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| 18 | <tr> |
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| 19 | <td width="100%" align="left" height="100%" valign="top"> |
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| 20 | <h2>Nonlinear operators</h2> |
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| 21 | <hr noshade size="1" color="#000000"> |
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| 22 | <p>YALMIP supports modeling of nonlinear, often non-differentiable, |
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| 23 | operators that typically occur in convex programming. Nine simple operators |
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| 24 | are currently supported: <b>min</b>, <b>max</b>, <b>abs</b>, <b>sqrt</b>, |
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| 25 | <b>norm</b>, <b>sumk</b>, <b>sumabsk</b>, <b>geomean</b> and <b>cpower</b>, |
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| 26 | and users can easily add their own (<a href="#operatorformat">see</a> the end of this page). The operators |
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| 27 | can be used intuitively, and YALMIP will automatically try to find out |
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| 28 | if they are used in a way that enables a convex representation. Although |
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| 29 | this can simplify the modeling phase significantly in some cases, it |
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| 30 | is recommended not to use these operators unless you know how to model |
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| 31 | them by your self using epigraphs and composition rules of convex and |
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| 32 | concave functions, why and when it can be done etc. The text-book |
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| 33 | <a href="readmore.htm#BOYDVAN2003">[S. Boyd and L. Vandenberghe]</a> |
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| 34 | should be a suitable introduction for the beginner. </p> |
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| 35 | <p>In addition to modeling convex and concave operators and perform |
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| 36 | automatic analysis and derivation of equivalent conic programs, YALMIP |
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| 37 | also uses the nonlinear operator framework for implementing |
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| 38 | <a href="logic.htm">logic expression</a> involving <b>or</b> and <b> |
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| 39 | and</b>, and in the same vein but on a higher level, to handle piecewise |
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| 40 | functions in <a href="reference.htm#pwf">pwf</a>.</p> |
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| 41 | <p>The nonlinear operator framework was initially implemented for |
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| 42 | functions that can be modelled rigorously using conic constraints |
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| 43 | and additional variables. However, there are many functions that |
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| 44 | cannot be exactly modelled using conic constraints, such as |
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| 45 | exponential functions and logarithms, but are convex or |
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| 46 | concave, and of course can be analyzed in terms of convexity |
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| 47 | preserving operations. These function are supported in a framework |
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| 48 | called evaluation based nonlinear operators. The models using these |
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| 49 | general convex functions will be analysed for convexity, but the |
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| 50 | resulting model will be a problem that only can be solved using |
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| 51 | a general nonlinear solver, such as <a href="solvers.htm#fmincon"> |
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| 52 | fmincon</a>. See <a href="#evaluationbased">evaluation based nonlinear operators</a>. Note that |
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| 53 | this extension still is experimental and not intended for large |
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| 54 | problems.</p> |
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| 55 | <h3><a name="propagation"></a>Convexity analysis in 10 lines</h3> |
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| 56 | <p>Without going into theoretical details, the convexity analysis is |
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| 57 | based on epi- and hypograph formulations, and composition rules. For |
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| 58 | the compound expression <b>f = h(g(x))</b>, it holds that (For |
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| 59 | simplicity, we write increasing, decreasing, convex and concave, but |
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| 60 | the correct notation would be nondecreasing, nonincreasing, convex |
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| 61 | or affine and concave or affine. This notation us used throughout |
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| 62 | this manual and inside YALMIP)</p> |
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| 63 | <div align="center"> |
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| 64 | <table border="1" bgcolor="#EEEEEE" bordercolor="#000000" id="table1"> |
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| 65 | <tr> |
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| 66 | <td bordercolorlight="#FFFFFF" bordercolordark="#FFFFFF"> |
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| 67 | <b>f</b> is <i>convex</i> if <b>h</b> is <i>convex</i> and |
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| 68 | <i>increasing</i> and <b>g</b> is <i>convex</i><br> |
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| 69 | <b>f</b> is <i>convex</i> if <b>h</b> is <i>convex</i> and |
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| 70 | <i>decreasing</i> and <b>g</b> is <i>concave</i> <br> |
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| 71 | <b>f</b> is <i>concave</i> if <b>h</b> is <i>concave</i> |
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| 72 | and <i>increasing</i> and <b>g</b> is <i>concave</i> |
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| 73 | <br> |
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| 74 | <b>f</b> is <i>concave</i> if <b>h</b> is <i>concave</i> |
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| 75 | and <i>decreasing</i> and <b>g</b> is <i>convex</i></td> |
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| 76 | </tr> |
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| 77 | </table> |
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| 78 | </div> |
