1 | clc |
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2 | echo on |
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3 | %********************************************************* |
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4 | % |
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5 | % Global bilinear programming |
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6 | % |
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7 | %********************************************************* |
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8 | % |
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9 | % YALMIP comes with a simple branch-and-bound code |
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10 | % for solving problems with bilinear/quadratic constraints. |
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11 | % |
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12 | % The code is under development, and is currently only |
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13 | % applicable to small problems. |
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14 | % |
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15 | % For the examples below to work, you must have an efficient |
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16 | % LP solver (cplex,clp,nag,...) insrtalled and a solver for |
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17 | % general polynomial problems (currently only fmincon or PENBMI) |
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18 | |
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19 | |
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20 | pause |
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21 | yalmip('clear') |
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22 | clc |
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23 | %********************************************************* |
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24 | % The first example is the problem from the moment |
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25 | % relaxation demo (concave quadratic constraint). |
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26 | %********************************************************* |
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27 | x1 = sdpvar(1,1); |
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28 | x2 = sdpvar(1,1); |
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29 | x3 = sdpvar(1,1); |
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30 | |
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31 | objective = -2*x1+x2-x3; |
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32 | |
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33 | F = set(x1*(4*x1-4*x2+4*x3-20)+x2*(2*x2-2*x3+9)+x3*(2*x3-13)+24>0); |
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34 | F = F + set(4-(x1+x2+x3)>0); |
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35 | F = F + set(6-(3*x2+x3)>0); |
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36 | F = F + set(x1>0); |
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37 | F = F + set(2-x1>0); |
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38 | F = F + set(x2>0); |
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39 | F = F + set(x3>0); |
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40 | F = F + set(3-x3>0); |
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41 | pause |
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42 | |
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43 | % Explicitly specify the global solver (this is currently recommended) |
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44 | ops = sdpsettings('solver','bmibnb'); % Global solver |
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45 | pause |
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46 | |
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47 | % Solve! |
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48 | solvesdp(F,objective,ops) |
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49 | pause |
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50 | yalmip('clear') |
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51 | clc |
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52 | %*********************************************************** |
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53 | % The second example is slightly larger. It is a problem |
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54 | % with a concave quadratic objective function, and two |
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55 | % concave quadratic constraints. |
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56 | %*********************************************************** |
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57 | x1 = sdpvar(1,1); |
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58 | x2 = sdpvar(1,1); |
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59 | x3 = sdpvar(1,1); |
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60 | x4 = sdpvar(1,1); |
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61 | x5 = sdpvar(1,1); |
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62 | x6 = sdpvar(1,1); |
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63 | |
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64 | objective = -25*(x1-2)^2-(x2-2)^2-(x3-1)^2-(x4-4)^2-(x5-1)^2-(x6-4)^2; |
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65 | |
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66 | F = set((x3-3)^2+x4>4) + set((x5-3)^2+x6>4); |
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67 | F = F + set(x1-3*x2<2) + set(-x1+x2<2) + set(x1+x2>2); |
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68 | F = F + set(6>x1+x2>2); |
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69 | F = F + set(1<x3<5) + set(0<x4<6)+set(1<x5<5)+set(0<x6<10)+set(x1>0)+set(x2>0); |
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70 | |
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71 | pause |
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72 | |
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73 | % The global solver in YALMIP can currently only deal with linear objectives. |
