[37] | 1 | function [model,changed] = bilinearize(model) |
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| 2 | |
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| 3 | % Assume we don't do anything |
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| 4 | changed = 0; |
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| 5 | |
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| 6 | % Are there really any non-quadratic terms? |
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| 7 | if any(model.variabletype > 2) |
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| 8 | % Bugger... |
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| 9 | changed = 1; |
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| 10 | |
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| 11 | % Find a higher order term |
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| 12 | first_polynomial = find(model.variabletype == 3); |
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| 13 | model = fixbounds(model); |
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| 14 | first_polynomial = first_polynomial(1); |
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| 15 | powers = model.monomtable(first_polynomial,:); |
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| 16 | if nnz(powers) == 1 |
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| 17 | model = univariate_bilinearize(model,first_polynomial,powers); |
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| 18 | else |
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| 19 | model = multivariable_bilinearize(model,first_polynomial,powers); |
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| 20 | end |
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| 21 | % |
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| 22 | % % Find inverses etc |
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| 23 | % first_sigmonial = find(model.variabletype == 4); |
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| 24 | % if ~isempty(first_sigmonial) |
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| 25 | % first_sigmonial = first_sigmonial(1); |
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| 26 | % powers = model.monomtable(first_sigmonial,:); |
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| 27 | % if any(powers ~= fix(powers) |
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| 28 | % error('model class not supported') |
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| 29 | % else |
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| 30 | % powers_new(powers>0) = 0; |
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| 31 | % powers_new(powers>0) = 0; |
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| 32 | % end |
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| 33 | % end |
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| 34 | end |
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| 35 | |
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| 36 | function model = univariate_bilinearize(model,first_polynomial,powers); |
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| 37 | |
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| 38 | % Fix initial? |
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| 39 | fix_initials = ~isempty(model.x0); |
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| 40 | |
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| 41 | % variable^power |
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| 42 | variable = find(powers); |
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| 43 | p1 = floor(powers(variable)/2); |
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| 44 | p2 = ceil(powers(variable)/2); |
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| 45 | |
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| 46 | powers_1 = powers;powers_1(variable) = p1; |
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| 47 | powers_2 = powers;powers_2(variable) = p2; |
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| 48 | |
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| 49 | % Only recursive if power>4 |
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| 50 | switch p1+p2 |
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| 51 | case 3 |
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| 52 | [model,index2] = findoradd(model,powers_2); |
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| 53 | % Now define new variable y, replace x^3 with x*y, and add |
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| 54 | % constraint y == x^2 |
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| 55 | model.monomtable(end+1,end+1) = 1; |
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| 56 | model.variabletype(end+1) = 0; |
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| 57 | model.monomtable(first_polynomial,variable) = 1; |
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| 58 | model.monomtable(first_polynomial,end) = 1; |
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| 59 | model.variabletype(first_polynomial) = 1; |
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| 60 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 61 | model.K.f = model.K.f + 1; |
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| 62 | model.F_struc(1,end+1) = 1; |
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| 63 | model.F_struc(1,1+index2) = -1; |
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| 64 | model.c(end+1) = 0; |
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| 65 | model.Q(end+1,end+1) = 0; |
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| 66 | if fix_initials |
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| 67 | model.x0(end+1) = initial(model.x0,powers_2); |
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| 68 | % model.x0(end+1) = 0; |
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| 69 | end |
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| 70 | bound = powerbound(model.lb,model.ub,powers_2); |
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| 71 | model.lb(end+1) = -bound; |
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| 72 | model.ub(end+1) = bound; |
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| 73 | |
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| 74 | case 4 |
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| 75 | [model,index2] = findoradd(model,powers_2); |
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| 76 | model.monomtable(end+1,end+1) = 1; |
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| 77 | model.variabletype(end+1) = 0; |
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| 78 | model.monomtable(first_polynomial,variable) = 0; |
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| 79 | model.monomtable(first_polynomial,end) = 2; |
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| 80 | model.variabletype(first_polynomial) = 2; |
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| 81 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 82 | model.K.f = model.K.f + 1; |
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| 83 | model.F_struc(1,end+1) = 1; |
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| 84 | model.F_struc(1,1+index2) = -1; |
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| 85 | model.c(end+1) = 0; |
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| 86 | if fix_initials |
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| 87 | model.x0(end+1) = initial(model.x0,powers_2); |
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| 88 | % model.x0(end+1) = 0; |
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| 89 | end |
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| 90 | model.Q(end+1,end+1) = 0; |
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| 91 | bound = powerbound(model.lb,model.ub,powers_2); |