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| 79 | <p>Based on this information, it is possible to recursively analyze |
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| 80 | convexity of a complex expression involving convex and concave functions. |
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| 81 | When <a href="reference.htm#solvesdp">solvesdp</a> is called, YALMIP |
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| 82 | checks the convexity of objective function and constraints by using |
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| 83 | information about the properties of the operators. If YALMIP manage |
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| 84 | to prove convexity, graph formulations of the operators are automatically |
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| 85 | introduced. This means that the operator is replaced with a graph, i.e., |
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| 86 | a set of constraints. </p> |
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| 87 | <div align="center"> |
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| 88 | <table border="1" bgcolor="#EEEEEE" bordercolor="#000000" id="table2"> |
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| 89 | <tr> |
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| 90 | <td bordercolorlight="#FFFFFF" bordercolordark="#FFFFFF"> |
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| 91 | <b>epigraph: t</b> represents convex function <b>f(x)</b> |
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| 92 | : replace with <b>t<font face="Tahoma">≥</font>f(x)</b><br> |
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| 93 | <b>hypograph</b>: <b>t</b> represents concave function <b> |
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| 94 | f(x)</b> : replace with <b>t<font face="Tahoma">≤</font>f(x)</b></td> |
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| 95 | </tr> |
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| 96 | </table> |
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| 97 | </div> |
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| 98 | <p align="left">Of course, in order for this to be useful, the epigraph |
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| 99 | representation has to be represented using standard constraints, such |
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| 100 | as conic constraints.</p> |
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| 101 | <h3 align="left">The operators</h3> |
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| 102 | <p align="left">The operators defined in the current release are |
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| 103 | described in the table below. This information might be useful to |
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| 104 | understand how and when YALMIP can derive convexity.</p> |
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| 105 | <div align="center"> |
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| 106 | <table border="1" width="86%" id="table7" bordercolor="#000000" style="border-collapse: collapse" bgcolor="#EEEEEE"> |
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| 107 | <tr> |
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| 108 | <td> |
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| 109 | <p align="center"><b>Name</b></td> |
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| 110 | <td> |
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| 111 | <p align="center"><b>Convex/Concave</b></td> |
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| 112 | <td> |
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| 113 | <p align="center"><b>Monotinicity</b></td> |
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| 114 | <td width="468"> |
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| 115 | <p align="center"><b>Comments</b></td> |
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| 116 | </tr> |
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| 117 | <tr> |
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| 118 | <td align="center">abs</td> |
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| 119 | <td bgcolor="#FFFFFF" align="center">convex</td> |
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| 120 | <td bgcolor="#FFFFFF" align="center">none</td> |
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| 121 | <td width="468" bgcolor="#FFFFFF" height="31"> </td> |
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| 122 | </tr> |
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| 123 | <tr> |
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| 124 | <td align="center">min</td> |
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| 125 | <td bgcolor="#FFFFFF" align="center">concave</td> |
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| 126 | <td bgcolor="#FFFFFF" align="center">increasing</td> |
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| 127 | <td width="468" bgcolor="#FFFFFF"> </td> |
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| 128 | </tr> |
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| 129 | <tr> |
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| 130 | <td align="center">max</td> |
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| 131 | <td bgcolor="#FFFFFF" align="center">convex</td> |
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| 132 | <td bgcolor="#FFFFFF" align="center">increasing</td> |
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| 133 | <td width="468" bgcolor="#FFFFFF"> </td> |
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| 134 | </tr> |
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| 135 | <tr> |
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| 136 | <td align="center">norm</td> |
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| 137 | <td bgcolor="#FFFFFF" align="center">convex</td> |
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| 138 | <td bgcolor="#FFFFFF" align="center">none</td> |
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| 139 | <td width="468" bgcolor="#FFFFFF">All standard norms (1,2, inf and Frobenius) can be used |
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| 140 | (applicable on both vectors and matrices.)</td> |
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| 141 | </tr> |
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| 142 | <tr> |
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| 143 | <td align="center">sumk</td> |
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| 144 | <td bgcolor="#FFFFFF" align="center">convex</td> |
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| 145 | <td bgcolor="#FFFFFF" align="center">See comment</td> |
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| 146 | <td width="468" bgcolor="#FFFFFF">Defines the sum of the |
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| 147 | k largest elements of a vector, or the sum of the k |
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| 148 | largest eigenvalues of a symmetric matrix. Increasing |
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| 149 | for vector arguments, no monotinicity defined for |
