📄 complex.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Language" content="en-us">
<title>YALMIP Example : Complex-valued problems</title>
<meta http-equiv="Content-Type" content="text/html; charset=windows-1251">
<meta content="Microsoft FrontPage 6.0" name="GENERATOR">
<meta name="ProgId" content="FrontPage.Editor.Document">
<link href="yalmip.css" type="text/css" rel="stylesheet">
<base target="_self">
</head>
<body leftMargin="0" topMargin="0">
<div align="left">
<table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
<tr>
<td width="100%" align="left" height="100%" valign="top">
<h2>Complex-valued problems</h2>
<hr noShade SIZE="1">
<p>YALMIP supports complex valued constraints for all solvers by
automatically converting complex-valued problems to real-valued problems.</p>
<p>To begin with, let us just define a simple linear complex problem to
illustrate how complex variables and constraints are generated and
interpreted.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>p = sdpvar(1,1,'full','complex'); % A complex scalar (4 arguments necessary)
s = sdpvar(1,1)+sqrt(-1)*sdpvar(1,1); % Alternative definition
F = set('0.9>imag(p)'); % Imaginary part constrained
F = F+set('0.01>real(p)'); % Real part constrained
F = F+set('0.1+0.5*sqrt(-1)>p'); % Both parts constrained
F = F+set('s+p==2+4*sqrt(-1)'); % Both parts constrained</pre>
</td>
</tr>
</table>
<p>To see how complex-valued constraints can be used in a more advanced
setting, we solve the covariance estimation problem from the <a href="solvers.htm#sedumi">SeDuMi</a>
manual. The problem is to find a positive-definite Hermitian Toeplitz matrix
<b>Z</b> such that the Frobenious norm of <b><font face="Tahoma">P-Z</font></b> is minimized (<b><font face="Tahoma">P</font></b> is a given complex
matrix.)</p>
<p>The matrix <b>P</b> is</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>P = [4 1+2*i 3-i;1-2*i 3.5 0.8+2.3*i;3+i 0.8-2.3*i 4];</pre>
</td>
</tr>
</table>
<p>We define a complex-valued Toeplitz matrix of the corresponding dimension</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>Z = sdpvar(3,3,'toeplitz','complex')</pre>
</td>
</tr>
</table>
<p>A complex Toeplitz matrix is not Hermitian, but we can make it Hermitian
if we remove the imaginary part on the diagonal.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>Z = Z-diag(imag(diag(Z)))*sqrt(-1);</pre>
</td>
</tr>
</table>
<p>Minimizing the Frobenious norm of <b><font face="Tahoma">P-Z</font></b> can be cast as minimizing the
Euclidean norm of the vectorized difference <b><font face="Tahoma">P(:)-Z(:)</font></b>. By using a Schur
complement, we see that this can be written as the following SDP.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>e = P(:)-Z(:)
t = sdpvar(1,1);
F = set(Z>0);
F = F+set([t e';e eye(9)]>0);
solvesdp(F,t);</pre>
</td>
</tr>
</table>
<p>The problem can be implemented more efficiently using a second order cone
constraint.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>e = Z(:)-P(:)
t = sdpvar(1,1);
F = set(Z>0);
F = F+set(cone(e,t));
solvesdp(F,t);</pre>
</td>
</tr>
</table>
<p>...or by using a quadratic objective function</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>e = Z(:)-P(:)
F = set(Z>0);
solvesdp(F,e'*e);</pre>
</td>
</tr>
</table>
</td>
</tr>
</table>
</div>
</body>
</html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -