |
---|
556 | Ai(diags,:) = 2*Ai(diags,:); |
---|
557 | end |
---|
558 | |
---|
559 | % if norm(Ai-Ai2,inf)>1e-12 |
---|
560 | % error |
---|
561 | % end |
---|
562 | |
---|
563 | |
---|
564 | vecA{j} = Ai; |
---|
565 | |
---|
566 | start = start + length(ind); |
---|
567 | else |
---|
568 | % Second order cone |
---|
569 | ind = 1:ns(j); |
---|
570 | |
---|
571 | % Get constraint AiX=b |
---|
572 | Fi = Fbase_X(1:length(b),start+(1:length(ind)))'; |
---|
573 | Ai = spalloc(ns(j),length(b),nnz(Fi)); |
---|
574 | Ai(ind,:) = Fi; |
---|
575 | vecA{j} = Ai; |
---|
576 | start = start + length(ind); |
---|
577 | end |
---|
578 | |
---|
579 | end |
---|
580 | % LP Cone |
---|
581 | if n_lp>0 |
---|
582 | Alp=Fbase_x(1:length(b),:)'; |
---|
583 | end |
---|
584 | % FREE VARIABLES |
---|
585 | start = 0; |
---|
586 | if n_free>0 |
---|
587 | Afree=Fbase_t(1:length(b),:)'; |
---|
588 | end |
---|
589 | |
---|
590 | % COST MATRIX |
---|
591 | % SDP CONE |
---|
592 | start = 0; |
---|
593 | for j = 1:1:n_cones |
---|
594 | if size(X{j},1)==size(X{j},2) |
---|
595 | %ind = reshape(1:ns(j)^2,ns(j),ns(j)); |
---|
596 | %ind = find(tril(ind)); |
---|
597 | %C{j} = spalloc(ns(j),ns(j),0); |
---|
598 | %C{j}(ind) = -Fbase_X(end,start+(1:length(ind))); |
---|
599 | %C{j} = (C{j}+ C{j}')/2; |
---|
600 | %start = start + length(ind); |
---|
601 | |
---|
602 | ind = reshape(1:ns(j)^2,ns(j),ns(j)); |
---|
603 | [indi,indj] = find(tril(ind)); |
---|
604 | C{j} = sparse(indi,indj,-Fbase_X(end,start+(1:length(indi))),ns(j),ns(j)); |
---|
605 | C{j} = (C{j}+ C{j}')/2; |
---|
606 | start = start + length(indi); |
---|
607 | else |
---|
608 | ind = 1:ns(j); |
---|
609 | C{j} = spalloc(ns(j),1,0); |
---|
610 | C{j}(ind) = -Fbase_X(end,start+(1:length(ind))); |
---|
611 | start = start + length(ind); |
---|
612 | end |
---|
613 | end |
---|
614 | % LP CONE |
---|
615 | for j = 1:1:n_lp |
---|
616 | Clp = -Fbase_x(end,:)'; |
---|
617 | end |
---|
618 | % FREE CONE |
---|
619 | if n_free>0 |
---|
620 | Cfree = -Fbase_t(end,1:end)'; |
---|
621 | end |
---|
622 | |
---|
623 | % Create dual |
---|
624 | y = sdpvar(length(b),1); |
---|
625 | yvars = getvariables(y); |
---|
626 | cf = []; |
---|
627 | Af = []; |
---|
628 | Fdual = set([]); |
---|
629 | for j = 1:n_cones |
---|
630 | if size(X{j},1)==size(X{j},2) |
---|
631 | % Yep, this is optimized... |
---|
632 | S = sdpvar(ns(j),ns(j),[],yvars,[C{j}(:) -vecA{j}]); |
---|
633 | % Fast call avoids symmetry check |
---|
634 | Fdual = Fdual + lmi(S,[],[],[],1); |
---|
635 | else |
---|
636 | Ay = reshape(vecA{j}*y,ns(j),1); |
---|
637 | S = C{j}-Ay; |
---|
638 | Fdual = Fdual + lmi(cone(S(2:end),S(1))); |
---|
639 | end |
---|
640 | end |
---|
641 | if n_lp > 0 |
---|
642 | keep = any(Alp,2); |
---|
643 | if ~all(keep) |
---|
644 | % Fix for unused primal cones |
---|
645 | preset=find(~keep); |
---|
646 | xfix = x(preset); |
---|
647 | assign(xfix(:),Clp(preset(:))); |
---|
648 | end |
---|
649 | keep = find(keep); |
---|
650 | if ~isempty(keep) |
---|
651 | z = Clp(keep)-Alp(keep,:)*y; |
---|
652 | if isa(z,'double') |
---|
653 | assign(x(:),z(:)); |
---|
654 | else |
---|
655 | Fdual = Fdual + set(z); |
---|
656 | x =x(keep); |
---|
657 | X{end+1} = x(:); % We have worked with a row for performance reasons |
---|
658 | end |
---|
659 | end |
---|
660 | end |
---|
661 | if n_free > 0 |
---|
662 | CfreeAfreey = Cfree-Afree*y; |
---|
663 | if isa(CfreeAfreey,'double') |
---|
664 | if nnz(CfreeAfreey)>0 |
---|
665 | error('Trivially unbounded!'); |
---|
666 | end |
---|
667 | else |
---|
668 | Fdual = Fdual + set(0 == CfreeAfreey); |
---|
669 | end |
---|
670 | end |
---|
671 | |
---|
672 | objdual = b'*y; |
---|
673 | if auto |
---|
674 | for i = 1:length(X) |
---|
675 | yalmip('associatedual',getlmiid(Fdual(i)),X{i}); |
---|
676 | end |
---|
677 | if n_free>0 |
---|
678 | yalmip('associatedual',getlmiid(Fdual(end)),t); |
---|
679 | end |
---|
680 | end |
---|