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6<title>YALMIP Example : Determinant maximization</title>
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17  <table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
18    <tr>
19      <td width="100%" align="left" height="100%" valign="top">
20      <h2>Determinant maximization </h2>
21      <hr noshade size="1">
22      <p>Let us solve a determinant maximization problem. Given two ellipsoids
23      </p>
24      <blockquote dir="ltr" style="MARGIN-RIGHT: 0px">
25        <p><span style="font-style: normal"><strong>E<sub>1</sub> = {x | x<sup>T</sup>P<sub>1</sub>x</strong></span><strong>&#8804;</strong><strong><span style="font-style: normal">1}</span></strong></p>
26        <p><span style="font-style: normal"><strong>E<sub>2</sub> = {x | x<sup>T</sup>P<sub>2</sub>x</strong></span><strong>&#8804;</strong><span style="font-style: normal"><strong>1}</strong></span></p>
27      </blockquote>
28      <p>Find the ellipsoid <strong>E = {x | x</strong><span style="font-style: normal"><strong><sup>T</sup></strong></span><strong>Px&#8804;1}</strong> 
29      with smallest possible volume that contain the union of <strong>E<sub>1</sub></strong> 
30      and <strong>E<sub>2</sub></strong>. By using the fact that the volume of the
31      ellipsoid is proportional to <strong>-det P </strong>and applying the S-procedure,
32      it can be shown that this problem can be written as</p>
33      <p><img border="0" src="ellips5.gif" hspace="45"></p>
34      <p>The objective function <b>-det P</b> (which is minimized) is not
35      convex, but monotonic transformations can render this problem convex. One
36      alternative is the logarithmic transform, leading to minimization of <b>-log
37      det P</b> instead. This operator was used in previous version of YALMIP,
38      but is not recommended any more. </p>
39      <p>Instead, YALMIP uses <b>-(det P)<sup>1/m</sup></b> where <b>m</b> is dimension of <b>P </b>(in other words, the geometric mean of the eigenvalues). The
40      concave function <b>(det
41      P)<sup>1/m</sup></b>, available by applying <b>geomean</b> on a Hermitian
42      matrix in YALMIP, can be modeled using semidefinite and second order
43      cones, hence any SDP solver can be used for solving determinant
44                maximization problems. See <a href="readmore.htm#NESNEM94">[Nesterov and
45      Nemirovskii]</a> for details.</p>
46      <table cellpadding="10" width="100%">
47        <tr>
48          <td class="xmpcode"><font face="Courier New" color="#0000c0">n = 2;<br>
49          P1 = randn(2);P1 = P1*P1&#39;; % Generate random ellipsoid<br>
50          P2 = randn(2);P2 = P2*P2&#39;; % Generate random ellipsoid<br>
51          t = sdpvar(2,1);<br>
52          P = sdpvar(n,n);<br>
53          F = set(1 &gt; t &gt; 0);<br>
54          F = F + set(t(1)*P1-P &gt; 0);<br>
55          F = F + set(t(2)*P2-P &gt; 0);<br>
56          sol = solvesdp(F,-geomean(P));<br>
57          ellipplot(double(P));hold on;<br>
58          ellipplot(double(P1));<br>
59          ellipplot(double(P2));</font></td>
60        </tr>
61      </table>
62      <p>If you have
63      the dedicated solver
64      <a href="solvers.htm#maxdet">MAXDET</a> installed and want to use it, you must use the dedicated command
65      <a href="reference.htm#logdet">logdet</a> for the objective and explicitly select
66      <a href="solvers.htm#maxdet">MAXDET</a>. This command can not be used
67      in any other construction than in the objective function, compared to the <b>geomean</b> operator that can be used as any other variable in YALMIP, since it
68      a so called <a href="extoperators.htm">extended operator</a>. </p>
69      <table cellpadding="10" width="100%">
70        <tr>
71          <td class="xmpcode"><font face="Courier New" color="#0000c0">solvesdp(F,-logdet(P),sdpsettings('solver','maxdet'));<br>
72          ellipplot(double(P));hold on;<br>
73          ellipplot(double(P1));<br>
74          ellipplot(double(P2));</font></td>
75        </tr>
76      </table>
77      <p>
78          <img border="0" src="demoicon.gif" width="16" height="16"> Note that
79                if you use the
80      <a href="reference.htm#logdet">logdet</a> command, if
81      <a href="solvers.htm#maxdet">MAXDET</a> not is
82      explicitly selected, YALMIP will use <b>-(det P)<sup>1/m</sup></b> as objective
83      function instead.
84      This will not cause any problems if your objective function is a simple
85      <a href="reference.htm#logdet">logdet</a> expression (since the two functions
86                are monotonically related). However, if you have a mixed objective
87                function such as <b>tr(P)-logdet(P)</b>, the conversion will change your
88                optimal solution. Hence, if you really want to optimize the mixed
89                expression, you must explicitly select
90      <a href="solvers.htm#maxdet">MAXDET</a>. Otherwise, YALMIP will change
91                your objective to <b>tr(P)-(det P)<sup>1/m</sup></b>.</td>
92    </tr>
93  </table>
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