[37] | 1 | function F = robustify_lp_conic(F_xw,Zmodel,x,w) |
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| 2 | |
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| 3 | % Creates robustified version of the uncertain set of linear inequalities |
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| 4 | % s.t A(w)*x <= b(w) for all F(w) >= 0 where F(w) is a conic set, here |
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| 5 | % given in SeDuMi (and YALMIP internal) numerical format. |
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| 6 | % |
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| 7 | % Based on Robust Optimization - Methodology and Applications. A. Ben-Tal |
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| 8 | % and A. Nemerovskii. Mathematical Programming (Series B), 92:453-480, 2002 |
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| 9 | % |
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| 10 | % Note, there are some sign errors in the paper. |
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| 11 | |
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| 12 | if length(F_xw) == 0 |
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| 13 | F = []; |
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| 14 | return |
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| 15 | end |
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| 16 | |
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| 17 | X = sdpvar(F_xw); |
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| 18 | b = []; |
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| 19 | A = []; |
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| 20 | % Some pre-calc |
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| 21 | xw = [x;w]; |
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| 22 | xind = find(ismembc(getvariables(xw),getvariables(x))); |
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| 23 | wind = find(ismembc(getvariables(xw),getvariables(w))); |
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| 24 | [Qs,cs,fs,dummy,nonquadratic] = vecquaddecomp(X,xw); |
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| 25 | for i = 1:length(X) |
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| 26 | %[Q,c,f,dummy,nonquadratic] = quaddecomp(X(i),xw); |
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| 27 | Q = Qs{i}; |
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| 28 | c = cs{i}; |
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| 29 | f = fs{i}; |
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| 30 | if nonquadratic |
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| 31 | error('Constraints can be at most quadratic, with the linear term uncertain'); |
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| 32 | end |
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| 33 | Q_ww = Q(wind,wind); |
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| 34 | Q_xw = Q(xind,wind); |
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| 35 | Q_xx = Q(xind,xind); |
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| 36 | c_x = c(xind); |
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| 37 | c_w = c(wind); |
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| 38 | |
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| 39 | b = [b;f + c_w'*w]; |
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| 40 | A = [A;-(c_x + 2*Q_xw*w)']; |
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| 41 | end |
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| 42 | |
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| 43 | % Try to find variables that only have simple bound constraints. These |
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| 44 | % variables can explicitly be optimized and thus speed up the construction, |
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| 45 | % and allow a model with fewer variables. |
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| 46 | [Zmodel2,lower,upper] = find_simple_variable_bounds(Zmodel); |
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| 47 | |
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| 48 | % Partition the uncertain variables |
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| 49 | simple_w = find( ~isinf(lower) & ~isinf(upper)); |
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| 50 | general_w = find( isinf(lower) | isinf(upper)); |
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| 51 | simple_w = recover(simple_w); |
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| 52 | general_w = recover(general_w); |
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| 53 | |
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| 54 | |
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| 55 | % Linear uncertain constraint is (Bbetai*x + cdi) >= 0 for all w, or |
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| 56 | % (bi' + (Bi*w)')*x + (ci'*w + di). |
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| 57 | cd = b; |
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| 58 | Bbeta = -A; |
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| 59 | |
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| 60 | F = set([]); |
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| 61 | top = 1; |
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| 62 | |
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| 63 | % To speed up the construction, compute the ci vectors for all constraints |
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| 64 | % in one call ci_basis = [c1 c2 ...] |
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| 65 | ci_basis = basis(cd',w); |
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| 66 | |
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| 67 | for i = 1:length(b) |
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| 68 | cdi = cd(i); |
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| 69 | Bbetai = Bbeta(i,:); |
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| 70 | |
