201 lines
6.0 KiB
Mathematica
201 lines
6.0 KiB
Mathematica
|
function [model,changed] = convert_polynomial_to_quadratic(model)
|
||
|
|
|
||
|
|
% Assume we don't do anything
|
||
|
|
changed = 0;
|
||
|
|
|
||
|
|
% Are there really any non-quadratic terms?
|
||
|
|
already_done = 0;
|
||
|
|
while any(model.variabletype > 2)
|
||
|
|
% Bugger...
|
||
|
|
changed = 1;
|
||
|
|
|
||
|
|
% Find a higher order term
|
||
|
|
polynomials = find(model.variabletype >= 3);
|
||
|
|
model = update_monomial_bounds(model,(already_done+1):size(model.monomtable,1));
|
||
|
|
already_done = size(model.monomtable,1);
|
||
|
|
% Start with the highest order monomial (the bilinearization is not
|
||
|
|
% unique, but by starting here, we get a reasonable small
|
||
|
|
% bilineared model
|
||
|
|
[i,j] = max(sum(abs(model.monomtable(polynomials,:)),2));
|
||
|
|
polynomials = polynomials(j);
|
||
|
|
|
||
|
|
powers = model.monomtable(polynomials,:);
|
||
|
|
if any(powers < 0)
|
||
|
|
model = eliminate_sigmonial(model,polynomials,powers);
|
||
|
|
elseif nnz(powers) == 1
|
||
|
|
model = bilinearize_recursive(model,polynomials,powers);
|
||
|
|
else
|
||
|
|
model = bilinearize_recursive(model,polynomials,powers);
|
||
|
|
end
|
||
|
|
end
|
||
|
|
|
||
|
|
function model = eliminate_sigmonial(model,polynomial,powers);
|
||
|
|
% Silly bug
|
||
|
|
if isempty(model.F_struc)
|
||
|
|
model.F_struc = zeros(0,length(model.c) + 1);
|
||
|
|
end
|
||
|
|
|
||
|
|
% x^(-p1)*w^(p2)
|
||
|
|
powers_pos = powers;
|
||
|
|
powers_neg = powers;
|
||
|
|
powers_pos(powers_pos<0) = 0;
|
||
|
|
powers_neg(powers_neg>0) = 0;
|
||
|
|
[model,index_neg] = findoraddlinearmonomial(model,-powers_neg);
|
||
|
|
if any(powers_pos)
|
||
|
|
[model,index_pos] = findoraddlinearmonomial(model,powers_pos);
|
||
|
|
else
|
||
|
|
index_pos = [];
|
||
|
|
end
|
||
|
|
% Now create a new variable y, used to model x^(-p1)*w^(p2)
|
||
|
|
% We will also add the constraint y*x^p1 = w^p2
|
||
|
|
model.monomtable(polynomial,:) = 0;
|
||
|
|
model.monomtable(polynomial,polynomial) = 1;
|
||
|
|
model.variabletype(polynomial) = 0;
|
||
|
|
model.high_monom_model = blockthem(model.high_monom_model,[polynomial powers_neg]);;
|
||
|
|
|
||
|
|
if ~isempty(model.x0);
|
||
|
|
model.x0(polynomial) = prod(model.x0(find(powers_neg))'.^powers_neg);
|
||
|
|
end
|
||
|
|
powers = -powers_neg;
|
||
|
|
powers(polynomial) = 1;
|
||
|
|
[model,index_xy] = findoraddmonomial(model,powers);
|
||
|
|
model.lb(index_xy) = 1;
|
||
|
|
model.ub(index_xy) = 1;
|
||
|
|
|
||
|
|
model = convert_polynomial_to_quadratic(model);
|
||
|
|
|
||
|
|
function model = bilinearize_recursive(model,polynomial,powers);
|
||
|
|
% Silly bug
|
||
|
|
if isempty(model.F_struc)
|
||
|
|
model.F_struc = zeros(0,length(model.c) + 1);
|
||
|
|
end
|
||
|
|
% variable^power
|
||
|
|
if nnz(powers) == 1
|
||
|
|
univariate = 1;
|
||
|
|
variable = find(powers);
|
||
|
|
p1 = floor(powers(variable)/2);
|
||
|
|
p2 = ceil(powers(variable)/2);
|
||
|
|
powers_1 = powers;powers_1(variable) = p1;
|
||
|
|
powers_2 = powers;powers_2(variable) = p2;
|
||
|
|
else
|
||
|
|
univariate = 0;
|
||
|
|
variables = find(powers);
|
||
|
|
mid = floor(length(variables)/2);
|
||
|
|
variables_1 = variables(1:mid);
|
||
|
|
variables_2 = variables(mid+1:end);
|
||
|
|
powers_1 = powers;
|
||
|
|
powers_2 = powers;
|
||
|
|
powers_1(variables_2) = 0;
|
||
|
|
powers_2(variables_1) = 0;
|
||
|
|
end
|
||
|
|
|
||
|
|
[model,index1] = findoraddlinearmonomial(model,powers_1);
|
||
|
|
[model,index2] = findoraddlinearmonomial(model,powers_2);
|
||
|
|
model.monomtable(polynomial,:) = 0;
|
||
|
|
model.monomtable(polynomial,index1) = model.monomtable(polynomial,index1) + 1;
|
||
|
|