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74 | % A simple epi-graph formulation solves this. |
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75 | t = sdpvar(1,1); |
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76 | F = F + set(objective<t); |
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77 | pause |
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78 | |
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79 | % Solve! |
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80 | solvesdp(F,t,ops); |
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81 | pause |
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82 | clc |
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83 | %*********************************************************** |
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84 | % Random bound-constrained indefinte quadratic program. |
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85 | %*********************************************************** |
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86 | |
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87 | n = 5; |
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88 | x = sdpvar(n,1); |
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89 | t = sdpvar(1,1); |
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90 | |
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91 | A = randn(n,n);A = A*A'; |
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92 | B = randn(n,n);B = B*B'; |
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93 | |
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94 | objective = x'*A*x-x'*B*x; |
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95 | |
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96 | F = set(-1 < x < 1) + set(objective<t); |
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97 | pause |
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98 | |
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99 | % Solve! |
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100 | solvesdp(F,t,ops); |
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101 | pause |
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102 | yalmip('clear') |
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103 | clc |
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104 | %*********************************************************** |
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105 | % Random boolean quadratic programming. |
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106 | % |
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107 | % By adding a nonlinear constraint x(x-1)==0, we can solve |
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108 | % boolean programming using global nonlinear programming. |
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109 | %*********************************************************** |
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110 | |
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111 | n = 10; |
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112 | x = sdpvar(n,1); |
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113 | t = sdpvar(1,1); |
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114 | |
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115 | Q = randn(n,n);Q = Q*Q'/norm(Q)^2; |
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116 | c = randn(n,1); |
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117 | |
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118 | objective = x'*Q*x+c'*x; |
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119 | |
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120 | F = set(x.*(x-1)==0) + set(objective < t); |
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121 | pause |
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122 | |
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123 | % Note, all complicating variables (variables involved in nonlinear terms) |
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124 | % must be bounded (either explicitely or implicetely via linear and relaxed |
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125 | % nonlinear constraints). |
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126 | % |
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127 | % The easible set for the relaxed problem is not bounded |
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128 | % so YALMIP will not be able to derive variable bounds. |
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129 | % We add some valid bounds to fix this |
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130 | F = F + set(0<x<1); |
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131 | pause |
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132 | |
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133 | % Solve! |
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134 | solvesdp(F,t,ops); |
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135 | double(x'*Q*x+c'*x) |
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136 | pause |
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137 | |
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138 | % Let us compare with an integer solver (use we use the internal MIQP solver) |
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139 | solvesdp(set(binary(x)),x'*Q*x+c'*x,sdpsettings(ops,'solver','bnb')); |
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140 | double(x'*Q*x+c'*x) |
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141 | pause |
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142 | |
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143 | yalmip('clear') |
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144 | clc |
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145 | %*********************************************************** |
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146 | % This example is a somewhat larger indefinite quadratic |
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147 | % programming problem, taken from the GAMS problem library. |
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148 | %*********************************************************** |
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149 | pause |
