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6<title>YALMIP Example : Advanced programming</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>Advanced YALMIP programming</h2>
21      <hr noShade SIZE="1" color="#000000">
22      <p>In some cases, it might be necessary to do more advanced YALMIP coding.
23      YALMIP comes with a number of commands to facilitate this.</p>
24      <p>Let us begin by defining a set of variables.</p>
25      <table cellPadding="10" width="100%">
26        <tr>
27          <td class="xmpcode">
28          <pre>x = sdpvar(2,1);
29y = x'*randn(2)*x;</pre>
30          </td>
31        </tr>
32      </table>
33      <p>A common task is to find variables used in an expression. To do this, there are
34      two important commands, <code>depends.m </code>and <code>getvariables.m</code>.</p>
35      <table cellPadding="10" width="100%">
36        <tr>
37          <td class="xmpcode">
38          <pre>depends(x)
39 <font color="#000000">ans =
40  1 2</font>
41
42depends(y)
43<font color="#000000"> ans =
44  1 2</font>
45
46getvariables(x)
47<font color="#000000"> ans =
48  1 2</font>
49
50getvariables(y)
51<font color="#000000"> ans =
52  3 4 5</font></pre>
53          </td>
54        </tr>
55      </table>
56      <p>The command <code>depends.m</code> gives the actual linear independent variables, while <code>getvariables.m</code> returns the
57      variable indices to the so called relaxed variables used in YALMIP. When
58      a nonlinear expression is defined in YALMIP, a new variable is introduced
59      for every monomial term. For our quadratic variable y, we need 3 monomials.</p>
60      <table cellPadding="10" width="100%">
61        <tr>
62          <td class="xmpcode">
63          <pre>getvariables(x(1)^2)
64<font color="#000000"> ans =
65  3</font>
66
67getvariables(x(2)^2)
68 <font color="#000000">ans =
69  4</font></pre>
70          <pre>getvariables(x(1)*x(2))
71<font color="#000000"> ans =
72  5</font></pre>
73          </td>
74        </tr>
75      </table>
76      <p>The commands above only give the internal identifiers. To create an
77      <a href="reference.htm#sdpvar">sdpvar</a> variable using these variables,
78      we use the command <code>recover.m</code></p>
79      <table cellPadding="10" width="100%">
80        <tr>
81          <td class="xmpcode">
82          <pre>variables_in_y = recover(depends(y))
83<font color="#000000">Linear matrix variable 2x1 (full, real)</font></pre>
84          </td>
85        </tr>
86      </table>
87      <p>All <a href="reference.htm#sdpvar">sdpvar</a> 
88      objects are defined by the variable indicies obtained with <code>getvariables.m</code>,
89      and a basis. Let us begin with scalar variables to explain the concepts.
90      The basis with respect to all variables returned from <code>getvariables.m </code>
91      is obtained using <code>getbase.m</code></p>
92      <table cellPadding="10" width="100%">
93        <tr>
94          <td class="xmpcode">
95          <pre>x = sdpvar(1,1);
96y = sdpvar(1,1);
97z = pi+2*x+3*y;
98full(getbase(z))
99<font color="#000000">
100 ans =
101
102  3.1416 2.0000 3.0000</font></pre>
103          </td>
104        </tr>
105      </table>
106      <p>A basis with respect to a particular variable is obtained using <code>getbasematrix.m.</code></p>
107      <table cellPadding="10" width="100%">
108        <tr>
109          <td class="xmpcode">
110          <pre>full(getbasematrix(z,getvariables(x)))
111<font color="#000000">
112 ans =
113
114  2
115</font>
116full(getbasematrix(z,0))
117<font color="#000000">
118 ans =
119
120  3.1416</font>
121
122full(getbasematrix(z,123456789))
123<font color="#000000">
124 ans =
125
126  0</font></pre>
127          </td>
128        </tr>
129      </table>
130      <p>Matrix variables are defined in the same way, except that all basis
131      matrices are stored in a vectorized format.</p>
132      <table cellPadding="10" width="100%">
133        <tr>
134          <td class="xmpcode">
135          <pre>x = pi*eye(2)+5*sdpvar(2,2);
136base = full(getbase(x))
137<font color="#000000"> base =
138
139 3.1416 5.0000      0      0
140      0      0 5.0000      0
141      0      0 5.0000      0
142 3.1416      0      0 5.0000</font>
143
144reshape(base(:,1),2,2)
145<font color="#000000"> ans =
146
147  3.1416      0
148       0 3.1416</font></pre>
149          </td>
150        </tr>
151      </table>
152      <p>The command <a href="reference.htm#is">is</a> can be useful to select
153      parts of <a href="reference.htm#set">set</a> objects</p>
154      <table cellPadding="10" width="100%">
155        <tr>
156          <td class="xmpcode">
157          <pre>F = set(x&gt;0) + lmi(x*x' &gt; 0);
158F_linear = F(find(is(F,'linear')))</pre>
159          </td>
160        </tr>
161      </table>
162      <p><code>replace.m</code> can be useful in some situations to replace an
163      <a href="reference.htm#sdpvar">sdpvar</a> inside an
164      <a href="reference.htm#sdpvar">sdpvar</a> or <a href="reference.htm#set">
165      set</a>.</p>
166      <table cellPadding="10" width="100%">
167        <tr>
168          <td class="xmpcode">
169          <pre>F0 = replace(F,x(1),0);
170y0 = replace(y,x(1),0);</pre>
171          </td>
172        </tr>
173      </table>
174      <p>The terms in a <a href="reference.htm#set">set</a> object can be
175      extracted and converted to <a href="reference.htm#sdpvar">sdpvar</a> 
176      objects.</p>
177      <table cellPadding="10" width="100%">
178        <tr>
179          <td class="xmpcode">
180          <pre>xtimesx = sdpvar(F(2));</pre>
181          </td>
182        </tr>
183      </table>
184      <p>More to come but do not hesitate to ask.</td>
185    </tr>
186  </table>
187</div>
188
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190
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