model.monomtable(polynomial,index2) = model.monomtable(polynomial,index2) + 1;
|
||
|
|
|
||
|
|
if index1 == index2
|
||
|
|
model.variabletype(polynomial) = 2;
|
||
|
|
else
|
||
|
|
model.variabletype(polynomial) = 1;
|
||
|
|
end
|
||
|
|
%model = convert_polynomial_to_quadratic(model);
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
function [model,index] = findoraddmonomial(model,powers);
|
||
|
|
|
||
|
|
if length(powers) < size(model.monomtable,2)
|
||
|
|
powers(size(model.monomtable,1)) = 0;
|
||
|
|
end
|
||
|
|
|
||
|
|
index = findrows(model.monomtable,powers);
|
||
|
|
|
||
|
|
if isempty(index)
|
||
|
|
model.monomtable = [model.monomtable;powers];
|
||
|
|
model.monomtable(end,end+1) = 0;
|
||
|
|
index = size(model.monomtable,1);
|
||
|
|
model.c(end+1) = 0;
|
||
|
|
model.Q(end+1,end+1) = 0;
|
||
|
|
if size(model.F_struc,1)>0
|
||
|
|
model.F_struc(1,end+1) = 0;
|
||
|
|
else
|
||
|
|
model.F_struc = zeros(0, size(model.F_struc,2)+1);
|
||
|
|
end
|
||
|
|
bound = powerbound(model.lb,model.ub,powers);
|
||
|
|
model.lb(end+1) = bound(1);
|
||
|
|
model.ub(end+1) = bound(2);
|
||
|
|
if ~isempty(model.x0)
|
||
|
|
model.x0(end+1) = 0;
|
||
|
|
end
|
||
|
|
switch sum(powers)
|
||
|
|
case 1
|
||
|
|
model.variabletype(end+1) = 0;
|
||
|
|
case 2
|
||
|
|
if nnz(powers)==1
|
||
|
|
model.variabletype(end+1) = 2;
|
||
|
|
else
|
||
|
|
model.variabletype(end+1) = 1;
|
||
|
|
end
|
||
|
|
otherwise
|
||
|
|
model.variabletype(end+1) = 3;
|
||
|
|
end
|
||
|
|
end
|
||
|
|
|
||
|
|
function [model,index] = findoraddlinearmonomial(model,powers);
|
||
|
|
|
||
|
|
if sum(powers) == 1
|
||
|
|
if length(powers)<size(model.monomtable,2)
|
||
|
|
powers(size(model.monomtable,2)) = 0;
|
||
|
|
end
|
||
|
|
index = findrows(model.monomtable,powers);
|
||
|
|
return
|
||
|
|
end
|
||
|
|
|
||
|
|
% We want to see if the monomial x^powers already is modelled by a linear
|
||
|
|
% variable. If not, we create the linear variable and add the associated
|
||
|
|
% constraint y == x^powers
|
||
|
|
index = [];
|
||
|
|
if ~isempty(model.high_monom_model)
|
||
|
|
if length(powers)>size(model.high_monom_model,2)+1
|
||
|
|
model.high_monom_model(1,end+1) = 0;
|
||
|
|
end
|
||
|
|
Z = model.high_monom_model(:,2:end);
|
||
|
|
index = findrows(Z,powers);
|
||
|
|
if ~isempty(index)
|
||
|
|
index = model.high_monom_model(index,1);
|
||
|
|
return
|
||
|
|
end
|
||
|
|
end
|
||
|
|
% OK, we could not find a linear model of this monomial. We thus create a
|
||
|
|
% linear variable, and add the constraint. Note that it is assumed that the
|
||
|
|
% nonlinear monomial x^power does exist
|
||
|
|
[model,index_nonlinear] = findoraddmonomial(model,powers);
|
||
|
|
model.monomtable(end+1,end+1) = 1;
|
||
|
|
model.variabletype(end+1) = 0;
|
||
|
|
model.F_struc = [zeros(1,size(model.F_struc,2));model.F_struc];
|
||
|
|
model.K.f = model.K.f + 1;
|
||
|
|
model.F_struc(1,end+1) = 1;
|
||
|
|
model.F_struc(1,1+index_nonlinear) = -1;
|
||
|
|
model.c(end+1) = 0;
|
||
|
|
model.Q(end+1,end+1) = 0;
|
||
|
|
model.high_monom_model = blockthem(model.high_monom_model,[length(model.c) powers]);;
|
||
|
|
if ~isempty(model.x0);
|
||
|
|
model.x0(end+1) = model.x0(index_nonlinear);
|
||
|
|
end
|
||
|
|
model.lb(end+1) = model.lb(index_nonlinear);
|
||
|
|
model.ub(end+1) = model.ub(index_nonlinear);
|
||
|
|
index = length(model.c);
|
||
|
|
|
||
|
|
function z = initial(x0,powers)
|
||
|
|
z = 1;
|
||
|
|
vars = find(powers);
|
||
|
|
for k = 1:length(vars)
|
||
|
|
z = z * x0(vars(k))^powers(vars(k));
|
||
|
|
end
|
||
|
|
|
||
|
|
function A = blockthem(A,B)
|
||
|
|
n = size(A,2);
|
||
|
|
m = size(B,2);
|
||
|
|
A = [A zeros(size(A,1),max([0 m-n]));B zeros(size(B,1),max([0 n-m]))];
|