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| 150 | eigenvalue operator.</td> |
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| 151 | </tr> |
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| 152 | <tr> |
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| 153 | <td align="center">sumabsk</td> |
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| 154 | <td bgcolor="#FFFFFF" align="center">convex</td> |
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| 155 | <td bgcolor="#FFFFFF" align="center">none</td> |
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| 156 | <td width="468" bgcolor="#FFFFFF">Defines the sum of the |
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| 157 | k largest absolute value elements of a vector, or the |
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| 158 | sum of the k largest absolute value eigenvalues of a |
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| 159 | symmetric matrix. </td> |
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| 160 | </tr> |
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| 161 | <tr> |
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| 162 | <td align="center">geomean</td> |
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| 163 | <td bgcolor="#FFFFFF" align="center">concave</td> |
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| 164 | <td bgcolor="#FFFFFF" align="center">See comment</td> |
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| 165 | <td width="468" bgcolor="#FFFFFF">For vector arguments, the operator is increasing. For |
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| 166 | symmetric matrix argument, the operator is defined as the |
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| 167 | geometric mean of the eigenvalues. No monotinicity |
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| 168 | defined for eigenvalue operator</td> |
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| 169 | </tr> |
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| 170 | <tr> |
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| 171 | <td align="center">cpower</td> |
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| 172 | <td bgcolor="#FFFFFF" align="center">See comment</td> |
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| 173 | <td bgcolor="#FFFFFF" align="center">See comment</td> |
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| 174 | <td width="468" bgcolor="#FFFFFF">Convexity-aware |
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| 175 | version of power. For negative powers, the operator is convex and decreasing. |
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| 176 | For positive powers less than one, the operator is concave |
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| 177 | and increasing. Positive powers larger than 1 gives a convex |
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| 178 | increasing operator.</td> |
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| 179 | </tr> |
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| 180 | <tr> |
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| 181 | <td align="center">sqrt</td> |
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| 182 | <td bgcolor="#FFFFFF" align="center">concave</td> |
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| 183 | <td bgcolor="#FFFFFF" align="center">increasing</td> |
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| 184 | <td width="468" bgcolor="#FFFFFF">Short for |
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| 185 | cpower(x,0.5)</td> |
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| 186 | </tr> |
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| 187 | </table> |
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| 188 | </div> |
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| 189 | <h3>Standard use</h3> |
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| 190 | <p>Consider once again the <a href="linearregression.htm">linear regression |
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| 191 | problem</a>.</p> |
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| 192 | <table cellpadding="10" width="100%"> |
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| 193 | <tr> |
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| 194 | <td class="xmpcode"> |
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| 195 | <pre>a = [1 2 3 4 5 6]'; |
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| 196 | t = (0:0.2:2*pi)'; |
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| 197 | x = [sin(t) sin(2*t) sin(3*t) sin(4*t) sin(5*t) sin(6*t)]; |
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| 198 | y = x*a+(-4+8*rand(length(x),1)); |
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| 199 | a_hat = sdpvar(6,1); |
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| 200 | residuals = y-x*a_hat;</pre> |
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| 201 | </td> |
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| 202 | </tr> |
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| 203 | </table> |
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| 204 | <p>Using <b>abs</b> and <b>max</b>, we can easily solve the L<sub>1</sub> |
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| 205 | and the L<sub>∞</sub> problem (Note that the <b>abs</b> operator currently |
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| 206 | has performance issues and should be avoided for large arguments. Moreover, explicitly creating |
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| 207 | absolute values when minimizing the L<sub>∞</sub> error is |
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| 208 | unnecessarily complicated). |
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| 209 | </p> |
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| 210 | <table cellpadding="10" width="100%"> |
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| 211 | <tr> |
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| 212 | <td class="xmpcode"> |
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| 213 | <pre>solvesdp([],sum(abs(residuals))); |
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| 214 | a_L1 = double(a_hat) |
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| 215 | solvesdp([],max(abs(residuals))); |
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| 216 | a_Linf = double(a_hat)</pre> |
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| 217 | </td> |
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| 218 | </tr> |
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| 219 | </table> |
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| 220 | <p>YALMIP automatically concludes that the objective functions can be |
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| 221 | modeled using some additional linear inequalities, adds these, and solves |
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| 222 | the problems. We can simplify the code even more by using the <b>norm</b> |
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| 223 | operator (this is much faster for large-scale problems due to implementation |
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| 224 | issues in YALMIP). Here we also compute the least-squares solution (note |
