📄 ellipsoidal.htm
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<title>YALMIP Example : Determinant maximization</title>
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<h2>Ellipsoidal calculus </h2>
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<p>Let us solve a determinant maximization problem. Given two ellipsoids
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<p><span style="font-style: normal"><strong>E<sub>1</sub> = {x | x<sup>T</sup>P<sub>1</sub>x</strong></span><strong>≤</strong><strong><span style="font-style: normal">1}</span></strong></p>
<p><span style="font-style: normal"><strong>E<sub>2</sub> = {x | x<sup>T</sup>P<sub>2</sub>x</strong></span><strong>≤</strong><span style="font-style: normal"><strong>1}</strong></span></p>
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<p>Find the ellipsoid <strong>E = {x | x</strong><span style="font-style: normal"><strong><sup>T</sup></strong></span><strong>Px≤1}</strong>
with smallest possible volume that contain the union of <strong>E<sub>1</sub></strong>
and <strong>E<sub>2</sub></strong>. By using the fact that the volume of the
ellipsoid is proportional to <strong>-det P </strong>and applying the S-procedure,
it can be shown that this problem can be written as</p>
<p><img border="0" src="ellips5.gif" hspace="45"></p>
<p>The objective function <b>-det P</b> (which is minimized) is not
convex, but monotonic transformations can render this problem convex. One
alternative is the logarithmic transform, leading to minimization of <b>-log
det P</b> instead. This operator was used in previous version of YALMIP,
but is not recommended any more. </p>
<p>Instead, YALMIP uses <b>-(det P)<sup>1/m</sup></b> where <b>m</b> is the
smallest power of 2 larger than or equal to the dimension of <b>P</b> (hence,
for matrices with dimension equal to a power of two, the function gives
the geometric mean of the eigenvalues of <b>P</b>). The function <b>(det
P)<sup>1/m</sup></b>, called <b>geomean2</b> in YALMIP, can be modeled using semidefinite and second order
cones, hence any SDP solver can be used for solving determinant
maximization problems. See <a href="readmore.htm#NESNEM94">[Nesterov and
Nemirovskii]</a> for details.</p>
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<td class="xmpcode"><font face="Courier New" color="#0000c0">n = 2;<br>
P1 = randn(2);P1 = P1*P1'; % Generate random ellipsoid<br>
P2 = randn(2);P2 = P2*P2'; % Generate random ellipsoid<br>
t = sdpvar(2,1);<br>
P = sdpvar(n,n);<br>
F = set(1 > t > 0);<br>
F = F + set(t(1)*P1-P > 0);<br>
F = F + set(t(2)*P2-P > 0);<br>
sol = solvesdp(F,-geomean2(P));<br>
ellipplot(double(P));hold on;<br>
ellipplot(double(P1));<br>
ellipplot(double(P2));</font></td>
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<p>If you have
the dedicated solver
<a href="solvers.htm#maxdet">MAXDET</a> installed and want to use it, you must use the dedicated command
<a href="reference.htm#logdet">logdet</a> for the objective and explicitly select
<a href="solvers.htm#maxdet">MAXDET</a>. This command can not be used
in any other construction than in the objective function, compared to the <b>geomean2</b> operator that can be used as any other variable in YALMIP, since it
a so called <a href="extoperators.htm">extended operator</a>. </p>
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<td class="xmpcode"><font face="Courier New" color="#0000c0">solvesdp(F,-logdet(P),sdpsettings('solver','maxdet'));<br>
ellipplot(double(P));hold on;<br>
ellipplot(double(P1));<br>
ellipplot(double(P2));</font></td>
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<p>
<img border="0" src="demoicon.gif" width="16" height="16"> Note that
if you use the
<a href="reference.htm#logdet">logdet</a> command, if
<a href="solvers.htm#maxdet">MAXDET</a> not is
explicitly selected, YALMIP will use <b>-(det P)<sup>1/m</sup></b> as objective
function instead.
This will not cause any problems if your objective function is a simple
<a href="reference.htm#logdet">logdet</a> expression (since the two functions
are monotonically related). However, if you have a mixed objective
function such as <b>tr(P)-logdet(P)</b>, the conversion will change your
optimal solution. Hence, if you really want to optimize the mixed
expression, you must explicitly select
<a href="solvers.htm#maxdet">MAXDET</a>. Otherwise, YALMIP will change
your objective to <b>tr(P)-(det P)<sup>1/m</sup></b></td>
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