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150 | |
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151 | % The problem has 11 variables |
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152 | x1 = sdpvar(1,1); |
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153 | x2 = sdpvar(1,1); |
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154 | x3 = sdpvar(1,1); |
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155 | x4 = sdpvar(1,1); |
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156 | x5 = sdpvar(1,1); |
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157 | x6 = sdpvar(1,1); |
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158 | x7 = sdpvar(1,1); |
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159 | x8 = sdpvar(1,1); |
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160 | x9 = sdpvar(1,1); |
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161 | x10 = sdpvar(1,1); |
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162 | x11 = sdpvar(1,1); |
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163 | t = sdpvar(1,1); |
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164 | x = [x1;x2;x3;x4;x5;x6;x7;x8;x9;x10;x11]; |
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165 | |
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166 | pause |
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167 | |
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168 | % ...and 22 linear constraints |
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169 | e1 = - x1 - 2*x2 - 3*x3 - 4*x4 - 5*x5 - 6*x6 - 7*x7 - 8*x8 - 9*x9 - 10*x10 - 11*x11 - 0; |
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170 | e2 = - 2*x1 - 3*x2 - 4*x3 - 5*x4 - 6*x5 - 7*x6 - 8*x7 - 9*x8 - 10*x9 - 11*x10 - x11 - 0; |
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171 | e3 = - 3*x1 - 4*x2 - 5*x3 - 6*x4 - 7*x5 - 8*x6 - 9*x7 - 10*x8 - 11*x9 - x10 - 2*x11 - 0; |
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172 | e4 = - 4*x1 - 5*x2 - 6*x3 - 7*x4 - 8*x5 - 9*x6 - 10*x7 - 11*x8 - x9 - 2*x10 - 3*x11 - 0; |
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173 | e5 = - 5*x1 - 6*x2 - 7*x3 - 8*x4 - 9*x5 - 10*x6 - 11*x7 - x8 - 2*x9 - 3*x10 - 4*x11 - 0; |
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174 | e6 = - 6*x1 - 7*x2 - 8*x3 - 9*x4 - 10*x5 - 11*x6 - x7 - 2*x8 - 3*x9 - 4*x10 - 5*x11 - 0; |
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175 | e7 = - 7*x1 - 8*x2 - 9*x3 - 10*x4 - 11*x5 - x6 - 2*x7 - 3*x8 - 4*x9 - 5*x10 - 6*x11 - 0; |
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176 | e8 = - 8*x1 - 9*x2 - 10*x3 - 11*x4 - x5 - 2*x6 - 3*x7 - 4*x8 - 5*x9 - 6*x10 - 7*x11 - 0; |
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177 | e9 = - 9*x1 - 10*x2 - 11*x3 - x4 - 2*x5 - 3*x6 - 4*x7 - 5*x8 - 6*x9 - 7*x10 - 8*x11 - 0; |
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178 | e10 = - 10*x1 - 11*x2 - x3 - 2*x4 - 3*x5 - 4*x6 - 5*x7 - 6*x8 - 7*x9 - 8*x10 - 9*x11 - 0; |
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179 | e11 = - 11*x1 - x2 - 2*x3 - 3*x4 - 4*x5 - 5*x6 - 6*x7 - 7*x8 - 8*x9 - 9*x10 - 10*x11 - 0; |
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180 | e12 = x1 + 2*x2 + 3*x3 + 4*x4 + 5*x5 + 6*x6 + 7*x7 + 8*x8 + 9*x9 + 10*x10 + 11*x11 - 66; |
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181 | e13 = 2*x1 + 3*x2 + 4*x3 + 5*x4 + 6*x5 + 7*x6 + 8*x7 + 9*x8 + 10*x9 + 11*x10 + x11 - 66; |
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182 | e14 = 3*x1 + 4*x2 + 5*x3 + 6*x4 + 7*x5 + 8*x6 + 9*x7 + 10*x8 + 11*x9 + x10 + 2*x11 - 66; |
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183 | e15 = 4*x1 + 5*x2 + 6*x3 + 7*x4 + 8*x5 + 9*x6 + 10*x7 + 11*x8 + x9 + 2*x10 + 3*x11 - 66; |
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184 | e16 = 5*x1 + 6*x2 + 7*x3 + 8*x4 + 9*x5 + 10*x6 + 11*x7 + x8 + 2*x9 + 3*x10 + 4*x11 - 66; |
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185 | e17 = 6*x1 + 7*x2 + 8*x3 + 9*x4 + 10*x5 + 11*x6 + x7 + 2*x8 + 3*x9 + 4*x10 + 5*x11 - 66; |
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186 | e18 = 7*x1 + 8*x2 + 9*x3 + 10*x4 + 11*x5 + x6 + 2*x7 + 3*x8 + 4*x9 + 5*x10 + 6*x11 - 66; |
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187 | e19 = 8*x1 + 9*x2 + 10*x3 + 11*x4 + x5 + 2*x6 + 3*x7 + 4*x8 + 5*x9 + 6*x10 + 7*x11 - 66; |
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188 | e20 = 9*x1 + 10*x2 + 11*x3 + x4 + 2*x5 + 3*x6 + 4*x7 + 5*x8 + 6*x9 + 7*x10 + 8*x11 - 66; |
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189 | e21 = 10*x1 + 11*x2 + x3 + 2*x4 + 3*x5 + 4*x6 + 5*x7 + 6*x8 + 7*x9 + 8*x10 + 9*x11 - 66; |
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190 | e22 = 11*x1 + x2 + 2*x3 + 3*x4 + 4*x5 + 5*x6 + 6*x7 + 7*x8 + 8*x9 + 9*x10 + 10*x11 - 66; |
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191 | |
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192 | pause |
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193 | |
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194 | |
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195 | % Define the objective function and the whole program |
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196 | obj = (0.5*x1*x2 - x1*x1 + 0.5*x2*x1 - x2*x2 + 0.5*x2*x3 + 0.5*x3*x2 - x3*x3 + 0.5*x3*x4 + 0.5*x4*x3 - x4*x4 + 0.5*x4*x5 + 0.5*x5*x4 - x5*x5 + 0.5*x5 *x6 + 0.5*x6*x5 - x6*x6 + 0.5*x6*x7 + 0.5*x7*x6 - x7*x7 + 0.5*x7*x8 + 0.5 *x8*x7 - x8*x8 + 0.5*x8*x9 + 0.5*x9*x8 - x9*x9 + 0.5*x9*x10 + 0.5*x10*x9 - x10*x10 + 0.5*x10*x11 + 0.5*x11*x10 - x11*x11); |
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197 | e = -[e1;e2;e3;e4;e5;e6;e7;e8;e9;e10;e11;e12;e13;e14;e15;e16;e17;e18;e19;e20;e21;e22]; |
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198 | F = set(10>x>0); |
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199 | F = F + set(e>0); |
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200 | F = F + set(obj==t); |
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201 | |
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202 | % We will nowe try to solve this using the same strategy as before |
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203 | solvesdp(F,t,ops) |
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204 | |
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205 | pause |
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206 | clc |