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| 225 | that this norm will generate a second-order cone constraint).</p> |
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| 226 | <table cellpadding="10" width="100%"> |
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| 227 | <tr> |
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| 228 | <td class="xmpcode"> |
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| 229 | <pre>solvesdp([],norm(residuals,1)); |
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| 230 | a_L1 = double(a_hat) |
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| 231 | solvesdp([],norm(residuals,2)); |
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| 232 | a_L2 = double(a_hat) |
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| 233 | solvesdp([],norm(residuals,inf)); |
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| 234 | a_Linf = double(a_hat)</pre> |
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| 235 | </td> |
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| 236 | </tr> |
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| 237 | </table> |
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| 238 | <p>The following piece of code shows how we easily can solve a regularized |
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| 239 | problem.</p> |
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| 240 | <table cellpadding="10" width="100%"> |
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| 241 | <tr> |
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| 242 | <td class="xmpcode"> |
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| 243 | <pre>solvesdp([],1e-3*norm(a_hat,2)+norm(residuals,inf)); |
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| 244 | a_regLinf = double(a_hat)</pre> |
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| 245 | </td> |
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| 246 | </tr> |
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| 247 | </table> |
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| 248 | <p>The <b>norm</b> operator is used exactly as the built-in norm function |
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| 249 | in MATLAB, both for vectors and matrices. Hence it can be used also |
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| 250 | to minimize the largest singular value (2-norm in matrix case), or the |
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| 251 | Frobenious norm of a matrix.</p> |
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| 252 | <p>The <code>double</code> command of-course applies also to the nonlinear |
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| 253 | operators (<b>double</b>(<b>OPERATOR</b>(<b>X</b>)) returns <b>OPERATOR</b>(<b>double</b>(<b>X</b>)).</p> |
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| 254 | <table cellpadding="10" width="100%"> |
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| 255 | <tr> |
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| 256 | <td class="xmpcode"> |
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| 257 | <pre>double(1e-3*norm(a_hat,2)+norm(residuals,inf)) |
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| 258 | <font color="#000000">ans = |
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| 259 | 3.1175</font></pre> |
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| 260 | </td> |
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| 261 | </tr> |
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| 262 | </table> |
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| 263 | <p><a name="geomean2"></a>A construction useful for maximizing determinants |
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| 264 | of positive definite matrices is the function <b>(det P)<sup>1/m</sup></b>, |
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| 265 | for positive definite matrix P, where <b>m</b> is the dimension of |
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| 266 | <b>P</b>. This concave function, called <b>geomean</b> in YALMIP, is |
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| 267 | supported as an extended operator. Note that the positive semidefiniteness |
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| 268 | constraint on <b>P</b> is added automatically by YALMIP.</p> |
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| 269 | <table cellpadding="10" width="100%"> |
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| 270 | <tr> |
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| 271 | <td class="xmpcode"> |
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| 272 | <pre>D = randn(5,5); |
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| 273 | P = sdpvar(5,5); |
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| 274 | solvesdp(set(P < D*D'),-geomean(P));</pre> |
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| 275 | </td> |
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| 276 | </tr> |
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| 277 | </table> |
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| 278 | <p>The command can be applied also on positive vectors, and will then |
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| 279 | model the geometric mean of the elements. We can use this to find the analytic |
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| 280 | center of a set of linear inequalities (note that this is absolutely |
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| 281 | not the recommended way to compute the analytic center.)</p> |
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| 282 | <table cellpadding="10" width="100%"> |
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| 283 | <tr> |
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| 284 | <td class="xmpcode"> |
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| 285 | <pre>A = randn(15,2); |
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| 286 | b = rand(15,1)*5;</pre> |
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| 287 | <pre>x = sdpvar(2,1); |
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| 288 | solvesdp([],-geomean(b-A*x)); % Maximize product of elements in b-Ax, s.t Ax < b</pre> |
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| 289 | </td> |
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| 290 | </tr> |
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| 291 | </table> |
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| 292 | <p>Rather advanced constructions are possible, and YALMIP will try derive |
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| 293 | an equivalent convex model.</p> |
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| 294 | <table cellpadding="10" width="100%" id="table3"> |
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| 295 | <tr> |
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| 296 | <td class="xmpcode"> |
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| 297 | <pre>sdpvar x y z |
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| 298 | F = set(max(1,x)+max(y^2,z) < 3)+set(max(1,-min(x,y)) < 5)+set(norm([x;y],2) < z); |
