[37] | 1 | function varargout = cpower(varargin) |
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| 2 | %CPOWER Power of SDPVAR variable with convexity knowledge |
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| 3 | % |
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| 4 | % CPOWER is recommended if your goal is to obtain |
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| 5 | % a convex model, since the function CPOWER is implemented |
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| 6 | % as a so called nonlinear operator. (For p/q ==2 you can |
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| 7 | % however just as well use the overloaded power) |
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| 8 | % |
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| 9 | % t = power(x,p/q) |
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| 10 | % |
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| 11 | % For negative p/q, the operator is convex. |
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| 12 | % For positive p/q with p>q, the operator is convex. |
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| 13 | % For positive p/q with p<q, the operator is concave. |
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| 14 | % |
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| 15 | % A domain constraint x>0 is automatically added if |
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| 16 | % p/q not is an even integer. |
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| 17 | % |
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| 18 | % Note, the complexity of generating the conic representation |
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| 19 | % of these variables are O(2^L) where L typically is the |
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| 20 | % smallest integer such that 2^L >= min(p,q) |
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| 21 | |
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| 22 | % Author Johan Löfberg |
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| 23 | % $Id: cpower.m,v 1.1 2006/03/30 13:36:39 joloef Exp $ |
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| 24 | |
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| 25 | switch class(varargin{1}) |
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| 26 | |
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| 27 | case 'double' |
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| 28 | varargout{1} = power(varargin{1},varargin{2}); |
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| 29 | case 'sdpvar' % Overloaded operator for SDPVAR objects. Pass on args and save them. |
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| 30 | X = varargin{1}; |
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| 31 | [n,m] = size(X); |
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| 32 | if isreal(X) & n*m==1 |
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| 33 | varargout{1} = yalmip('addextendedvariable',mfilename,varargin{:}); |
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| 34 | else |
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| 35 | error('CPOWER can only be applied to real vectors.'); |
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| 36 | end |
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| 37 | |
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| 38 | case 'char' % YALMIP send 'model' when it wants the epigraph or hypograph |
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| 39 | if isequal(varargin{1},'graph') |
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| 40 | t = varargin{2}; % Second arg is the extended operator variable |
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| 41 | X = varargin{3}; % Third arg and above are the args user used when defining t. |
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| 42 | p = varargin{4}; |
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| 43 | |
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| 44 | if p>0 |
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| 45 | [p,q] = rat(abs(p)); |
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| 46 | F = pospower(set([]),X,t,p,q); |
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| 47 | if p>q |
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| 48 | convexity = 'convex'; |
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| 49 | monotonicity = 'increasing'; |
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| 50 | else |
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| 51 | convexity = 'concave'; |
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| 52 | monotonicity = 'decreasing'; |
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| 53 | |
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| 54 | end |
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| 55 | else |
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| 56 | [p,q] = rat(abs(p)); |
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| 57 | F = negpower(set([]),X,t,p,q); |
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| 58 | convexity = 'convex'; |
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| 59 | monotonicity = 'decreasing'; |
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| 60 | end |
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| 61 | |
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| 62 | varargout{1} = F; |
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| 63 | varargout{2} = struct('convexity',convexity,'monotonicity',monotonicity,'definiteness','positive'); |
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| 64 | varargout{3} = X; |
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| 65 | end |
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| 66 | otherwise |
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| 67 | end |
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| 68 | |
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| 69 | function F = pospower(F,x,t,p,q) |
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| 70 | if p>q |
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| 71 | l = ceil(log2(abs(p))); |
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| 72 | r = 2^l-p; |
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| 73 | y = [ones(r,1)*x;ones(q,1)*t;ones(2^l-r-q,1)]; |
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| 74 | F = detset(x,y); |
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| 75 | else |
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| 76 | l = ceil(log2(abs(q))); |
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| 77 | y = [ones(p,1)*x;ones(2^l-q,1)*t;ones(q-p,1)]; |
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| 78 | F = detset(t,y); |
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| 79 | end |
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| 80 | |
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| 81 | function F = negpower(F,x,t,p,q) |
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| 82 | l = ceil(log2(abs(p+q))); |
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| 83 | p = abs(p); |
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| 84 | q = abs(q); |
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| 85 | y = [ones(2^l-p-q,1);ones(p,1)*x;ones(q,1)*t]; |
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| 86 | F = detset(1,y); |
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| 87 | |
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