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207 | |
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208 | % As an alternative, we might use additional cuts using the CUT functionality. |
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209 | % These additional constraints are only used in domain reduction and |
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210 | % solution of lower bound programs, but not in the local solver. |
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211 | |
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212 | % We square all linear constraints and add these as cuts |
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213 | % |
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214 | % Note that the triu operator leaves returns a matrix with |
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215 | % zeros below the diagonal. These are however neglected by YALMIP |
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216 | % in the call to SET. |
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217 | % |
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218 | f = sdpvar(F(find(is(F,'linear')))); |
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219 | F = F + cut(triu(f*f') > 0); |
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220 | pause |
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221 | |
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222 | % ...and solve (To speed up things, we turn off LP based domain reduction) |
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223 | % |
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224 | % NOTE : The first ~15 iterations are rather slow, but things starts to |
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225 | % speed up when YALMIP starts pruning redundnant linear constraints. |
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226 | % |
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227 | solvesdp(F,t,sdpsettings(ops,'bmibnb.lpreduce',0)) |
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228 | pause |
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229 | clc |
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230 | |
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231 | % The global solver in YALMIP can only solve bilinear programs, |
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232 | % but a pre-processor is capable of transforming higher order problems |
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233 | % to bilinear programs. As an example, the variable x^3y^2 will be replaced |
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234 | % with the the variable w and the constraints w == uv, u == zx, z == x^2, v == y^2. |
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235 | % The is done automatically if the global solver, or PENBMI, is called with |
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236 | % a higher order polynomial problem. Note that this conversion is rather |
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237 | % inefficient, and only very small problems can be addressed using this simple approach. |
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238 | % Additionally, this module has not been tested in any serious sense. |
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239 | pause |
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240 | |
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241 | % Let us solve a small trivial polynomial program |
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242 | sdpvar x y |
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243 | F = set(x^3+y^5 < 5) + set(y > 0); |
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244 | F = F + set(-100 < [x y] < 100) % Always bounded domain |
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245 | pause |
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246 | |
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247 | solvesdp(F,-x,ops) |
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248 | pause |
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249 | |
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250 | %*********************************************************** |
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251 | % The main motivation for implementing a bilinear solver in |
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252 | % YALMIP is matrix-valued problems, i.e. BMI problems. |
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253 | % |
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254 | % Of-course, BMI problems are even harder than the nonconvex |
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255 | % quadratic problems solved above. However, in some cases, |
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256 | % it is actually possible to solve BMI problems globally. |
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257 | % |
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258 | % Don't expect too much though... |
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259 | %*********************************************************** |
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260 | |
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261 | pause |
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262 | clc |
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263 | |
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264 | %*********************************************************** |
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265 | % The following is a classical problem |
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266 | %*********************************************************** |
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267 | A0 = [-10 -0.5 -2;-0.5 4.5 0;-2 0 0]; |
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268 | A1 = [9 0.5 0;0.5 0 -3 ; 0 -3 -1]; |
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269 | A2 = [-1.8 -0.1 -0.4;-0.1 1.2 -1;-0.4 -1 0]; |
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270 | K12 = [0 0 2;0 -5.5 3;2 3 0]; |
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271 | |
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272 | x = sdpvar(1,1); |
