reference.htm

optimization toolbox

              <p>Giving a second argument controls what variables to differentiate 
              with respect to </p>
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                  <pre>sdisplay(hessian(f,x1))
<font color="#000000">ans = </font></pre>
                  <pre><font color="#000000">    &#39;12x1^2&#39;</font></pre>
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              Related commands</th>
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              <td class="tabxpl"><a href="reference.htm#jacobian">jacobian</a>,
              <a href="reference.htm#sdisplay">sdisplay</a>,
              <a href="reference.htm#sdpvar">sdpvar</a></td>
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      <p>&nbsp;</p>
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          <p class="tableheader"><a name="dual">DUAL</a></p>
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              Syntax</th>
              <td class="code" valign="top" nowrap width="100%"><code>Z = dual(F)</code></td>
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                  <p align="right"><font face="Courier New" size="2">Z:</font></p>
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                  <td>double</td>
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                  <p align="right"><font face="Courier New" size="2">F:</font></p>
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                  <td>set object (with one constraint)</td>
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              Description</th>
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              <td class="tabxpl">dual is used to extract the dual variable related 
              to a constraint</td>
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              Examples</th>
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              <td class="tabxpl"><p>After solving an optimization problem we might, 
              e.g., extract the dual variable of the 2nd constraint.</p><table cellpadding="10" width="100%">
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                  <pre>solvesdp(F,obj);
Z2 = dual(F(2));</pre>
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              <p>If the constraints in the set object have been tagged, we can use 
              the tag instead </p>
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                  <td class="xmpcode">
                  <pre>solvesdp(F,obj);
Z2 = dual(F(&#39;Lyapunov constraint&#39;));</pre>
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              Related commands</th>
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              <td class="tabxpl"><a href="reference.htm#set">set</a>,
              <a href="reference.htm#solvesdp">solvesdp</a>,
              <a href="reference.htm#sdpvar">sdpvar</a></td>
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      <p>&nbsp;</p>
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          <p class="tableheader"><a name="dualize">DUALIZE</a></p>
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              Syntax</th>
              <td class="code" valign="top" nowrap width="100%"><code>
              [Fd,objd,X,free] = dualize(F,obj)</code></td>
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                  <p align="right"><font face="Courier New" size="2">F:</font></p>
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                  <td>set object in primal SDP form</td>
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                  <p align="right"><font face="Courier New" size="2">obj:</font></p>
                  </td>
                  <td>sdpvar object (primal cost)</td>
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                  <p align="right"><font face="Courier New" size="2">Fd:</font></p>
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                  <td>set object in dual SDP form</td>
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                  <p align="right"><font face="Courier New">objd</font><font face="Courier New" size="2">:</font></p>
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                  <td>sdpvar object (dual cost)</td>
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                  <p align="right"><font face="Courier New" size="2">X:</font></p>
                  </td>
                  <td>cell of sdpvar object (primal cone variables)</td>
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                  <p align="right"><font face="Courier New" size="2">free:</font></p>
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                  <td>sdpvar objects (free primal variables)</td>
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              <th class="doc" valign="top" nowrap align="left" bgcolor="#eeeeee" rowspan="2">
              Description</th>
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              <td class="tabxpl">dualize is used to convert a SDP problem given 
              in primal form to the corresponding dual problem.</td>
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              Examples</th>
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              <td class="tabxpl">Consider the following SDP problem in primal 
              form.<table cellpadding="10" width="100%">
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                  <td class="xmpcode">
                  <pre>X = sdpvar(3,3);
Y = sdpvar(3,3);
F = set(X&gt;0) + set(Y&gt;0);
F = F + set(X(1,3)==9) + set(Y(1,1)==X(2,2));
F = F + set(sum(sum(X))+sum(sum(Y)) == 20);
obj = trace(X)+trace(Y);</pre>
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              <p>We can solve this in the given format. This will however be 
              very inefficient in YALMIP, since the matrices X and Y will be 
              explicitely parameterized, and the problem will be solved in a 
              dual form with equality constraints. Instead, we note that the 
              problem is an SDP in standard primal form. We therefor let YALMIP 
              extract this primal model, and return the dual of this. If we 
              solve this problem, the dual of this will be the original primal. 
              Confused yet? (note that the dual objective should be maximized)</p>
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                  <td class="xmpcode">
                  <pre>[Fd,objd,XX,y] = dualize(F,obj);
solvesdp(Fd,-objd);</pre>
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              <p>To obtain our original variables, we extract the duals</p>
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                  <td class="xmpcode">
                  <pre>assign(XX{1},dual(Fd(1));
assign(XX{2},dual(Fd(2));</pre>
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              <p>Check out the tutorial for more <a href="dual.htm#dualize">
				examples</a></td>
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              Related commands</th>
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              <td class="tabxpl">
              <a href="reference.htm#dual">dual</a>,
              <a href="reference.htm#set">set</a>,
              <a href="reference.htm#solvesdp">solvesdp</a>,
              <a href="reference.htm#sdpvar">sdpvar</a></td>
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      <p>&nbsp;</p>
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          <p class="tableheader"><a name="export">EXPORT</font></a></p>
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              Syntax<p>&nbsp;</p>
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              <td class="code" valign="top" nowrap width="100%"><code>[model,recoverymodel] 
              = export(F,h,ops)</code></td>
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                  <p align="right"><font face="Courier New" size="2">model:</font></p>
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                  <td>Output structure.</td>
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                  <td>
                  <font face="Courier New" size="2">recoverymodel:</font></td>
                  <td>Output structure.</td>
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                  <p align="right"><font face="Courier New" size="2">F:</font></p>
                  </td>
                  <td>set object describing the constraints. </td>
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                  <p align="right"><font face="Courier New" size="2">h:</font></p>
                  </td>
                  <td>sdpvar-object describing the objective function.</td>
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                  <td>
                  <p align="right"><font face="Courier New" size="2">ops:</font></p>
                  </td>
                  <td>options structure from sdpsettings.</td>
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              <th class="doc" valign="top" nowrap align="left" bgcolor="#eeeeee" rowspan="2">
              Description</th>
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              <td class="tabxpl">export is used to export YALMIP models to 
              various solver formats (<font color="#FF0000">note : not all 
              solvers are supported yet</font>)</td>
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              Examples</th>
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              <td class="tabxpl"><p>Consider a Lyapunov stability problem</p><table cellpadding="10" width="100%">
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                  <td class="xmpcode">
                  <pre>A = randn(5,5);A = -A*A';
P = sdpvar(5,5);
F = set(A'*P+P*A &lt; 0) + set(P&gt;eye(5));
obj = trace(P);</pre>
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              <p>Exporting this to a model in SeDuMi format is done by 
              specifying the solver as <code>'sedumi'</code> and calling export in the same way 
              as <a href="reference.htm#solvesdp">solvesdp</a> would have ben called.</p>
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                  <td class="xmpcode">
                  <pre>[model,recoverymodel] = export(F,obj,sdpsettings('solver','sedumi'));</pre>
                  <pre><font color="#000000">model = </font></pre>
                  <pre><font color="#000000">       A: [50x15 double]
       b: [15x1 double]
       C: [50x1 double]
       K: [1x1 struct]
    pars: [1x1 struct]</font></pre>
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              <p>The data in <code>recoverymodel</code> can be used to relate a solution 
              obtained from using the exported model, to the actual variables in 
              YALMIP. </p>
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                  <td class="xm