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| 92 | model.lb(end+1) = 0; |
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| 93 | model.ub(end+1) = bound; |
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| 94 | otherwise |
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| 95 | [model,index1] = findoradd(model,powers_1); |
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| 96 | [model,index2] = findoradd(model,powers_2); |
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| 97 | model.monomtable(end+1,end+1) = 1; |
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| 98 | model.monomtable(end+1,end+1) = 1; |
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| 99 | model.variabletype(end+1) = 0; |
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| 100 | model.variabletype(end+1) = 0; |
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| 101 | model.monomtable(first_polynomial,variable) = 0; |
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| 102 | model.monomtable(first_polynomial,end) = 1; |
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| 103 | model.monomtable(first_polynomial,end-1) = 1; |
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| 104 | model.variabletype(first_polynomial) = 1; |
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| 105 | |
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| 106 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 107 | model.K.f = model.K.f + 1; |
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| 108 | model.F_struc(1,end+1) = 1; |
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| 109 | model.F_struc(1,1+index1) = -1; |
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| 110 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 111 | model.K.f = model.K.f + 1; |
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| 112 | model.F_struc(1,end+1) = 1; |
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| 113 | model.F_struc(1,1+index2) = -1; |
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| 114 | |
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| 115 | model.c(end+1) = 0; |
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| 116 | model.Q(end+1,end+1) = 0; |
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| 117 | model.c(end+1) = 0; |
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| 118 | model.Q(end+1,end+1) = 0; |
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| 119 | |
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| 120 | if fix_initials |
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| 121 | x0 = model.x0; |
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| 122 | model.x0(end+1) = initial(x0,powers_1); |
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| 123 | model.x0(end+1) = initial(x0,powers_2); |
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| 124 | % model.x0(end+1) = 0; |
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| 125 | % model.x0(end+1) = 0; |
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| 126 | end |
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| 127 | bound1 = powerbound(model.lb,model.ub,powers_1); |
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| 128 | bound2 = powerbound(model.lb,model.ub,powers_2), |
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| 129 | |
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| 130 | model.lb(end+1) = -bound1; |
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| 131 | model.ub(end+1) = bound1; |
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| 132 | model.lb(end+1) = -bound2; |
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| 133 | model.ub(end+1) = bound2; |
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| 134 | end |
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| 135 | model = bilinearize(model); |
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| 136 | |
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| 137 | |
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| 138 | |
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| 139 | |
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| 140 | function model = multivariable_bilinearize(model,first_polynomial,powers); |
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| 141 | |
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| 142 | % Fix initial? |
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| 143 | fix_initials = ~isempty(model.x0); |
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| 144 | |
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| 145 | variables = find(powers); |
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| 146 | mid = floor(length(variables)/2); |
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| 147 | variables_1 = variables(1:mid); |
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| 148 | variables_2 = variables(mid+1:end); |
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| 149 | powers_1 = powers; |
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| 150 | powers_2 = powers; |
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| 151 | powers_1(variables_2) = 0; |
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| 152 | powers_2(variables_1) = 0; |
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| 153 | |
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| 154 | [model,index1] = findoradd(model,powers_1); |
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| 155 | if sum(powers_1)>1 |
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| 156 | model.monomtable(end+1,end+1) = 1; |
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| 157 | pos1 = size(model.monomtable,2); |
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| 158 | model.variabletype(end+1) = 0; |
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| 159 | bound = powerbound(model.lb,model.ub,powers_1); |
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| 160 | model.lb(end+1) = -bound; |
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| 161 | model.ub(end+1) = bound; |
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| 162 | if fix_initials;model.x0(end+1) = initial(model.x0,powers_1);end |
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| 163 | end |
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| 164 | |
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| 165 | [model,index2] = findoradd(model,powers_2); |
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| 166 | if sum(powers_2)>1 |
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| 167 | model.monomtable(end+1,end+1) = 1; |
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| 168 | pos2 = size(model.monomtable,2); |
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| 169 | model.variabletype(end+1) = 0; |
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| 170 | bound = powerbound(model.lb,model.ub,powers_2); |
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| 171 | model.lb(end+1) = -bound; |
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| 172 | model.ub(end+1) = bound; |
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| 173 | if fix_initials;model.x0(end+1) = initial(model.x0,powers_2);end |
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| 174 | end |
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| 175 | |
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| 176 | model.monomtable(first_polynomial,:) = 0; |
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| 177 | if sum(powers_1)>1 |
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| 178 | model.monomtable(first_polynomial,pos1) = 1; |
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| 179 | else |
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| 180 | model.monomtable(first_polynomial,variables_1) = 1; |
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| 181 | end |