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273 | y = sdpvar(1,1); |
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274 | t = sdpvar(1,1); |
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275 | |
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276 | F = set(A0+x*A1+y*A2+x*y*K12-t*eye(3)<0); |
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277 | F = F + set(-0.5 < x < 2) + set(-3 < y < 7); |
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278 | pause |
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279 | |
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280 | % Specify solvers (this requires PENBMI!) |
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281 | ops = sdpsettings('solver','bmibnb'); % Global solver |
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282 | ops = sdpsettings(ops,'bmibnb.uppersolver','penbmi'); % Local solver |
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283 | |
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284 | pause |
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285 | |
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286 | % Solve! |
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287 | solvesdp(F,t,ops); |
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288 | pause |
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289 | clc |
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290 | %*********************************************************** |
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291 | % The next example shows how to solve a standard LQ control |
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292 | % problem with constraints on the feedback gain |
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293 | %*********************************************************** |
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294 | |
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295 | A = [-1 2;-3 -4];B = [1;1]; |
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296 | |
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297 | P = sdpvar(2,2); |
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298 | K = sdpvar(1,2); |
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299 | |
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300 | F = set(P>0)+set((A+B*K)'*P+P*(A+B*K) < -eye(2)-K'*K); |
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301 | F = F + set(K<0.1)+set(K>-0.1); |
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302 | F = F + set(1000> P(:)>-1000); |
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303 | solvesdp(F,trace(P),ops); |
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304 | pause |
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305 | clc |
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306 | %*********************************************************** |
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307 | % Decay-rate estimation as a global BMI example ... |
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308 | % |
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309 | % Note, this is a quasi-convex problem that PENBMI actually |
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310 | % solves to global optimality, so the use of a global solver |
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311 | % is pretty stupid. It is only included here to show some |
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312 | % properties of the global solver. |
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313 | % |
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314 | % may run for up to 100 iterations depending on the solvers |
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315 | %*********************************************************** |
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316 | |
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317 | A = [-1 2;-3 -4]; |
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318 | t = sdpvar(1,1); |
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319 | P = sdpvar(2,2); |
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320 | F = set(P>eye(2))+set(A'*P+P*A < -2*t*P); |
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321 | F = F + set(-1e4 < P(:) < 1e4) + set(100 > t > 0) ; |
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322 | pause |
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323 | solvesdp(F,-t,ops); |
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324 | pause |
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325 | |
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326 | % Now, that sucked, right! 100 iterations and still 10% gap... |
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327 | % |
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328 | % The way we model a problem can make a huge difference. |
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329 | % The decay-rate problem can be substantially improved by |
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330 | % using a smarter model. The original problem is homogenous |
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331 | % w.r.t P, and a better way to make the feasible set bounded |
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332 | % is to constrain the trace of P |
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333 | A = [-1 2;-3 -4]; |
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334 | t = sdpvar(1,1); |
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335 | P = sdpvar(2,2); |
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336 | F = set(P>0)+set(A'*P+P*A < -2*t*P); |
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337 | F = F + set(trace(P)==1) + set(100 > t > 0) ; |
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338 | pause |
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339 | solvesdp(F,-t,ops); |
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340 | pause |
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341 | |
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342 | % Nothing prevents us from adding an additional |
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343 | % constraint derived from trace(P)=1 |
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344 | F = F + set(trace(A'*P+P*A) < -2*t); |
