125 lines
3.6 KiB
Mathematica
125 lines
3.6 KiB
Mathematica
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function [exponent_m,changed,no_lp_solved,keep] = newtonreduce(exponent_m,exponent_p,ops,interfacedata)
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% NEWTONREDUCE Removes monomials outside half Newton polytope
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%
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% V = NEWTONREDUCE(V,P)
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%
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% Input
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% P : Scalar SDPVAR object
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% V : Vector with SDPVAR objects
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%
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% Output
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% V : Vector with SDPVAR objects
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%
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% Example:
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%
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% sdpvar x y
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% p = 1+x^4*y^2+x^2*y^4;
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% v = monolist([x y],degree(p)/2);
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% sdisplay(v)
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% v = newtonreduce(v,p);
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% sdisplay(v)
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%
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% See also NEWTONMONOMS, CONSISTENT, CONGRUENCEBLOCKS
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% Higher level call with SDPVAR polynomials
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% Convert to exponent form
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sdpvarout = 0;
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if isa(exponent_m,'sdpvar')
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z = depends(exponent_p);
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z = recover(unique([depends(exponent_p) depends(exponent_m)]));
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[exponent_p,p_base] = getexponentbase(exponent_p,z);
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[m,m_base] = getexponentbase(exponent_m,z);
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exponent_m = cell(1);exponent_m{1} = m;
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sdpvarout = 1;
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end
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% Put exponents in cell (compability with monomialreduction.m
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doubleout = 0;
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if isa(exponent_m,'double')
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temp = exponent_m;
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exponent_m = cell(1);
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exponent_m{1} = temp;
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doubleout = 1;
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end
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if nargin<3
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ops = sdpsettings;
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end
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ops.solver = 'cdd,glpk,*'; % CDD is generally robust on these problems
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ops.verbose = 0;
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ops.saveduals = 0;
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% Create an LP structure
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if nargin < 4
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x=sdpvar(1,1);
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[aux1,aux2,aux3,interfacedata] = export((x>=0),x,ops);
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interfacedata.getsolvertime = 0;
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end
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bfixed = [zeros(size(exponent_p,1),1);1];
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interfacedata.K.l = length(bfixed);
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interfacedata.lb = [];
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interfacedata.ub = [];
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no_lp_solved = 0;
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for j = 1:length(exponent_m)
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try
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% Basic idea : Try to find separating hyperplane between Newton polytope and candidate,
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% for all candidates.
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% Pro : Numerical stability, polynomial (does NOT calculate H-representation of Newton polytope)
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% Con : Slightly slower than explicit convex hull calculation in most cases.
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keep = ismember(exponent_m{j}*2,exponent_p,'rows');
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dontkeep = 0*keep;
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if 0
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% Requires that cdd is installed.
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H=cddmex('hull',struct('V',full(exponent_p)));
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for i = 1:size(exponent_m{j},1)
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if ~all(H.A*exponent_m{j}(i,:)'*2 <= H.B)
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dontkeep(i) = 1;
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end
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end
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else
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ii = 0;
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for i = 1:length(keep)
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if ~keep(i) & ~dontkeep(i)
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q = exponent_m{j}(i,:)'*2;
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interfacedata.c = ([-q' 1])';
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interfacedata.Q = spalloc(length(interfacedata.c),length(interfacedata.c),0);
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interfacedata.F_struc = ([bfixed -[exponent_p -ones(size(exponent_p,1),1);q' -1]]);
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solution = feval(interfacedata.solver.call,interfacedata);
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no_lp_solved = no_lp_solved + 1;
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ii = ii + 1;
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if (solution.problem == 0 & ([-q' 1]*solution.Primal < 0)) | solution.problem == 2
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a = solution.Primal(1:end-1);
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b = solution.Primal(end);
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u = find(a'*2*exponent_m{j}' - b > sqrt(eps));
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dontkeep(u) = 1;
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end
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end
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if keep(i)
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q = exponent_m{j}(i,:)'*2;
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end
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end
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end
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keep = find(~dontkeep);
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catch
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keep = newtonpolytope(exponent_m{j}*2,exponent_p);
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end
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changed = length(keep)<size(exponent_m{j},1);
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exponent_m{j} = exponent_m{j}(keep,:);
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end
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if doubleout
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exponent_m = exponent_m{1};
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end
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if sdpvarout
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exponent_m = recovermonoms(exponent_m{1},z);
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end
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