114 lines
3.8 KiB
Mathematica
114 lines
3.8 KiB
Mathematica
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function varargout = pnorm(varargin)
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%PNORM P-Norm of SDPVAR variable with convexity knowledge
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%
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% PNORM is recommended if your goal is to obtain
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% a convex model, since the function PNORM is implemented
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% as a so called nonlinear operator. (For p/q ==1,2,inf you should use the
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% overloaded norm)
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%
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% t = pnorm(x,p/q), p/q >= 1
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%
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% Note, the pnorm is implemented using cpower, which adds
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% a large number of variables and constraints
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switch class(varargin{1})
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case {'double', 'gem', 'sgem'}
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varargout{1} = sum(varargin{1}.^varargin{2}).^(1/varargin{2});
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case 'sdpvar' % Overloaded operator for SDPVAR objects. Pass on args and save them.
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X = varargin{1};
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[n,m] = size(X);
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if isreal(X) & min(n,m)==1
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if varargin{2}>=1
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varargout{1} = yalmip('define',mfilename,varargin{:});
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else
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error('PNORM only applicable for p>=1');
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end
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else
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error('PNORM can only be applied to real vectors.');
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end
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case 'char' % YALMIP send 'model' when it wants the epigraph or hypograph
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if isequal(varargin{1},'graph')
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t = varargin{2}; % Second arg is the extended operator variable
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X = varargin{3}; % Third arg and above are the args user used when defining t.
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p = varargin{4};
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% sum(|xi|^(l/m))^(l/m) < t
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% si>0
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% -t^({l-m}/m)si^(m/l) < -xi
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% -t^({l-m}/m)si^(m/l) < xi
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%sum(si) < t
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% t>0
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if 0
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[p,q] = rat(p);
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absX = sdpvar(length(X),1);
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y = sdpvar(length(X),1);
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F = [-absX < X < absX];
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for i = 1:length(y)
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F = [F,pospower(absX(i),y(i),p,q)];
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end
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F = [F,pospower(t,sum(y),q,p)];
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else
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[l,m] = rat(p);
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% l=4
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% m = 1
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% x<(t^(l-m)*s^m)^1/l
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% -x<(t^(l-m)*s^m)^1/l
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% x < t t t s
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if 2^fix(log2(l))==l & m == 1
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s=sdpvar(length(X),1);
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absX = sdpvar(length(X),1);
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F = [-absX <= X <= absX];
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F = [F,sum(s)<= t,s>=0];
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for i = 1:length(X)
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F = [F,detset(absX(i),[repmat(t,1,l-m) s(i)])];
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F = [F,detset(-absX(i),[repmat(t,1,l-m) s(i)])];
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end
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else
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% l = 7
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% m = 2
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% x < (t t t t t s s)^(1/7)
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% x^7 < (t t t t t s s)
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% x^8 < (t t t t t s s x)
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% x < (t t t t t s s x)^(1/8
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s=sdpvar(length(X),1);
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w = 2^(ceil(log2(l)));
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absX = sdpvar(length(X),1);
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F = [-absX <= X <= absX];
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F = [F,sum(s)<= t,s>=0];
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for i = 1:length(X)
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F = [F,detset(absX(i),[repmat(t,1,l-m) repmat(s(i),1,m) repmat(absX(i),1,w-l)])];
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F = [F,detset(-absX(i),[repmat(t,1,l-m) repmat(s(i),1,m) repmat(absX(i),1,w-l)])];
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end
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end
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end
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varargout{1} = F;
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varargout{2} = struct('convexity','convex','monotonicity','none','definiteness','positive','model','graph');
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varargout{3} = X;
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end
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otherwise
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end
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function F = pospower(x,t,p,q)
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if p>q
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l = ceil(log2(abs(p)));
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r = 2^l-p;
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y = [ones(r,1)*x;ones(q,1)*t;ones(2^l-r-q,1)];
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F = detset(x,y);
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else
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l = ceil(log2(abs(q)));
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y = [ones(p,1)*x;ones(2^l-q,1)*t;ones(q-p,1)];
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F = detset(t,y);
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end
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