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345 | pause |
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346 | |
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347 | solvesdp(F,-t,ops); |
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348 | pause |
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349 | |
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350 | % When we add redundant constraints, we improve the relaxations |
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351 | % and obtain better lower bounds. For the upper bounds however, |
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352 | % these cuts only add additional computational burden. |
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353 | % To overcome this, the command CUT can be used. |
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354 | % Constraints defined with CUT are not used when the upper |
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355 | % bound problems are solved, but are only used in relaxations |
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356 | F = set(P>0)+set(A'*P+P*A < -2*t*P); |
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357 | F = F + set(trace(P)==1) + set(100 > t > 0) ; |
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358 | F = F + cut(trace(A'*P+P*A) < -2*t); |
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359 | pause |
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360 | |
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361 | solvesdp(F,-t,ops); |
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362 | pause |
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363 | |
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364 | % Finally, it should be mentioned that the branch & bound |
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365 | % algorithm can run without any installed local BMI solver. |
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366 | % This version of the algorithms is obtained by specifying |
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367 | % the upper bound solver as 'none' |
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368 | ops = sdpsettings(ops,'bmibnb.uppersolver','none'); |
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369 | F = set(P>0)+set(A'*P+P*A < -2*t*P); |
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370 | F = F + set(trace(P)==1) + set(100 > t > 0) ; |
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371 | F = F + cut(trace(A'*P+P*A) < -2*t); |
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372 | pause |
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373 | |
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374 | solvesdp(F,-t,ops); |
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375 | pause |
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376 | |
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377 | echo off |
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378 | |
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379 | |
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380 | |
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381 | break |
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382 | |
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383 | % ******************************** |
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384 | % Sahinidis (-6.666 in 1) |
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385 | % ******************************** |
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386 | yalmip('clear'); |
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387 | x1 = sdpvar(1,1); |
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388 | x2 = sdpvar(1,1); |
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389 | F = set(0 < x1 < 6)+set(0 < x2 < 4) + set(x1*x2 < 4) |
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390 | solvesdp(F,-x1-x2,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bmibnb.maxiter',100,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','cdd','cdd.method','dual-simplex')) |
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391 | |
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392 | % **************************** |
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393 | % Decay-rate (-2.5) |
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394 | % **************************** |
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395 | yalmip('clear'); |
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396 | A = [-1 2;-3 -4]; |
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397 | t = sdpvar(1,1); |
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398 | P = sdpvar(2,2); |
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399 | F = set(P>eye(2))+set(A'*P+P*A < -2*t*P); |
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400 | F = F + set(-1e4 < P(:) < 1e4) + set(100 > t > -100) ; |
---|
401 | ops = sdpsettings('bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bmibnb.maxiter',100,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','cdd','cdd.method','dual-simplex'); |
---|
402 | solvesdp(F,-t,ops); |
---|
403 | |
---|
404 | % ******************************** |
---|
405 | % Classical example from Safonov etc |
---|
406 | % ******************************** |
---|
407 | yalmip('clear') |
---|
408 | x = sdpvar(1,1); |
---|
409 | y = sdpvar(1,1); |
---|
410 | t = sdpvar(1,1); |
---|
411 | A0 = [-10 -0.5 -2;-0.5 4.5 0;-2 0 0]; |
---|
412 | A1 = [9 0.5 0;0.5 0 -3 ; 0 -3 -1]; |
---|
413 | A2 = [-1.8 -0.1 -0.4;-0.1 1.2 -1;-0.4 -1 0]; |
---|
414 | K12 = [0 0 2;0 -5.5 3;2 3 0]; |
---|
415 | F = lmi(x>-0.5)+lmi(x<2) + lmi(y>-3)+lmi(y<7) + lmi(t>-1000)+lmi(t<1000);%set(x+y<2.47)+set(x+y>2.47); |
---|
416 | F = F + lmi(A0+x*A1+y*A2+x*y*K12-t*eye(3)<0); |
---|
417 | solvesdp(F,t,sdpsettings('bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
418 | |
---|
419 | % ******************************** |
---|
420 | % Lassere 1 |
---|
421 | % optimal : -4, takes 13 |
---|
422 | % ******************************** |
---|
423 | clear all |
---|
424 | x1 = sdpvar(1,1); |
---|
425 | x2 = sdpvar(1,1); |
---|
426 | x3 = sdpvar(1,1); |
---|
427 | x = [x1;x2;x3]; |
---|
428 | B = [0 0 1;0 -1 0;-2 1 -1]; |
---|
429 | b = [3;0;-4]; |
---|
430 | v = [0;-1;-6]; |
---|
431 | r = [1.5;-0.5;-5]; |
---|
432 | p = -2*x1+x2-x3; |
---|
433 | F = set(x1+x2+x3 < 4) + set(x1<2)+set(x3<3) + set(3*x2+x3 < 6); |
---|
434 | F = F + set(x>0) + set(x'*B'*B*x-2*r'*B*x+r'*r-0.25*(b-v)'*(b-v) >0); |
---|
435 | solvesdp(F,p,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
436 | |
---|
437 | % ******************************** |
---|
438 | % Lassere 2 |
---|
439 | % optimal : -310 in 3 |
---|
440 | % ******************************** |
---|
441 | clear all |
---|
442 | x1 = sdpvar(1,1); |
---|
443 | x2 = sdpvar(1,1); |
---|
444 | x3 = sdpvar(1,1); |
---|
445 | x4 = sdpvar(1,1); |
---|
446 | x5 = sdpvar(1,1); |
---|
447 | x6 = sdpvar(1,1); |
---|
448 | t = sdpvar(1,1); |
---|
449 | p = -25*(x1-2)^2-(x2-2)^2-(x3-1)^2-(x4-4)^2-(x5-1)^2-(x6-4)^2; |
---|
450 | F = set((x3-3)^2+x4>4)+set((x5-3)^2+x6>4); |
---|
451 | F = F + set(x1-3*x2<2)+set(-x1+x2<2); |
---|
452 | F = F + set(x1-3*x2 <2)+set(x1+x2>2); |
---|
453 | F = F + set(6>x1+x2>2); |
---|
454 | F = F + set(1<x3<5) + set(0<x4<6)+set(1<x5<5)+set(0<x6<10)+set(x1>0)+set(x2>0); |
---|
455 | |
---|
456 | solvesdp(F,p,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
457 | F = F + set(p<t); |
---|
458 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
459 | % ******************************** |
---|
460 | % Lassere 2.6 |
---|
461 | % optimal : -39 in 6 |
---|
462 | % ******************************** |
---|
463 | clear all |
---|
464 | Q = 100*eye(10); |
---|
465 | c = [48, 42, 48, 45, 44, 41, 47, 42, 45, 46]'; |
---|
466 | b = [-4, 22,-6,-23,-12]'; |
---|
467 | A =[-2 -6 -1 0 -3 -3 -2 -6 -2 -2; |
---|
468 | 6 -5 8 -3 0 1 3 8 9 -3; |
---|
469 | -5 6 5 3 8 -8 9 2 0 -9; |
---|
470 | 9 5 0 -9 1 -8 3 -9 -9 -3; |
---|
471 | -8 7 -4 -5 -9 1 -7 -1 3 -2]; |
---|
472 | x = sdpvar(10,1); |
---|
473 | t = sdpvar(1,1); |
---|
474 | p = c'*x-0.5*x'*Q*x; |
---|
475 | F = set(0<x<1)+set(A*x<b); |
---|
476 | solvesdp(F,p,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
477 | F = F + set(p<t); |
---|
478 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
479 | % ******************************** |
---|
480 | % Lassere |
---|
481 | % -0.37? >100 |
---|
482 | % ******************************** |
---|
483 | clear all |
---|
484 | x = sdpvar(10,1); |
---|
485 | t = sdpvar(1,1); |
---|
486 | x(1) = 1-sum(x(2:end)); |
---|
487 | obj = 0; |
---|
488 | for i = 1:9 |
---|
489 | obj = obj+x(i)*x(i+1); |
---|
490 | end |
---|
491 | for i = 1:8 |
---|
492 | obj = obj+x(i)*x(i+2); |
---|
493 | end |
---|
494 | obj = obj + x(1)*x(7)+ x(1)*x(9)+ x(2)*x(10)+ x(4)*x(7); |
---|
495 | F = set(1>x>0); |
---|
496 | F = F + set(-obj<t); |
---|
497 | solvesdp(F,-obj,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
498 | F = F + set(-obj<t); |
---|
499 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
500 | |
---|
501 | % ******************************** |
---|
502 | % Is J co-positive ? |
---|
503 | % Optimal : 0 |
---|
504 | % ******************************** |
---|
505 | clear all |
---|
506 | t = sdpvar(1,1); |
---|
507 | x = sdpvar(5,1); |
---|
508 | J = [1 -1 1 1 -1; |
---|
509 | -1 1 -1 1 1; |
---|
510 | 1 -1 1 -1 1; |
---|
511 | 1 1 -1 1 -1; |
---|
512 | -1 1 1 -1 1]; |
---|
513 | F = set(x>0) + set(sum(x)<1); |
---|
514 | F = F+set(-10<t<10); |
---|
515 | solvesdp(F,x'*J*x,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
516 | F = F + set(x'*J*x<t); |
---|
517 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
518 | |
---|
519 | % ******************************** |
---|
520 | % BARON example problems |
---|
521 | % -17 in 9 |
---|
522 | % ******************************** |
---|
523 | clear all |
---|
524 | x1 = sdpvar(1,1); |
---|
525 | x2 = sdpvar(1,1); |
---|
526 | x3 = sdpvar(1,1); |
---|
527 | x4 = sdpvar(1,1); |
---|
528 | x5 = sdpvar(1,1); |
---|
529 | t = sdpvar(1,1); |
---|
530 | p = 42*x1-50*x1^2+44*x2-50*x2^2+45*x3-50*x3^2+47*x4-50*x4^2+47.5*x5-50*x5^2; |
---|
531 | F = set([20 12 11 7 4]*[x1;x2;x3;x4;x5] < 40) + set(0<[x1;x2;x3;x4;x5]<1); |
---|
532 | solvesdp(F,p,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','nag','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
533 | F = F+set(p<t); |
---|
534 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','nag','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
535 | |
---|
536 | % ******************************** |
---|
537 | % BARON example |
---|
538 | % -13 in 1 |
---|
539 | % ******************************** |
---|
540 | clear all |
---|
541 | x1 = sdpvar(1,1); |
---|
542 | x2 = sdpvar(1,1); |
---|
543 | x3 = sdpvar(1,1); |
---|
544 | x4 = sdpvar(1,1); |
---|
545 | t = sdpvar(1,1); |
---|
546 | p = x1 - x2 - x3 - x1*x3 + x1*x4 + x2*x3 - x2*x4; |
---|
547 | F = set(x1+4*x2 <= 8) + set(4*x1+x2 <= 12) + set(3*x1+4*x2 <= 12) + set(2*x3+x4 <= 8) + set(x3+2*x4 <= 8) + set(x3+x4 <= 5)+set([x1;x2;x3;x4]>0); |
---|
548 | F = F+set(p<t); |
---|
549 | solvesdp(F,p,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
550 | F = F+set(p<t); |
---|
551 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.09e-1,'bmibnb.lpsolver','glpk')) |
---|
552 | |
---|
553 | % ******************************** |
---|
554 | % Control design |
---|
555 | % 5.46 in 12 |
---|
556 | % ******************************** |
---|
557 | A = [1 2;-3 0];B = [1;1]; |
---|
558 | [K0,P0] = lqr(A,B,eye(2),1); |
---|
559 | P = sdpvar(2,2);setsdpvar(P,2*P0);K0(K0>1)=1;K0(K0<-1)=-1; |
---|
560 | K = sdpvar(1,2);setsdpvar(K,-K0); |
---|
561 | F = set(K<1)+set(K>-1)+set(P>0)+set((A+B*K)'*P+P*(A+B*K) < -eye(2)-K'*K); |
---|
562 | %F = F + set([-(A+B*K)'*P-P*(A+B*K)-eye(2) K';K eye(1)] > 0) |
---|
563 | F = F+lmi(diag(P)>0)+lmi(P(:)>-151) + lmi(P(:)<150) + lmi(P>P0)+lmi(K>-100) + lmi(K<100); |
---|
564 | solvesdp(F,trace(P),sdpsettings('usex0',1,'bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk','bmibnb.lpreduce',1)) |
---|
565 | |
---|
566 | % ******************************** |
---|
567 | % GAMS st_qpc_m1 |
---|
568 | % optimal -473.77 in 3 |
---|
569 | % ******************************** |
---|
570 | clear all |
---|
571 | x1 = sdpvar(1,1); |
---|
572 | x2 = sdpvar(1,1); |
---|
573 | x3 = sdpvar(1,1); |
---|
574 | x4 = sdpvar(1,1); |
---|
575 | x5 = sdpvar(1,1); |
---|
576 | t = sdpvar(1,1); |
---|
577 | |
---|
578 | F = set([x1;x2;x3;x4;x5]>0); |
---|
579 | F = F + set(x1 + x2 + 2*x3 + x4 + x5 > 10); |
---|
580 | F = F + set(2*x1 + 3*x2 + x5 > 8); |
---|
