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| 6 | <title>YALMIP Example : Complex-valued problems</title> |
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| 19 | <tr> |
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| 20 | <td width="100%" align="left" height="100%" valign="top"> |
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| 21 | |
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| 22 | <h2>Complex-valued problems</h2> |
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| 23 | <hr noShade SIZE="1"> |
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| 24 | <p>YALMIP supports complex valued constraints for all solvers by |
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| 25 | automatically converting complex-valued problems to real-valued problems.</p> |
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| 26 | <p>To begin with, let us just define a simple linear complex problem to |
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| 27 | illustrate how complex variables and constraints are generated and |
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| 28 | interpreted.</p> |
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| 29 | <table cellPadding="10" width="100%"> |
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| 30 | <tr> |
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| 31 | <td class="xmpcode"> |
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| 32 | <pre>p = sdpvar(1,1,'full','complex'); % A complex scalar (4 arguments necessary) |
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| 33 | s = sdpvar(1,1)+sqrt(-1)*sdpvar(1,1); % Alternative definition |
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| 34 | F = set('0.9>imag(p)'); % Imaginary part constrained |
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| 35 | F = F+set('0.01>real(p)'); % Real part constrained |
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| 36 | |
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| 37 | F = F+set('0.1+0.5*sqrt(-1)>p'); % Both parts constrained |
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| 38 | |
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| 39 | F = F+set('s+p==2+4*sqrt(-1)'); % Both parts constrained</pre> |
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| 40 | </td> |
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| 41 | </tr> |
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| 42 | </table> |
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| 43 | <p>To see how complex-valued constraints can be used in a more advanced |
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| 44 | setting, we solve the covariance estimation problem from the <a href="solvers.htm#sedumi">SeDuMi</a> |
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| 45 | manual. The problem is to find a positive-definite Hermitian Toeplitz matrix |
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| 46 | <b>Z</b> such that the Frobenious norm of <b><font face="Tahoma">P-Z</font></b> is minimized (<b><font face="Tahoma">P</font></b> is a given complex |
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| 47 | matrix.)</p> |
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| 48 | <p>The matrix <b>P</b> is</p> |
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| 49 | <table cellPadding="10" width="100%"> |
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| 50 | <tr> |
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| 51 | <td class="xmpcode"> |
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| 52 | <pre>P = [4 1+2*i 3-i;1-2*i 3.5 0.8+2.3*i;3+i 0.8-2.3*i 4];</pre> |
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| 53 | </td> |
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| 54 | </tr> |
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| 55 | </table> |
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| 56 | <p>We define a complex-valued Toeplitz matrix of the corresponding dimension</p> |
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| 57 | <table cellPadding="10" width="100%"> |
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| 58 | <tr> |
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| 59 | <td class="xmpcode"> |
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| 60 | <pre>Z = sdpvar(3,3,'toeplitz','complex')</pre> |
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| 61 | </td> |
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| 62 | </tr> |
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| 63 | </table> |
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| 64 | <p>A complex Toeplitz matrix is not Hermitian, but we can make it Hermitian |
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| 65 | if we remove the imaginary part on the diagonal.</p> |
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| 66 | <table cellPadding="10" width="100%"> |
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| 67 | <tr> |
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| 68 | <td class="xmpcode"> |
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| 69 | <pre>Z = Z-diag(imag(diag(Z)))*sqrt(-1);</pre> |
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| 70 | </td> |
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| 71 | </tr> |
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| 72 | </table> |
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| 73 | <p>Minimizing the Frobenious norm of <b><font face="Tahoma">P-Z</font></b> can be cast as minimizing the |
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| 74 | Euclidean norm of the vectorized difference <b><font face="Tahoma">P(:)-Z(:)</font></b>. By using a Schur |
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| 75 | complement, we see that this can be written as the following SDP.</p> |
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| 76 | <table cellPadding="10" width="100%"> |
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| 77 | <tr> |
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| 78 | <td class="xmpcode"> |
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| 79 | <pre>e = P(:)-Z(:) |
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| 80 | t = sdpvar(1,1); |
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| 81 | F = set(Z>0); |
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| 82 | F = F+set([t e';e eye(9)]>0); |
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| 83 | solvesdp(F,t);</pre> |
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| 84 | </td> |
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| 85 | </tr> |
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| 86 | </table> |
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| 87 | <p>The problem can be implemented more efficiently using a second order cone |
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| 88 | constraint.</p> |
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| 89 | <table cellPadding="10" width="100%"> |
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| 90 | <tr> |
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| 91 | <td class="xmpcode"> |
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| 92 | <pre>e = Z(:)-P(:) |
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| 93 | t = sdpvar(1,1); |
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| 94 | F = set(Z>0); |
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| 95 | F = F+set(cone(e,t)); |
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| 96 | solvesdp(F,t);</pre> |
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| 97 | </td> |
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| 98 | </tr> |
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| 99 | </table> |
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| 100 | <p>...or by using a quadratic objective function</p> |
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| 101 | <table cellPadding="10" width="100%"> |
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| 102 | <tr> |
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| 103 | <td class="xmpcode"> |
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| 104 | <pre>e = Z(:)-P(:) |
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| 105 | F = set(Z>0); |
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| 106 | solvesdp(F,e'*e);</pre> |
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| 107 | </td> |
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| 108 | </tr> |
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| 109 | </table> |
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| 110 | </td> |
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| 111 | </tr> |
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| 112 | </table> |
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