99 lines
2.3 KiB
Mathematica
99 lines
2.3 KiB
Mathematica
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function varargout = plog(varargin)
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%PLOG
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%
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% y = PLOG(x)
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%
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% Computes concave perspective log, x(1)*log(x(2)/x(1)) on x>0
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%
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% Implemented as evalutation based nonlinear operator. Hence, the concavity
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% of this function is exploited to perform convexity analysis and rigorous
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% modelling.
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switch class(varargin{1})
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case 'double'
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if ~isequal(prod(size(varargin{1})),2)
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error('PLOG only defined for 2x1 arguments');
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end
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x = varargin{1};
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% Safe version with defined negative values (helps fmincon when
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% outside feasible region)
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if isequal(x(1),[0])
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varargout{1} = 0;
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else
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varargout{1} = x(1)*log(x(2)/x(1));
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end
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case 'sdpvar'
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if ~isequal(prod(size(varargin{1})),2)
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error('PLOG only defined for 2x1 arguments');
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else
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varargout{1} = yalmip('define',mfilename,varargin{1});
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end
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case 'char'
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X = varargin{3};
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operator = struct('convexity','concave','monotonicity','none','definiteness','none','model','callback');
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operator.range = [-inf inf];
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operator.domain = [0 inf];
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operator.bounds = @bounds;
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operator.convexhull = @convexhull;
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operator.derivative = @derivative;
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varargout{1} = [];
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varargout{2} = operator;
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varargout{3} = X;
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otherwise
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error('SDPVAR/PLOG called with CHAR argument?');
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end
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function dp = derivative(x)
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z = x(2)/x(1);
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dp = [log(z)-1;1./z];
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function [L,U] = bounds(xL,xU)
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xU(isinf(xU)) = 1e12;
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x1 = xL(1)*log(xL(2)/xL(1));
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x2 = xU(1)*log(xU(2)/xU(1));
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x3 = xL(1)*log(xU(2)/xL(1));
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x4 = xU(1)*log(xL(2)/xU(1));
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L = min([x1 x2 x3 x4]);
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% Stationary in x1 when x1 = x2*exp(-1)
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% Increasing in x2, so max at border
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p1 = [exp(-1)*xU(2);xU(2)];
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x5 = p1(1)*log(p1(2)/p1(1));
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if x5(1) >= xL(1) && x5(1) <= xU(1)
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U = max([x1 x2 x3 x4 x5]);
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else
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U = max([x1 x2 x3 x4]);
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end
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function [Ax,Ay,b] = convexhull(xL,xU)
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x1 = [xL(1);xL(2)];
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x2 = [xU(1);xL(2)];
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x3 = [xL(1);xU(2)];
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x4 = [xU(1);xU(2)];
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x5 = (xL+xU)/2;
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f1 = plog(x1);
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f2 = plog(x2);
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f3 = plog(x3);
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f4 = plog(x4);
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f5 = plog(x5);
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df1 = derivative(x1);
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df2 = derivative(x2);
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df3 = derivative(x3);
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df4 = derivative(x4);
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df5 = derivative(x5);
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[Ax,Ay,b] = convexhullConcave2D(x1,f1,df1,x2,f2,df2,x3,f3,df3,x4,f4,df4,x5,f5,df5);
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