581 | F = F + set(x2 + 4*x3 - x4 + 2*x5 > 12); |
---|
582 | F = F + set(8*x1 - x2 - x3 + 6*x4 > 20); |
---|
583 | F = F + set(- 2*x1 - x2 - 3*x3 - x4 - x5 > -30); |
---|
584 | obj = -(- (10*x1 - 0.34*x1*x1 - 0.28*x1*x2 + 10*x2 - 0.22*x1*x3 + 10*x3 - 0.24*x1*x4 + 10*x4 - 0.51*x1*x5 + 10*x5 - 0.28*x2*x1 - 0.34*x2*x2 - 0.23*x2*x3 -0.24*x2*x4 - 0.45*x2*x5 - 0.22*x3*x1 - 0.23*x3*x2 - 0.35*x3*x3 - 0.22*x3*x4 - 0.34*x3*x5 - 0.24*x4*x1 - 0.24*x4*x2 - 0.22*x4*x3 - 0.2*x4*x4 - 0.38*x4*x5 - 0.51*x5*x1 - 0.45*x5*x2 - 0.34*x5*x3 - 0.38*x5*x4 - 0.99*x5*x5)); |
---|
585 | |
---|
586 | solvesdp(F,obj,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
587 | F = F + set(obj<t); |
---|
588 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
589 | |
---|
590 | % ******************************** |
---|
591 | % GAMS st_qpk1 |
---|
592 | % optimal -3 in 7 |
---|
593 | % ******************************** |
---|
594 | clear all |
---|
595 | x1 = sdpvar(1,1); |
---|
596 | x2 = sdpvar(1,1); |
---|
597 | t = sdpvar(1,1); |
---|
598 | |
---|
599 | F = set([x1;x2]>0); |
---|
600 | F = F + set(- x1 + x2 < 1); |
---|
601 | F = F + set(x1 - x2 < 1); |
---|
602 | F = F + set(- x1 + 2*x2 < 3); |
---|
603 | F = F + set(2*x1 - x2 < 3); |
---|
604 | obj = (2*x1 - 2*x1*x1 + 2*x1*x2 + 3*x2 - 2*x2*x2); |
---|
605 | F = F + set(obj<t); |
---|
606 | solvesdp(F,t,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
607 | |
---|
608 | |
---|
609 | % ******************************** |
---|
610 | % GAMS st_robot |
---|
611 | % optimal -3.52 in 4 |
---|
612 | % ******************************** |
---|
613 | yalmip('clear') |
---|
614 | x1 = sdpvar(1,1); |
---|
615 | x2 = sdpvar(1,1); |
---|
616 | x3 = sdpvar(1,1); |
---|
617 | x4 = sdpvar(1,1); |
---|
618 | x5 = sdpvar(1,1); |
---|
619 | x6 = sdpvar(1,1); |
---|
620 | x7 = sdpvar(1,1); |
---|
621 | x8 = sdpvar(1,1); |
---|
622 | x = [x1;x2;x3;x4;x5;x6;x7;x8]; |
---|
623 | F = set(-1<x<1); |
---|
624 | F = F + set( 0.004731*x1*x3 - 0.1238*x1 - 0.3578*x2*x3 - 0.001637*x2 - 0.9338*x4 + x7 == 0.3571); |
---|
625 | F = F + set(0.2238*x1*x3 + 0.2638*x1 + 0.7623*x2*x3 - 0.07745*x2 - 0.6734*x4 - x7 == 0.6022); |
---|
626 | F = F + set( x6*x8 + 0.3578*x1 + 0.004731*x2 == 0); |
---|
627 | F = F + set( - 0.7623*x1 + 0.2238*x2 == -0.3461); |
---|
628 | F = F + set(x1^2 + x2^2 == 1); |
---|
629 | F = F + set(x3^2 + x4^2 == 1); |
---|
630 | F = F + set(x5^2 + x6^2 == 1); |
---|
631 | F = F + set(x7^2 + x8^2 == 1); |
---|
632 | |
---|
633 | solvesdp(F,sum(x),sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','pensdp','verbose',2,'solver','bmibnb','bnb.maxiter',30,'penbmi.P0',0.9e-1,'bmibnb.lpsolver','glpk')) |
---|
634 | |
---|
635 | |
---|
636 | % ******************************** |
---|
637 | % GAMS st_robot |
---|
638 | % optimal -382 in 5 |
---|
639 | % ******************************** |
---|
640 | yalmip('clear') |
---|
641 | x1 = sdpvar(1,1); |
---|
642 | x2 = sdpvar(1,1); |
---|
643 | x3 = sdpvar(1,1); |
---|
644 | x4 = sdpvar(1,1); |
---|
645 | x5 = sdpvar(1,1); |
---|
646 | x6 = sdpvar(1,1); |
---|
647 | x7 = sdpvar(1,1); |
---|
648 | x8 = sdpvar(1,1); |
---|
649 | x9 = sdpvar(1,1); |
---|
650 | x10 = sdpvar(1,1); |
---|
651 | t = sdpvar(1,1); |
---|
652 | x = [x1;x2;x3;x4;x5;x6;x7;x8;x9;x10]; |
---|
653 | |
---|
654 | e1= 20*x1 + 20*x2 + 60*x3 + 60*x4 + 60*x5 + 60*x6 + 5*x7 + 45*x8 + 55*x9 + 65*x10 - 600.1; |
---|
655 | |
---|
656 | e2= 5*x1 + 7*x2 + 3*x3 + 8*x4 + 13*x5 + 13*x6 + 2*x7 + 14*x8 + 14*x9 + 14*x10 - 310.5; |
---|
657 | |
---|
658 | e3= 100*x1 + 130*x2 + 50*x3 + 70*x4 + 70*x5 + 70*x6 + 20*x7 + 80*x8 + 80*x9 + 80*x10 - 1800; |
---|
659 | |
---|
660 | e4= 200*x1 + 280*x2 + 100*x3 + 200*x4 + 250*x5 + 280*x6 + 100*x7 + 180*x8 + 200*x9 + 220*x10 - 3850; |
---|
661 | |
---|
662 | e5= 2*x1 + 2*x2 + 4*x3 + 4*x4 + 4*x5 + 4*x6 + 2*x7 + 6*x8 + 6*x9 + 6*x10 - 18.6; |
---|
663 | |
---|
664 | e6= 4*x1 + 8*x2 + 2*x3 + 6*x4 + 10*x5 + 10*x6 + 5*x7 + 10*x8 + 10*x9 + 10*x10 - 198.7; |
---|
665 | |
---|
666 | e7= 60*x1 + 110*x2 + 20*x3 + 40*x4 + 60*x5 + 70*x6 + 10*x7 + 40*x8 + 50*x9 + 50*x10 - 882; |
---|
667 | |
---|
668 | e8= 150*x1 + 210*x2 + 40*x3 + 70*x4 + 90*x5 + 105*x6 + 60*x7 + 100*x8 + 140*x9 + 180*x10 - 4200; |
---|
669 | |
---|
670 | e9= 80*x1 + 100*x2 + 6*x3 + 16*x4 + 20*x5 + 22*x6 + 20*x8 + 30*x9 + 30*x10 - 40.25; |
---|
671 | |
---|
672 | e10= 40*x1 + 40*x2 + 12*x3 + 20*x4 + 24*x5 + 28*x6 + 40*x9 + 50*x10 - 327; |
---|
673 | |
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674 | obj = (10*x1 - 6.8*x1*x1 - 4.6*x1*x2 + 10*x2 - 7.9*x1*x3 + 10*x3 - 5.1*x1*x4 + 10*x4 - 6.9*x1*x5 + 10*x5 - 6.8*x1*x6 + 10*x6 - 4.6*x1*x7 + 10*x7 - 7.9*x1*x8 + 10*x8 - 5.1*x1*x9 + 10*x9 - 6.9*x1*x10 + 10*x10 - 4.6*x2*x1 - 5.5*x2*x2 - 5.8*x2*x3 - 4.5*x2*x4 - 6*x2*x5 - 4.6*x2*x6 - 5.5*x2*x7 - 5.8*x2*x8 - 4.5*x2*x9 - 6*x2*x10 - 7.9*x3*x1 - 5.8*x3*x2 - 13.3*x3*x3 - 6.7*x3*x4 - 8.9*x3*x5 - 7.9*x3*x6 - 5.8*x3*x7 - 13.3*x3*x8 - 6.7*x3*x9 - 8.9*x3*x10 - 5.1*x4*x1 - 4.5*x4*x2 - 6.7*x4*x3 - 6.9*x4*x4 - 5.8*x4*x5 - 5.1*x4*x6 - 4.5*x4*x7 - 6.7*x4*x8 - 6.9*x4*x9 - 5.8*x4*x10 - 6.9*x5*x1 - 6*x5*x2 - 8.9*x5*x3 - 5.8*x5*x4 - 11.9*x5*x5 - 6.9*x5*x6 - 6*x5*x7 - 8.9* x5*x8 - 5.8*x5*x9 - 11.9*x5*x10 - 6.8*x6*x1 - 4.6*x6*x2 - 7.9*x6*x3 - 5.1 *x6*x4 - 6.9*x6*x5 - 6.8*x6*x6 - 4.6*x6*x7 - 7.9*x6*x8 - 5.1*x6*x9 - 6.9* x6*x10 - 4.6*x7*x1 - 5.5*x7*x2 - 5.8*x7*x3 - 4.5*x7*x4 - 6*x7*x5 - 4.6*x7 *x6 - 5.5*x7*x7 - 5.8*x7*x8 - 4.5*x7*x9 - 6*x7*x10 - 7.9*x8*x1 - 5.8*x8* x2 - 13.3*x8*x3 - 6.7*x8*x4 - 8.9*x8*x5 - 7.9*x8*x6 - 5.8*x8*x7 - 13.3*x8 *x8 - 6.7*x8*x9 - 8.9*x8*x10 - 5.1*x9*x1 - 4.5*x9*x2 - 6.7*x9*x3 - 6.9*x9 *x4 - 5.8*x9*x5 - 5.1*x9*x6 - 4.5*x9*x7 - 6.7*x9*x8 - 6.9*x9*x9 - 5.8*x9* x10 - 6.9*x10*x1 - 6*x10*x2 - 8.9*x10*x3 - 5.8*x10*x4 - 11.9*x10*x5 - 6.9 *x10*x6 - 6*x10*x7 - 8.9*x10*x8 - 5.8*x10*x9 - 11.9*x10*x10); |
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675 | |
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676 | F = set(0<x<10000); |
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677 | F = F + set([e1;e2;e3;e4;e5;e6;e7;e8;e9;e10]<0); |
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678 | %F = F + set(obj==t); |
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679 | solvesdp(F,obj,sdpsettings('bmibnb.uppersolver','penbmi','bmibnb.lowersolver','glpk','solver','bmibnb','bmibnb.maxiter',15,'penbmi.P0',0.01,'bmibnb.lpsolver','glpk')) |
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