[37] | 1 | <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> |
---|
| 2 | <html> |
---|
| 3 | |
---|
| 4 | <head> |
---|
| 5 | <meta http-equiv="Content-Language" content="en-us"> |
---|
| 6 | <title>YALMIP Example : Polynomial expressions</title> |
---|
| 7 | <meta http-equiv="Content-Type" content="text/html; charset=windows-1251"> |
---|
| 8 | <meta content="Microsoft FrontPage 6.0" name="GENERATOR"> |
---|
| 9 | <meta name="ProgId" content="FrontPage.Editor.Document"> |
---|
| 10 | <link href="yalmip.css" type="text/css" rel="stylesheet"> |
---|
| 11 | <base target="_self"> |
---|
| 12 | </head> |
---|
| 13 | |
---|
| 14 | <body leftMargin="0" topMargin="0"> |
---|
| 15 | |
---|
| 16 | <div align="left"> |
---|
| 17 | |
---|
| 18 | <table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%"> |
---|
| 19 | <tr> |
---|
| 20 | <td width="100%" align="left" height="100%" valign="top"> |
---|
| 21 | <h2>Polynomial expressions</h2> |
---|
| 22 | <hr noShade SIZE="1"> |
---|
| 23 | <p>Starting from YALMIP 3, polynomial expressions are supported. These |
---|
| 24 | nonlinear expressions can be used for, e.g., SDPs with BMI constraints, |
---|
| 25 | quadratic programming, or |
---|
| 26 | to solve sum-of-squares problems.</p> |
---|
| 27 | <p>Nonlinear expressions are built using |
---|
| 28 | <a href="reference.htm#sdpvar"> |
---|
| 29 | sdpvar</a> objects, and are manipulated in same way</p> |
---|
| 30 | <table cellPadding="10" width="100%"> |
---|
| 31 | <tr> |
---|
| 32 | <td class="xmpcode"> |
---|
| 33 | <pre>x = sdpvar(1,1); |
---|
| 34 | y = sdpvar(1,1); |
---|
| 35 | p = 1+x*y+x^2+y^3; |
---|
| 36 | Y = sdpvar(3,3); |
---|
| 37 | Z = Y*Y+Y.*Y;</pre> |
---|
| 38 | </td> |
---|
| 39 | </tr> |
---|
| 40 | </table> |
---|
| 41 | <p>A convenient command is |
---|
| 42 | <a href="reference.htm#sdisplay"> |
---|
| 43 | sdisplay</a> (symbolic display)</p> |
---|
| 44 | <table cellPadding="10" width="100%"> |
---|
| 45 | <tr> |
---|
| 46 | <td class="xmpcode"> |
---|
| 47 | <pre>sdisplay(p) |
---|
| 48 | <font color="#000000"> ans = |
---|
| 49 | '1+xy+x^2+y^3'</font></pre> |
---|
| 50 | </td> |
---|
| 51 | </tr> |
---|
| 52 | </table> |
---|
| 53 | <p>Some simple operators for polynomials have been implemented, such as |
---|
| 54 | differentiation.</p> |
---|
| 55 | <table cellPadding="10" width="100%"> |
---|
| 56 | <tr> |
---|
| 57 | <td class="xmpcode"> |
---|
| 58 | <pre>dp = jacobian(p); |
---|
| 59 | d2p = jacobian(jacobian(p)');</pre> |
---|
| 60 | </td> |
---|
| 61 | </tr> |
---|
| 62 | </table> |
---|
| 63 | <p>Checking the degree is easily done</p> |
---|
| 64 | <table cellPadding="10" width="100%"> |
---|
| 65 | <tr> |
---|
| 66 | <td class="xmpcode"> |
---|
| 67 | <pre>degree(p) |
---|
| 68 | <font color="#000000"> ans = |
---|
| 69 | 3 |
---|
| 70 | </font> |
---|
| 71 | degree(p,x) |
---|
| 72 | <font color="#000000"> ans = |
---|
| 73 | 2</font></pre> |
---|
| 74 | </td> |
---|
| 75 | </tr> |
---|
| 76 | </table> |
---|
| 77 | <p>Of course, all standard operators applies to the nonlinear objects.</p> |
---|
| 78 | <table cellPadding="10" width="100%"> |
---|
| 79 | <tr> |
---|
| 80 | <td class="xmpcode"> |
---|
| 81 | <pre>x = sdpvar(3,1); |
---|
| 82 | p = 5*trace(x*x') + jacobian(sum(x.^4))</pre> |
---|
| 83 | </td> |
---|
| 84 | </tr> |
---|
| 85 | </table> |
---|
| 86 | <p><img border="0" src="demoicon.gif" width="16" height="16"> Clear the internals of YALMIP on a regular basis with |
---|
| 87 | the command <code>yalmip('clear')</code> when working with polynomial |
---|
| 88 | expressions. The reason is that every time a nonlinear variable is defined, |
---|
| 89 | a description on how it is created is saved inside YALMIP (all monomials |
---|
| 90 | generate new variables). With many |
---|
| 91 | nonlinear terms this list grows fast, making YALMIP slower and slower since |
---|
| 92 | the list has to be searched in when polynomial expressions are manipulated.<br> |
---|
| 93 | <br> |
---|
| 94 | <img border="0" src="demoicon.gif" width="16" height="16"> The current implementation of the |
---|
| 95 | polynomial objects is inefficient for large problems. Multiplying two |
---|
| 96 | matrices of dimension, say 20, takes several seconds. But if you have a |
---|
| 97 | problem with this type of non-linearity, the solver will probably be the |
---|
| 98 | bottle-neck anyway... |
---|
| 99 | </td> |
---|
| 100 | </tr> |
---|
| 101 | </table> |
---|
| 102 | |
---|
| 103 | </div> |
---|
| 104 | |
---|
| 105 | </body> |
---|
| 106 | |
---|
| 107 | </html> |
---|