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| 299 | sol = solvesdp(F,max(x,z)-min(y,z)-z);</pre> |
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| 300 | </td> |
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| 301 | </tr> |
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| 302 | </table> |
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| 303 | <h3><a name="polynomials"></a>Polynomial and sigmonial expressions</h3> |
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| 304 | <p>By default, polynomial expressions (except quadratics) are not analyzed |
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| 305 | with respect to convexity and conversion to a conic model is not performed. |
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| 306 | Hence, if you add a constraint such as <code>set(x^4 + y^8-x^0.5 < 10)</code>, |
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| 307 | YALMIP may complain about convexity, even though we can see that the |
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| 308 | expression is convex and can be represented using conic constraints. |
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| 309 | More importantly, YALMIP will not try to derive an equivalent conic |
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| 310 | model. However, by using the command <b>cpower</b> instead, rational |
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| 311 | powers can be used. </p> |
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| 312 | <p>To illustrate this, first note the difference between a monomial |
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| 313 | generated using overloaded power and a variable generated <b>cpower</b>.</p> |
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| 314 | <table cellpadding="10" width="100%" id="table4"> |
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| 315 | <tr> |
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| 316 | <td class="xmpcode"> |
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| 317 | <pre>sdpvar x |
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| 318 | x^4 |
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| 319 | <font color="#000000">Polynomial scalar (real, homogeneous, 1 variable)</font> |
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| 320 | cpower(x,4) |
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| 321 | <font color="#000000">Linear scalar (real, derived, 1 variable)</font></pre> |
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| 322 | </td> |
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| 323 | </tr> |
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| 324 | </table> |
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| 325 | <p>The property <i>derived</i> indicates that YALMIP will try to replace the |
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| 326 | variable with its epigraph formulation when the problem is solved. Working |
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| 327 | with these convexity-aware monomials is no different than usual.</p> |
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| 328 | <table cellpadding="10" width="100%" id="table5"> |
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| 329 | <tr> |
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| 330 | <td class="xmpcode"> |
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| 331 | <pre>sdpvar x y |
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| 332 | F = set(cpower(x,4) + cpower(y,4) < 10) + set(cpower(x,2/3) + cpower(y,2/3) > 1); |
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| 333 | plot(F,[x y]);</pre> |
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| 334 | </td> |
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| 335 | </tr> |
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| 336 | </table> |
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| 337 | <p>Note that when you plot sets with constraints involving nonlinear |
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| 338 | operators and polynomials, it is recommended that you specify the variables |
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| 339 | of interest in the second argument (YALMIP may otherwise plot the set |
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| 340 | with respect to auxiliary variables introduced during the construction |
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| 341 | of the conic model.)</p> |
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| 342 | <p>Do not use these operators unless you really need them. The conic |
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| 343 | representation of rational powers easily grow large.</p> |
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| 344 | <h3>Limitations</h3> |
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| 345 | <p>If the convexity propagation fails, an error will be issued (error |
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| 346 | code 14). Note that this does not imply that the model is nonconvex, |
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| 347 | but only means that the simple sufficient conditions used for checking |
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| 348 | convexity were violated. Failure is however typically an indication |
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| 349 | of a bad model, and most often due to an actual nonconvex part in the |
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| 350 | model. The problems above are all convex, but not this problem below, |
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| 351 | due to the way <b>min</b> enters in the constraint.</p> |
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| 352 | <table cellpadding="10" width="100%"> |
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| 353 | <tr> |
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| 354 | <td class="xmpcode"> |
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| 355 | <pre>sdpvar x y z |
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| 356 | F = set(max(1,x)+max(y^2,z) < 3)+set(max(1,<font color="#FF0000">min</font>(x,y)) < 5)+set(norm([x;y],2) < z); |
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| 357 | sol = solvesdp(F,max(x,z)-min(y,z)-z); |
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| 358 | sol.info |
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| 359 | |
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| 360 | <font color="#000000">ans = |
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| 361 | Convexity check failed (Expected convex function in constraint #2 at level 2)</font></pre> |
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| 362 | </td> |
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| 363 | </tr> |
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| 364 | </table> |
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| 365 | <p>In the same sense, this problem fails due to a nonconvex objective |
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| 366 | function.</p> |
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| 367 | <table cellpadding="10" width="100%"> |