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| 182 | if sum(powers_2)>1 |
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| 183 | model.monomtable(first_polynomial,pos2) = 1; |
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| 184 | else |
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| 185 | model.monomtable(first_polynomial,variables_2) = 1; |
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| 186 | end |
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| 187 | model.variabletype(first_polynomial) = 1; |
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| 188 | |
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| 189 | if sum(powers_1)>1 |
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| 190 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 191 | model.K.f = model.K.f + 1; |
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| 192 | model.F_struc(1,end+1) = 1; |
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| 193 | model.F_struc(1,1+index1) = -1; |
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| 194 | model.c(end+1) = 0; |
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| 195 | model.Q(end+1,end+1) = 0; |
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| 196 | end |
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| 197 | |
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| 198 | if sum(powers_2)>1 |
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| 199 | model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc]; |
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| 200 | model.K.f = model.K.f + 1; |
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| 201 | model.F_struc(1,end+1) = 1; |
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| 202 | model.F_struc(1,1+index2) = -1; |
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| 203 | model.c(end+1) = 0; |
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| 204 | model.Q(end+1,end+1) = 0; |
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| 205 | end |
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| 206 | model = bilinearize(model); |
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| 207 | |
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| 208 | |
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| 209 | function [model,index1] = findoradd(model,powers_1); |
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| 210 | |
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| 211 | if length(powers_1) < size(model.monomtable,2) |
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| 212 | powers_1(size(model.monomtable,1)) = 0; |
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| 213 | end |
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| 214 | |
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| 215 | index1 = findrows(model.monomtable,powers_1); |
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| 216 | |
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| 217 | if isempty(index1) |
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| 218 | model.monomtable = [model.monomtable;powers_1]; |
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| 219 | model.monomtable(end,end+1) = 0; |
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| 220 | index1 = size(model.monomtable,1); |
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| 221 | model.c(end+1) = 0; |
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| 222 | model.Q(end+1,end+1) = 0; |
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| 223 | model.F_struc(1,end+1) = 0; |
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| 224 | bound = powerbound(model.lb,model.ub,powers_1); |
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| 225 | model.lb(end+1) = -bound; |
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| 226 | model.ub(end+1) = bound; |
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| 227 | if ~isempty(model.x0) |
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| 228 | model.x0(end+1) = 0; |
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| 229 | end |
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| 230 | switch sum(powers_1) |
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| 231 | case 1 |
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| 232 | model.variabletype(end+1) = 0; |
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| 233 | case 2 |
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| 234 | model.variabletype(end+1) = 2; |
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| 235 | otherwise |
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| 236 | model.variabletype(end+1) = 3; |
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| 237 | end |
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| 238 | end |
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| 239 | |
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| 240 | |
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| 241 | |
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| 242 | function model = fixbounds(model); |
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| 243 | polynomials = find(model.variabletype > 0); |
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| 244 | LU = max([abs(model.lb) abs(model.ub)],[],2); |
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| 245 | for i = 1:length(polynomials) |
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| 246 | j = polynomials(i); |
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| 247 | if j<=length(model.lb) |
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| 248 | vars = find(model.monomtable(j,:)); |
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| 249 | bound = 1; |
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| 250 | for k = 1:length(vars) |
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| 251 | bound = bound * LU(vars(k))^model.monomtable(j,vars(k)); |
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| 252 | end |
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| 253 | model.lb(j) = max(model.lb(j),-bound); |
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| 254 | model.ub(j) = min(model.ub(j),bound); |
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| 255 | end |
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| 256 | end |
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| 257 | |
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| 258 | function bound = powerbound(lb,ub,powers) |
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| 259 | LU = max([abs(lb) abs(ub)],[],2); |
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| 260 | vars = find(powers); |
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| 261 | bound = 1; |
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| 262 | for k = 1:length(vars) |
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| 263 | bound = bound * LU(vars(k))^powers(vars(k)); |
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| 264 | end |
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| 265 | |
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| 266 | function z = initial(x0,powers) |
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| 267 | z = 1; |
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| 268 | vars = find(powers); |
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| 269 | for k = 1:length(vars) |
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| 270 | z = z * x0(vars(k))^powers(vars(k)); |
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| 271 | end |
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| 272 | |
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| 273 | |
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| 274 | |
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| 275 | |
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| 276 | |
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