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<title>YALMIP Example : Solving BMIs locally</title>
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<h2>Local solution of bilinear matrix inequalities (BMIs)</h2>
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<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
example requires the solver <a href="solvers.htm#penbmi">PENBMI</a></p>
<p>BMI problems are solved locally just as easy as standard SDP
problems, by using the solver <a href="solvers.htm#penbmi">PENBMI</a>. As an example, the <a href="gevp.htm">decay-rate estimation problem</a> that we
earlier solved using a bisection, is solved with the following code.</p>
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<pre>A = [-1 2;-3 -4];
t = sdpvar(1,1);
P = sdpvar(2,2);
F = set(P>eye(2))+set(A'*P+P*A < -2*t*P);
solvesdp(F,-t);</pre>
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<p> <a href="solvers.htm#penbmi">PENBMI</a> seem to perform better if
variables are constrained, so the following code is often recommended (see
the <a href="advanced.htm">advanced programming examples</a>).</p>
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<pre>A = [-1 2;-3 -4];
t = sdpvar(1,1);
P = sdpvar(2,2);
F = set(P>eye(2))+set(A'*P+P*A < -2*t*P);
F = F + set(-100 < recover(depends(F)) < 100);
solvesdp(F,-t);</pre>
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<p>As another example, the code below calculates an LQ optimal feedback.
(of-course, this can be solved much more efficiently by first recasting it
as a convex problem)</p>
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<pre>A = [-1 2;-3 -4];B = [1;1];
P = sdpvar(2,2);K = sdpvar(1,2);
F = set(P>0)+set((A+B*K)'*P+P*(A+B*K) < -eye(2)-K'*K);
solvesdp(F,trace(P));</pre>
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<p>More interesting, we can solve the LQ problem with constraints on the
feedback matrix.</p>
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<pre>A = [-1 2;-3 -4];B = [1;1];
P = sdpvar(2,2);K = sdpvar(1,2);
F = set(K<0.1)+set(K>-0.1)+set(P>0)+set((A+B*K)'*P+P*(A+B*K) < -eye(2)-K'*K);
solvesdp(F,trace(P));</pre>
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<p>Reasonable initial guesses is often crucial in non-convex optimization.
The easiest way to specify initial guesses in YALMIP is to use the field
<code>usex0</code> in
<a href="reference.htm#sdpsettings">
sdpsettings</a> and the command
<a href="reference.htm#assign">
assign</a>. Consider the constrained LQ example above, and let us specify
an initial guess using a standard LQ feedback. </p>
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<pre>A = [-1 2;-3 -4];B = [1;1];
[K0,P0] = lqr(A,B,eye(2),1);
P = sdpvar(2,2);assign(P,P0);
K = sdpvar(1,2);assign(K,-K0);
F = set(K<0.1)+set(K>-0.1)+set(P>0)+set((A+B*K)'*P+P*(A+B*K) < -eye(2)-K'*K);
solvesdp(F,trace(P),sdpsettings('usex0',1));</pre>
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<p><img border="0" src="demoicon.gif" width="16" height="16"><a href="solvers.htm#penbmi">PENBMI</a>
supports non-convex quadratic objective functions. Hence you can
(locally) solve non-convex quadratically constrained quadratic
programming using PENBMI.
<!-- <p><img border="0" src="demoicon.gif" width="16" height="16">You can
actually (try to...) solve the BMI above without PENBMI. The internal solver <code>'bmilin'</code> is one
alternative. Note though that this solver is slow and unstable. It is
based on a simple sequential linearization approach with a
trust-region, and is merely implemented to show how a simple
BMI-solver can be coded using a few lines of YALMIP code. For more
information, check out Example 14 in
<a href="reference.htm#yalmipdemo">yalmipdemo</a>.-->
<p><img border="0" src="demoicon.gif" width="16" height="16">YALMIP
will automatically convert higher order polynomial programs to
bilinear programs, hence YALMIP+<a href="solvers.htm#penbmi">PENBMI</a>
can be used to address general polynomial problems.
<!-- <p><img border="0" src="demoicon.gif" width="16" height="16">You can
actually (try to...) solve the BMI above without PENBMI. The internal solver <code>'bmilin'</code> is one
alternative. Note though that this solver is slow and unstable. It is
based on a simple sequential linearization approach with a
trust-region, and is merely implemented to show how a simple
BMI-solver can be coded using a few lines of YALMIP code. For more
information, check out Example 14 in
<a href="reference.htm#yalmipdemo">yalmipdemo</a>.-->
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