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| 368 | <tr> |
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| 369 | <td class="xmpcode"> |
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| 370 | <pre>sdpvar x y z |
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| 371 | F = set(max(1,x)+max(y^2,z) < 3); |
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| 372 | sol = solvesdp(F,-norm([x;y])); |
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| 373 | sol.info |
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| 374 | <font color="#000000"> |
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| 375 | ans = |
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| 376 | Convexity check failed (Expected concave function in objective at level 1)</font></pre> |
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| 377 | </td> |
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| 378 | </tr> |
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| 379 | </table> |
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| 380 | <p>This following problem is however convex, but convexity propagation |
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| 381 | fails.</p> |
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| 382 | <table cellpadding="10" width="100%" id="table6"> |
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| 383 | <tr> |
---|
| 384 | <td class="xmpcode"> |
---|
| 385 | <pre>sdpvar x |
---|
| 386 | sol = solvesdp([],norm(max([1 1-x 1+x]))) |
---|
| 387 | sol.info |
---|
| 388 | <font color="#000000"> |
---|
| 389 | ans = |
---|
| 390 | Convexity check failed (Monotonicity required at objective at level 1)</font></pre> |
---|
| 391 | </td> |
---|
| 392 | </tr> |
---|
| 393 | </table> |
---|
| 394 | <p>The described operators cannot be used in polynomial expressions |
---|
| 395 | in the current implementation. The following problem is trivially convex |
---|
| 396 | but fails.</p> |
---|
| 397 | <table cellpadding="10" width="100%" id="table8"> |
---|
| 398 | <tr> |
---|
| 399 | <td class="xmpcode"> |
---|
| 400 | <pre>sdpvar x y |
---|
| 401 | sol = solvesdp([],norm([x;y])^2); |
---|
| 402 | sol.info |
---|
| 403 | |
---|
| 404 | <font color="#000000">ans = |
---|
| 405 | Convexity check failed (Operator in polynomial in objective)</font></pre> |
---|
| 406 | </td> |
---|
| 407 | </tr> |
---|
| 408 | </table> |
---|
| 409 | <p>Another limitation is that the operators not are allowed in cone |
---|
| 410 | and semidefinite constraints.</p> |
---|
| 411 | <table cellpadding="10" width="100%" id="table10"> |
---|
| 412 | <tr> |
---|
| 413 | <td class="xmpcode"> |
---|
| 414 | <pre>sdpvar x y |
---|
| 415 | sol = solvesdp(set(cone(max(x,y,1),2)),x+y); |
---|
| 416 | sol.info |
---|
| 417 | |
---|
| 418 | <font color="#000000">ans = |
---|
| 419 | Convexity propagation failed (YALMIP)</font></pre> |
---|
| 420 | </td> |
---|
| 421 | </tr> |
---|
| 422 | </table> |
---|
| 423 | <p>In practice, these limitations should not pose a major problem. A |
---|
| 424 | better model is possible (and probably recommended) in most cases if |
---|
| 425 | these situations occur. </p> |
---|
| 426 | <h3><a name="milp"></a>Mixed integer models</h3> |
---|
| 427 | <p>In some cases when the convexity analysis fails, it is possible |
---|
| 428 | to tell YALMIP to switch from a graph based approach to a mixed |
---|
| 429 | integer model based approach. In other words, if no graph model can |
---|
| 430 | be derived, YALMIP introduces integer variables to model the |
---|
| 431 | operators. This is currently implemented for <b>min</b>, <b> |
---|
| 432 | max</b>, <b>abs</b> and linear <b>norm</b> for real arguments. By default, this feature |
---|
| 433 | is not invoked, but can be activated by <code>sdpsettings('allowmilp',1)</code>.</p> |
---|
| 434 | <p>Consider the following simple example which fails due to the |
---|
| 435 | non-convex use of the convex <b>abs</b> operator</p> |
---|
| 436 | <table cellpadding="10" width="100%" id="table12"> |
---|
| 437 | <tr> |
---|
| 438 | <td class="xmpcode"> |
---|
| 439 | <pre>sdpvar x y |
---|
| 440 | F = set(abs(abs(x+1)+3) > y)+set(0<x<3); |
---|
| 441 | sol = solvesdp(F,-y); |
---|
| 442 | sol.info |
---|
| 443 | <font color="#000000"> Convexity check failed (Expected concave function in constraint #1 at level 1)</font></pre> |
---|
| 444 | </td> |
---|
| 445 | </tr> |
---|
| 446 | </table> |
---|
| 447 | <p>By turning on the mixed integer fall back model, a mixed integer |
---|
| 448 | LP is generated and the problem is easily solved.</p> |
---|
| 449 | <table cellpadding="10" width="100%" id="table13"> |
---|
| 450 | <tr> |
---|
| 451 | <td class="xmpcode"> |
---|
| 452 | <pre>sdpvar x y |
---|
| 453 | F = set(abs(abs(x+1)+3) > y)+set(0<x<3); |
---|
| 454 | sol = solvesdp(F,-y,sdpsettings('allowmilp',1)); |
---|
| 455 | double([x y]) |
---|
| 456 | <font color="#000000">ans = |
---|
| 457 | 3.0000 7.0000</font></pre> |
---|
| 458 | </td> |
---|
| 459 | </tr> |
---|
| 460 | </table> |
---|
| 461 | <p>If you know that your model is non-convex and will require a |
---|
| 462 | mixed integer model, you can bypass the initial attempt to generate |
---|
| 463 | the graph model by using <code>sdpsettings('allowmilp',2)</code>.</p> |
---|
| 464 | <h3><a name="evaluationbased"></a>Evaluation based nonlinear operators</h3> |
---|
| 465 | <p>YALMIP now also supports experimental support for general |
---|
| 466 | convex/concave functions that cannot be modelled using conic |
---|
| 467 | representations. The main difference when working with these |
---|
| 468 | operators is that the problem always requires a general nonlinear |
---|
| 469 | solver to solved, such as<a href="solvers.htm#fmincon"> fmincon</a>. |
---|
| 470 | All convexity analysis is still performed tough.</p> |
---|
| 471 | <table cellpadding="10" width="100%" id="table14"> |
---|
| 472 | <tr> |
---|
| 473 | <td class="xmpcode"> |
---|
| 474 | <pre>sdpvar x |
---|
| 475 | solvesdp(set(exp(2*x + 1) < 1),-x,sdpsettings('solver','fmincon')); |
---|
| 476 | double(x) |
---|
| 477 | <font color="#000000">ans = |
---|
| 478 | -0.5000</font></pre> |
---|
| 479 | <pre>double(exp(2*x + 1)) |
---|
| 480 | <font color="#000000">ans = |
---|
| 481 | 1</font></pre> |
---|
| 482 | </td> |
---|
| 483 | </tr> |
---|
| 484 | </table> |
---|
| 485 | <p>Note that this feature still is very limited and experimental. |
---|
| 486 | Too see how you can add our own function, see the |
---|
| 487 | <a href="#entropy">example for scalar entropy</a>. </p> |
---|
| 488 | <p>As a word of caution, note that<a href="solvers.htm#fmincon"> fmincon</a> |
---|
| 489 | performs pretty bad on problems with functions that aren't defined |
---|
| 490 | everywhere, such as logarithms. Hence, solving problem involving |
---|
| 491 | these functions can easily lead to problems. It is highly |
---|
| 492 | recommended to at least provide a feasible solution, or even better, |
---|
| 493 | to use the inverse operator to formulate the problem. Consider the |
---|
| 494 | following trivial example of finding the analytic center of a unit |
---|
| 495 | cube centered at the point (3,3,3)</p> |
---|
| 496 | <table cellpadding="10" width="100%" id="table16"> |