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| 71 | if (nnz(ci_basis(:,i))==0) & isa(Bbetai,'double') |
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| 72 | % This constraint row is constant |
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| 73 | F = F + set(Bbetai*x + cdi >= 0); |
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| 74 | else |
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| 75 | |
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| 76 | if isempty(general_w) |
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| 77 | |
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| 78 | ci = ci_basis(:,i); |
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| 79 | |
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| 80 | di = basis(cdi,0); |
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| 81 | if isa(Bbetai,'double') |
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| 82 | Bi = zeros(1,length(w)); |
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| 83 | else |
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| 84 | Bi = basis(Bbetai,w)'; |
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| 85 | end |
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| 86 | bi = basis(Bbeta(i,:),0)'; |
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| 87 | % Scale to -1,1 uncertainty |
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| 88 | T = diag((upper-lower))/2; |
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| 89 | e = (upper+lower)/2; |
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| 90 | if nnz(Bi) == 0 |
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| 91 | F = F + set(bi'*x + (di-e'*T*ci) - norm(T*ci,1) > 0); |
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| 92 | else |
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| 93 | non_zeroBirow = find(sum(abs(Bi'),2)); |
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| 94 | zeroBirow = find(sum(abs(Bi'),2) == 0); |
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| 95 | t = sdpvar(length(non_zeroBirow),1); |
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| 96 | F = F + set((bi'-e'*T*Bi')*x + (di-e'*T*ci) - norm(T(zeroBirow,:)*ci,1) - sum(t) >= 0) + set(-t < T(non_zeroBirow,:)*(ci+Bi'*x) < t); |
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| 97 | end |
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| 98 | |
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| 99 | else |
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| 100 | |
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| 101 | lhs1 = 0; |
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| 102 | lhs2 = 0; |
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| 103 | top = 1; |
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| 104 | |
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| 105 | if Zmodel.K.f > 0 |
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| 106 | zeta = sdpvar(Zmodel.K.f,1); |
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| 107 | F = F + set(zeta >= 0); |
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| 108 | lhs1 = lhs1 + Zmodel.F_struc(top:top + Zmodel.K.f-1,2:end)'*zeta; |
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| 109 | lhs2 = lhs2 - Zmodel.F_struc(top:top + Zmodel.K.f-1,1)'*zeta; |
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| 110 | top = top + Zmodel.K.f; |
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| 111 | end |
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| 112 | |
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| 113 | if Zmodel.K.l > 0 |
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| 114 | zeta = sdpvar(Zmodel.K.l,1); |
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| 115 | F = F + set(zeta >= 0); |
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| 116 | lhs1 = lhs1 + Zmodel.F_struc(top:top + Zmodel.K.l-1,2:end)'*zeta; |
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| 117 | lhs2 = lhs2 - Zmodel.F_struc(top:top + Zmodel.K.l-1,1)'*zeta; |
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| 118 | top = top + Zmodel.K.l; |
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| 119 | end |
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| 120 | |
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| 121 | if Zmodel.K.q(1) > 0 |
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| 122 | for j = 1:length(Zmodel.K.q) |
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| 123 | zeta = sdpvar(Zmodel.K.q(j),1); |
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| 124 | F = F + set(cone(zeta)); |
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| 125 | lhs1 = lhs1 + Zmodel.F_struc(top:top + Zmodel.K.q(j)-1,2:end)'*zeta(:); |
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| 126 | lhs2 = lhs2 - Zmodel.F_struc(top:top + Zmodel.K.q(j)-1,1)'*zeta(:); |
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| 127 | top = top + Zmodel.K.q(j); |
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| 128 | end |
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| 129 | end |
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| 130 | |
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| 131 | if Zmodel.K.s(1) > 0 |
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| 132 | for j = 1:length(Zmodel.K.s) |
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| 133 | zeta = sdpvar(Zmodel.K.s(j)); |
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| 134 | F = F + set(zeta >= 0); |
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| 135 | lhs1 = lhs1 + Zmodel.F_struc(top:top + Zmodel.K.s(j)^2-1,2:end)'*zeta(:); |
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| 136 | lhs2 = lhs2 - Zmodel.F_struc(top:top + Zmodel.K.s(j)^2-1,1)'*zeta(:); |
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| 137 | top = top + Zmodel.K.s(j)^2; |
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| 138 | end |
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| 139 | end |
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| 140 | |
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| 141 | % if isempty(simple_w) |
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| 142 | ci = basis(cd(i),w); |
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| 143 | di = basis(cd(i),0); |
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| 144 | Bi = basis(Bbeta(i,:),w)'; |
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| 145 | bi = basis(Bbeta(i,:),0)'; |
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| 146 | F = F + set(lhs1 == Bi'*x + ci); |
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| 147 | F = F + set(lhs2 >= - (bi'*x + di)); |
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| 148 | end |
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| 149 | end |
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| 150 | end |
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| 151 | |
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| 152 | |
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| 153 | function b = basis(p,w) |
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| 154 | |
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| 155 | if isequal(w,0) |
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| 156 | b = getbasematrix(p,0); |
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| 157 | else |
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| 158 | n = length(w); |
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| 159 | if isequal(getbase(w),[zeros(n,1) eye(n)]) |
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| 160 | b = []; |
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| 161 | lmi_variables = getvariables(w); |
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| 162 | for i = 1:length(w) |
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| 163 | b = [b ; getbasematrix(p,lmi_variables(i))]; |
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| 164 | end |
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| 165 | else |
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| 166 | b = []; |
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| 167 | for i = 1:length(w) |
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| 168 | b = [b ; getbasematrix(p,getvariables(w(i)))]; |
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| 169 | end |
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| 170 | end |
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| 171 | end |
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| 172 | b = full(b); |
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| 173 | |
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| 174 | |
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| 175 | function [p,lower,upper] = find_simple_variable_bounds(p); |
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| 176 | |
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| 177 | if any(p.K.q > 0) | any(p.K.s > 0) |
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| 178 | lower = -inf(length(p.c),1); |
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| 179 | upper = inf(length(p.c),1); |
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| 180 | else |
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| 181 | [lower,upper,used_rows] = findulb(p.F_struc,p.K); |
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| 182 | used_rows = used_rows(~any(full(p.F_struc(used_rows,1+find(p.variabletype~=0))),2)); |
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| 183 | if ~isempty(used_rows) |
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| 184 | p_temp = p; |
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| 185 | p_temp.F_struc(p.K.f+used_rows,:)=[]; |
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| 186 | p_temp.K.l = p.K.l - length(used_rows); |
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| 187 | |
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| 188 | if size(p.F_struc,1) > 0 |
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| 189 | % These variables are still used in some other constraints |
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| 190 | still_used = find(sum(abs(p_temp.F_struc(:,2:end)),1) > 0); |
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| 191 | if ~isempty(still_used) |
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| 192 | % we have to keep these variables |
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| 193 | lower(still_used) = -inf; |
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| 194 | upper(still_used) = inf; |
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| 195 | % They are used here |
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| 196 | keep_rows = find(sum(abs(p.F_struc(:,1+still_used)),2) > 0); |
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| 197 | if any(keep_rows>(p.K.l + p.K.f)) |
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| 198 | % error('Tell johan to fix the SDP case in find_simple_variable_bounds!') |
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| 199 | end |
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| 200 | used_rows = used_rows + p.K.f; |
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| 201 | used_rows = setdiff(used_rows,keep_rows); |
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| 202 | p.F_struc(used_rows,:)=[]; |
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| 203 | p.K.l = p.K.l - nnz(used_rows>p.K.f); |
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| 204 | p.K.f = p.K.f - nnz(used_rows<=p.K.f); |
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| 205 | else |
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| 206 | p = p_temp; |
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| 207 | end |
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| 208 | else |
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| 209 | p = p_temp; |
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| 210 | end |
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| 211 | end |
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| 212 | end |
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