[37] | 1 | clc |
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| 2 | echo on |
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| 3 | |
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| 4 | % A recent and devloping extension in YALMIP is support |
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| 5 | % for nonlinear operators such as min, max, norm and more. |
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| 6 | % |
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| 7 | % Although nonlinear, and often non-differentiable, the resulting |
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| 8 | % optmization problem is in many cases still convex, and |
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| 9 | % can be modelled using suitable additional variables and constraits. |
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| 10 | % |
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| 11 | % If these operators are used, YALMIP will derive a suitable |
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| 12 | % convex model and solve the resulting problem. It may also happen |
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| 13 | % that YALMIP fails to build a convex model, since the rules to detect |
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| 14 | % convexity only are sufficient but not necessary. |
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| 15 | % |
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| 16 | % These extended operators should only be used of you now how |
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| 17 | % to model them manually, why it can be done and when it can be done. |
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| 18 | |
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| 19 | pause % Strike any key to continue. |
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| 20 | yalmip('clear') |
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| 21 | clc |
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| 22 | |
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| 23 | % To begin with, define some scalars |
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| 24 | sdpvar x y z |
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| 25 | |
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| 26 | % Nonlinear expressions are easily defined |
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| 27 | p = min(x+y+norm([x;y],1),x)-max(min(x,y),max(x,y)); |
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| 28 | pause |
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| 29 | |
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| 30 | % The result is a linear variable, but it is special. |
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| 31 | % This can be seen when displayed (note the "derived") |
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| 32 | p |
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| 33 | pause |
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| 34 | |
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| 35 | % These expressions can be used as any other expression |
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| 36 | % in YALMIP. The difference is when optmization problems |
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| 37 | % are solved. YALMIP will start by trying to expand the |
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| 38 | % definitions of the derived variables, and try to maintain |
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| 39 | % convexity while doing so. |
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| 40 | pause |
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| 41 | |
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| 42 | % Let us solve the linear regression again (from DEMO 2) |
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| 43 | a = [1 2 3 4 5 6]; |
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| 44 | t = (0:0.2:2*pi)'; |
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| 45 | x = [sin(t) sin(2*t) sin(3*t) sin(4*t) sin(5*t) sin(6*t)]; |
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| 46 | y = x*a'+(-4+8*rand(length(x),1)); |
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| 47 | a_hat = sdpvar(1,6); |
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| 48 | residuals = y-x*a_hat'; |
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| 49 | pause % Strike any key to continue. |
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| 50 | |
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| 51 | |
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| 52 | % Minimize L1 error (uses ABS operator) |
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| 53 | solvesdp([],sum(abs(residuals))); |
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| 54 | double(a_hat) |
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| 55 | pause |
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| 56 | |
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| 57 | % Minimize Linf error (uses both MAX and ABS) |
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| 58 | solvesdp([],max(abs(residuals))); |
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| 59 | double(a_hat) |
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| 60 | pause |
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| 61 | |
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| 62 | % Minimize L1 error even easier (uses NORM operator) |
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| 63 | % NOTE : This is much faster than explicitely |
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| 64 | % introducing the absolute values. |
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| 65 | solvesdp([],norm(residuals,1)); |
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| 66 | double(a_hat) |
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| 67 | pause |
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| 68 | |
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| 69 | % Minimize Linf error even easier (uses NORM operator) |
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| 70 | solvesdp([],norm(residuals,inf)); |
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| 71 | double(a_hat) |
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| 72 | pause |
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| 73 | |
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| 74 | % Regularized solution! |
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| 75 | solvesdp([],1e-2*norm(a_hat)+norm(residuals,inf)); |
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| 76 | double(a_hat) |
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| 77 | pause |
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| 78 | |
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| 79 | % Minimize Linf with performance constraint on L2 |
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| 80 | % |
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| 81 | % First, get the best possible L2 cost |
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| 82 | solvesdp([],norm(residuals)); |
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| 83 | optL2 = double(norm(residuals)); |
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| 84 | |
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| 85 | % Now optimize Linf with performance deteriation constraint on L2 |
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| 86 | F = set(norm(residuals) < optL2*1.2); |
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| 87 | obj = norm(residuals,inf); |
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| 88 | pause |
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| 89 | |
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| 90 | solvesdp(set(norm(residuals) < optL2*1.2),norm(residuals,inf)); |
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| 91 | |
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| 92 | double(a_hat) |
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| 93 | pause |
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| 94 | |
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| 95 | % Well, you get the picture... |
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| 96 | % |
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| 97 | % Here is an example were the convexity check (correctly) fails |
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| 98 | solvesdp(set(norm(residuals) < norm(residuals,1)),norm(residuals,inf)) |
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| 99 | pause |
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| 100 | |
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| 101 | % ...and here is an example where YALMIP fails to prove convexity |
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| 102 | % (this problem is convex) |
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| 103 | solvesdp([],norm(max(0,residuals))) |
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| 104 | pause |
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| 105 | |
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| 106 | % The rules for convexity preserving operations are currently very simple, |
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| 107 | % and will most likely be improved in a future version. |
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| 108 | % |
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| 109 | % Still, rather complicated construction are possible. |
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| 110 | sdpvar x y z |
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| 111 | F = set(max(1,x)+max(y^2,z)<3)+set(max(1,-min(x,y))<5)+set(norm([x;y],2)<z); |
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| 112 | sol = solvesdp(F,max(x,z)-min(y,z)-z); |
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| 113 | pause |
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| 114 | |
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| 115 | % The nonlinear operators currently supported are |
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| 116 | % |
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| 117 | % ABS : Absolute value of matrix |
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| 118 | % MIN : Minimum of column values |
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| 119 | % MAX : Minimum of column values |
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| 120 | % SUMK : Sum of k largest (eigen-) values |
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| 121 | % SUMABSK : Sum of k largest (by magniture) (eigen-) values |
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| 122 | % GEOMEAN2 : (Almost) Geometric mean of (eigen-) values (used for determinant maximization). |
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| 123 | % |
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| 124 | % Adding new operators is rather straightforward and is |
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| 125 | % described in the HTML-manual |
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| 126 | pause |
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| 127 | echo off |
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| 128 | |
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