---|
| 497 | <tr> |
---|
| 498 | <td class="xmpcode"> |
---|
| 499 | <pre>x = sdpvar(3,1); |
---|
| 500 | p = [1-(x-3);(x-3)+1] |
---|
| 501 | |
---|
| 502 | % Not recommended |
---|
| 503 | solvesdp([],-sum(log(p))); |
---|
| 504 | |
---|
| 505 | % Better |
---|
| 506 | assign(x,[3.1;3.2;3.3]); |
---|
| 507 | solvesdp([],-sum(log(p)),sdpsettings('usex0',1)); |
---|
| 508 | |
---|
| 509 | % Best (well, adding initials on x and t would be even better) |
---|
| 510 | t = sdpvar(3,1); |
---|
| 511 | solvesdp(exp(t) < p ,-sum(t));</pre> |
---|
| 512 | <pre> |
---|
| 513 | </pre> |
---|
| 514 | </td> |
---|
| 515 | </tr> |
---|
| 516 | </table> |
---|
| 517 | <h3>Behind the scene</h3> |
---|
| 518 | <p>If you want to look at the model that YALMIP generates, you can use |
---|
| 519 | the two commands <code>model</code> and <code>expandmodel</code>. Please |
---|
| 520 | note that these expanded models never should be used manually. The commands |
---|
| 521 | described below should only be used for illustrating the process that |
---|
| 522 | goes on behind the scenes.</p> |
---|
| 523 | <p>With the command <code>model</code>, the epi- or hypograph model |
---|
| 524 | of the variable is returned. As an example, to model the maximum of |
---|
| 525 | two scalars <b>x</b> and <b>y</b>, YALMIP generates two linear inequalities.</p> |
---|
| 526 | <table cellpadding="10" width="100%"> |
---|
| 527 | <tr> |
---|
| 528 | <td class="xmpcode"> |
---|
| 529 | <pre>sdpvar x y |
---|
| 530 | t = max([x y]); |
---|
| 531 | F = model(t) |
---|
| 532 | <font color="#000000">++++++++++++++++++++++++++++++++++++++++++++ |
---|
| 533 | | ID| Constraint| Type| |
---|
| 534 | ++++++++++++++++++++++++++++++++++++++++++++ |
---|
| 535 | | #1| Numeric value| Element-wise 1x2| |
---|
| 536 | ++++++++++++++++++++++++++++++++++++++++++++</font> |
---|
| 537 | sdisplay(sdpvar(F(1))) |
---|
| 538 | <font color="#000000">ans = |
---|
| 539 | '-x+t' '-y+t'</font></pre> |
---|
| 540 | </td> |
---|
| 541 | </tr> |
---|
| 542 | </table> |
---|
| 543 | <p>For more advanced models with recursively used nonlinear operators, |
---|
| 544 | the function <code>model</code> will not generate the complete model |
---|
| 545 | since it does not recursively expand the arguments. For this case, use |
---|
| 546 | the command <code>expandmodel</code>. This command takes two arguments, |
---|
| 547 | a set of constraints and an objective function. To expand an expression, |
---|
| 548 | just let the expression take the position as the objective function. |
---|
| 549 | Note that the command assumes that the expansion is performed in order |
---|
| 550 | to prove a convex function, hence if you expression is meant to be concave, |
---|
| 551 | you need to negate it. To illustrate this, let us expand the objective |
---|
| 552 | function in an extension of the geometric mean example above.</p> |
---|
| 553 | <table cellpadding="10" width="100%"> |
---|
| 554 | <tr> |
---|
| 555 | <td class="xmpcode"> |
---|
| 556 | <pre>A = randn(15,2); |
---|
| 557 | b = rand(15,1)*5;</pre> |
---|
| 558 | <pre>x = sdpvar(2,1); |
---|
| 559 | expandmodel([],-geomean([b-A*x;min(x)])) |
---|
| 560 | <font color="#000000">+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
---|
| 561 | | ID| Constraint| Type| |
---|
| 562 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
---|
| 563 | | #1| Numeric value| Element-wise 2x1| |
---|
| 564 | | #2| Numeric value| Second order cone constraint 3x1| |
---|
| 565 | | #3| Numeric value| Second order cone constraint 3x1| |
---|
| 566 | | #4| Numeric value| Second order cone constraint 3x1| |
---|
| 567 | | #5| Numeric value| Second order cone constraint 3x1| |
---|
| 568 | | #6| Numeric value| Second order cone constraint 3x1| |
---|
| 569 | | #7| Numeric value| Second order cone constraint 3x1| |
---|
| 570 | | #8| Numeric value| Second order cone constraint 3x1| |
---|
| 571 | | #9| Numeric value| Second order cone constraint 3x1| |
---|
| 572 | | #10| Numeric value| Second order cone constraint 3x1| |
---|
| 573 | | #11| Numeric value| Second order cone constraint 3x1| |
---|
| 574 | | #12| Numeric value| Second order cone constraint 3x1| |
---|
| 575 | | #13| Numeric value| Second order cone constraint 3x1| |
---|
| 576 | | #14| Numeric value| Second order cone constraint 3x1| |
---|
| 577 | | #15| Numeric value| Second order cone constraint 3x1| |
---|
| 578 | | #16| Numeric value| Second order cone constraint 3x1| |
---|
| 579 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre> |
---|
| 580 | </td> |
---|
| 581 | </tr> |
---|
| 582 | </table> |
---|
| 583 | <p>The result is two linear inequalities related to the <b>min</b> operator |
---|
| 584 | and 15 second order cone constraints used for the conic representation |
---|
| 585 | of the geometric mean.</p> |
---|
| 586 | <h3><a name="operatorformat"></a>Adding new operators</h3> |
---|
| 587 | <p>If you want to add your own operator, all you need to do is to create |
---|
| 588 | 1 file. This file should be able to return the numerical value of the |
---|
| 589 | operator for a numerical input, and return the epigraph (or hypograph) |
---|
| 590 | and a descriptive structure of the operator when the first input is |
---|
| 591 | <code>'graph'</code>. As an example, the following file implements the |
---|
| 592 | nonlinear operator <b>tracenorm</b>. This convex operator returns <b> |
---|
| 593 | sum(svd(X))</b> for matrices <b>X</b>. This value can also be described |
---|
| 594 | as the minimizing argument of the optimization problem <b>min<sub>t,A,B</sub> |
---|
| 595 | t</b> subject to <b>set([A X;X' B] > 0) + set(trace(A)+trace(B) < 2*t)</b>.</p> |
---|
| 596 | <table cellpadding="10" width="100%"> |
---|
| 597 | <tr> |
---|
| 598 | <td class="xmpcode"> |
---|
| 599 | <pre>function varargout = tracenorm(varargin) |
---|
| 600 | |
---|
| 601 | switch class(varargin{1}) |
---|
| 602 | |
---|
| 603 | case 'double' % What is the <font color="#FF0000">numerical value</font> (needed for displays etc) |
---|
| 604 | varargout{1} = sum(svd(varargin{1})); |
---|
| 605 | |
---|
| 606 | case 'char' % YALMIP send 'graph' when it wants the epigraph or hypograph |
---|
| 607 | switch varargin{1} |
---|
| 608 | case 'graph' |
---|
| 609 | t = varargin{2}; % 2nd arg is the extended operator variable |
---|
| 610 | X = varargin{3}; % 3rd arg and above are args user used when defining t. |
---|
| 611 | A = sdpvar(size(X,1)); |
---|
| 612 | B = sdpvar(size(X,2)); |
---|
| 613 | F = set([A X;X' B] > 0) + set(trace(A)+trace(B) < 2*t); |
---|
| 614 | |
---|
| 615 | % <font color="#FF0000">Return epigraph model |
---|
| 616 | </font>varargout{1} = F; |
---|
| 617 | % <font color="#FF0000">a description </font> |
---|
| 618 | properties.convexity = 'convex';<font color="#FF0000"> % convex | none | concave</font> |
---|
| 619 | properties.monotonicity = 'none';<font color="#FF0000"> % increasing | none | decreasing</font> |
---|
| 620 | properties.definiteness = 'positive';<font color="#FF0000"> % negative | none | positive</font> |
---|
| 621 | varargout{2} = properties; |
---|
| 622 | % <font color="#FF0000">and the argument |
---|
| 623 | </font> varargout{3} = X; |
---|
| 624 | |
---|
| 625 | case 'milp' |
---|
| 626 | varargout{1} = []; |
---|
| 627 | varargout{2} = []; |
---|
| 628 | varargout{3} = []; |
---|
| 629 | |
---|
| 630 | otherwise |
---|
| 631 | error('Something is very wrong now...') |
---|
| 632 | end |
---|
| 633 | |
---|
| 634 | case 'sdpvar' % Always the same. |
---|
| 635 | varargout{1} = yalmip('addextendedvariable',mfilename,varargin{:}); |
---|
| 636 | |
---|
| 637 | otherwise |
---|
| 638 | end</pre> |
---|
| 639 | </td> |
---|
| 640 | </tr> |
---|
| 641 | </table> |
---|
| 642 | <p>The function <code>sumk.m</code> in YALMIP is implemented using this |
---|
| 643 | framework and might serve as an additional fairly simple example. The |
---|
| 644 | overloaded operator <code>norm.m</code> is also defined using this method, |
---|
| 645 | but is a bit more involved, since it supports different norms. Note |
---|
| 646 | that we with a slight abuse of notation use the terms increasing and |
---|
| 647 | decreasing instead of nondecreasing and nonincreasing.</p> |
---|
| 648 | <h3><a name="operatorformat0"></a>Adding new operators with mixed |
---|
| 649 | integer models</h3> |
---|
| 650 | <p>If the convexity analysis fails, it is possible to have fall back |
---|
| 651 | alternative models based on integer variables. If the operator is |
---|
| 652 | called with the flag <code>milp</code>, a mixed integer exact model |
---|
| 653 | can be |
---|
| 654 | returned. As an illustration, here is a stripped down version of the |
---|
| 655 | epigraph and MILP model of the absolute value of a real scalar.</p> |
---|
| 656 | <table cellpadding="10" width="100%" id="table11"> |
---|
| 657 | <tr> |
---|
| 658 | <td class="xmpcode"> |
---|
| 659 | <pre>function varargout = scalarrealabs(varargin) |
---|
| 660 | |
---|
| 661 | switch class(varargin{1}) |
---|
| 662 | |
---|
| 663 | case 'double' |
---|
| 664 | varargout{1} = abs(varargin{1}); |
---|
| 665 | |
---|
| 666 | case 'char' % YALMIP send 'graph' when it wants the epigraph or hypograph |
---|
| 667 | switch varargin{1} |
---|
| 668 | case 'graph' |
---|
| 669 | t = varargin{2}; |
---|
| 670 | X = varargin{3}; |
---|
| 671 | <font color="#FF0000"> </font>varargout{1} = set(-t <= X <= t); |
---|
| 672 | properties.convexity = 'convex';<font color="#FF0000"> </font> |
---|
| 673 | properties.monotonicity = 'none';<font color="#FF0000"> |
---|
| 674 | </font> properties.definiteness = 'positive';<font color="#FF0000"> </font> |
---|
| 675 | varargout{2} = properties; |
---|
| 676 | varargout{3} = X; |
---|
| 677 | |
---|
| 678 | case <font color="#FF0000">'milp' |
---|
| 679 | </font> t = varargin{2}; |
---|
| 680 | X = varargin{3}; |
---|
| 681 | d = binvar(1,1); % d=1 means x>0, d=0 means x<0 |
---|
| 682 | F = set([]); |
---|
| 683 | M = 1e4; % Big-M constant |
---|
| 684 | F = F + set(x >= -M*(1-d)) % d=1 means x >= 0 |
---|
| 685 | F = F + set(x <= M*d) % d=0 means x <= 0 |
---|
| 686 | F = F + set(-M*(1-d) <= t-x <= M*(1-d); % d=1 means t = X |
---|
| 687 | F = F + set(-M*d <= t+x <= M*d; % d=0 means t = -X |
---|
| 688 | |
---|
| 689 | varargout{1} = F; |
---|
| 690 | properties.convexity = '<font color="#FF0000">milp</font>';<font color="#FF0000"> </font> |
---|
| 691 | properties.monotonicity = '<font color="#FF0000">milp</font>';<font color="#FF0000"> |
---|
| 692 | </font> properties.definiteness = '<font color="#FF0000">milp</font>';<font color="#FF0000"> </font> |
---|
| 693 | varargout{2} = properties; |
---|
| 694 | varargout{3} = X; |
---|
| 695 | |
---|
| 696 | otherwise |
---|
| 697 | error('Something is very wrong now...') |
---|
| 698 | end |
---|
| 699 | |
---|
| 700 | case 'sdpvar' % Always the same. |
---|
| 701 | varargout{1} = yalmip('addextendedvariable',mfilename,varargin{:}); |
---|
| 702 | |
---|
| 703 | otherwise |
---|
| 704 | end</pre> |
---|
| 705 | </td> |
---|
| 706 | </tr> |
---|
| 707 | </table> |
---|
| 708 | <p>MILP models are most often based on so called big-M models. For |
---|
| 709 | these methods to work well, it is important to have as small |
---|
| 710 | constants M as possible, but in the code above, we just picked a |
---|
| 711 | number. For the MILP models defined by default in YALMIP (<b>min</b>, |
---|
| 712 | <b>max</b>, <b>abs</b> and linear <b>norms</b>), more effort is spent on |
---|
| 713 | choosing the |
---|
| 714 | constants. To learn more about how you can do this for your model, |
---|
| 715 | please check out the code for these models.</p> |
---|
| 716 | <h3><a name="entropy"></a>Adding evaluation based nonlinear |
---|
| 717 | operators</h3> |
---|
| 718 | <p>General convex and concave functions are support in YALMIP by the |
---|
| 719 | evaluation based nonlinear operator framework. The definition of |
---|
| 720 | these operators are almost identical to the definition of standard |
---|
| 721 | nonlinear operators. The following code implements a (simplified |
---|
| 722 | version) of a scalar entropy measure <b>-xlog(x)</b>.</p> |
---|
| 723 | <table cellpadding="10" width="100%" id="table15"> |
---|
| 724 | <tr> |
---|
| 725 | <td class="xmpcode"> |
---|
| 726 | <pre>function varargout = entropy(varargin) |
---|
| 727 | |
---|
| 728 | switch class(varargin{1})</pre> |
---|
| 729 | <pre> case 'double' % What is the numerical value of this argument (needed for displays etc) |
---|
| 730 | varargout{1} = <font color="#FF0000">-varargin{1}*log(varargin{1})</font>; |
---|
| 731 | |
---|
| 732 | case 'sdpvar' % Overloaded operator for SDPVAR objects. |
---|
| 733 | varargout{1} = yalmip('addEvalVariable',mfilename,varargin{1}); </pre> |
---|
| 734 | <pre> case 'char' % YALMIP sends 'graph' when it wants the epigraph, hypograph or domain definition |
---|
| 735 | switch varargin{1} |
---|
| 736 | case 'graph' |
---|
| 737 | t = varargin{2}; |
---|
| 738 | X = varargin{3}; </pre> |
---|
| 739 | <pre><font color="#FF0000"> </font>% This is different from standard extended operators. |
---|
| 740 | % Just do it!<font color="#FF0000"> |
---|
| 741 | </font>F = SetupEvaluationVariable(varargin{:});<font color="#FF0000"> |
---|
| 742 | </font> |
---|
| 743 | % Now add your own code, typically domain constraints |
---|
| 744 | <font color="#FF0000">F = F + set(X > 0);</font> |
---|
| 745 | |
---|
| 746 | % Let YALMIP know about convexity etc |
---|
| 747 | varargout{1} = F; |
---|
| 748 | varargout{2} = struct('convexity','concave','monotonicity','none','definiteness','none'); |
---|
| 749 | varargout{3} = X; </pre> |
---|
| 750 | <pre> case 'milp' % No MILP model available for entropy |
---|
| 751 | varargout{1} = []; |
---|
| 752 | varargout{2} = []; |
---|
| 753 | varargout{3} = []; |
---|
| 754 | otherwise |
---|
| 755 | error('ENTROPY called with CHAR argument?'); |
---|
| 756 | end |
---|
| 757 | otherwise |
---|
| 758 | error('ENTROPY called with invalid argument.'); |
---|
| 759 | end</pre> |
---|
| 760 | </td> |
---|
| 761 | </tr> |
---|
| 762 | </table> |
---|
| 763 | <p>The evaluation based framework is primarily intended for scalar |
---|
| 764 | functions, but can be extended to support element-wise vector functions. See the |
---|
| 765 | implementation of the overloaded log operator for details.</p> |
---|
| 766 | </td> |
---|
| 767 | </tr> |
---|
| 768 | </table> |
---|
| 769 | <p> </div> |
---|
| 770 | |
---|
| 771 | </body> |
---|
| 772 | |
---|
| 773 | </html> |
---|