1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
|
<html><head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>lsqlin</title>
<style type="text/css" media="all">
@import url("scilab_code.css");
@import url("xml_code.css");
@import url("c_code.css");
@import url("style.css");
</style>
</head>
<body>
<div class="manualnavbar">
<table width="100%"><tr>
<td width="30%">
<span class="previous"><a href="linprog.html"><< linprog</a></span>
</td>
<td width="40%" class="center">
<span class="top"><a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a></span>
</td>
<td width="30%" class="next">
<span class="next"><a href="lsqnonlin.html">lsqnonlin >></a></span>
</td>
</tr></table>
<hr />
</div>
<span class="path"><a href="index.html">FOSSEE Optimization Toolbox</a> >> <a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a> > lsqlin</span>
<br /><br />
<div class="refnamediv"><h1 class="refname">lsqlin</h1>
<p class="refpurpose">Solves a linear quadratic problem.</p></div>
<div class="refsynopsisdiv"><h3 class="title">Calling Sequence</h3>
<div class="synopsis"><pre><span class="default">xopt</span><span class="default"> = </span><span class="functionid">lsqlin</span><span class="default">(</span><span class="default">C</span><span class="default">,</span><span class="default">d</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">lsqlin</span><span class="default">(</span><span class="default">C</span><span class="default">,</span><span class="default">d</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">lsqlin</span><span class="default">(</span><span class="default">C</span><span class="default">,</span><span class="default">d</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">lsqlin</span><span class="default">(</span><span class="default">C</span><span class="default">,</span><span class="default">d</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">x0</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">lsqlin</span><span class="default">(</span><span class="default">C</span><span class="default">,</span><span class="default">d</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">param</span><span class="default">)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">resnorm</span><span class="default">,</span><span class="default">residual</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">,</span><span class="default">lambda</span><span class="default">] = </span><span class="functionid">lsqlin</span><span class="default">( ... )</span></pre></div></div>
<div class="refsection"><h3 class="title">Parameters</h3>
<dl><dt><span class="term">C :</span>
<dd><p class="para">a matrix of double, represents the multiplier of the solution x in the expression C⋅x - d. Number of columns in C is equal to the number of elements in x.</p></dd></dt>
<dt><span class="term">d :</span>
<dd><p class="para">a vector of double, represents the additive constant term in the expression C⋅x - d. Number of elements in d is equal to the number of rows in C matrix.</p></dd></dt>
<dt><span class="term">A :</span>
<dd><p class="para">a matrix of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt>
<dt><span class="term">b :</span>
<dd><p class="para">a vector of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt>
<dt><span class="term">Aeq :</span>
<dd><p class="para">a matrix of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt>
<dt><span class="term">beq :</span>
<dd><p class="para">a vector of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt>
<dt><span class="term">lb :</span>
<dd><p class="para">a vector of double, contains lower bounds of the variables.</p></dd></dt>
<dt><span class="term">ub :</span>
<dd><p class="para">a vector of double, contains upper bounds of the variables.</p></dd></dt>
<dt><span class="term">x0 :</span>
<dd><p class="para">a vector of double, contains initial guess of variables.</p></dd></dt>
<dt><span class="term">param :</span>
<dd><p class="para">a list containing the parameters to be set.</p></dd></dt>
<dt><span class="term">xopt :</span>
<dd><p class="para">a vector of double, the computed solution of the optimization problem.</p></dd></dt>
<dt><span class="term">resnorm :</span>
<dd><p class="para">a double, objective value returned as the scalar value norm(C⋅x-d)^2.</p></dd></dt>
<dt><span class="term">residual :</span>
<dd><p class="para">a vector of double, solution residuals returned as the vector d-C⋅x.</p></dd></dt>
<dt><span class="term">exitflag :</span>
<dd><p class="para">The exit status. See below for details.</p></dd></dt>
<dt><span class="term">output :</span>
<dd><p class="para">The structure consist of statistics about the optimization. See below for details.</p></dd></dt>
<dt><span class="term">lambda :</span>
<dd><p class="para">The structure consist of the Lagrange multipliers at the solution of problem. See below for details.</p></dd></dt></dl></div>
<div class="refsection"><h3 class="title">Description</h3>
<p class="para">Search the minimum of a constrained linear least square problem specified by :</p>
<p class="para"><span><img src='./_LaTeX_lsqlin.xml_1.png' style='position:relative;top:41px;width:230px;height:90px'/></span></p>
<p class="para">The routine calls Ipopt for solving the linear least square problem, Ipopt is a library written in C++.</p>
<p class="para">The options allows the user to set various parameters of the Optimization problem.
It should be defined as type "list" and contains the following fields.
<ul class="itemizedlist"><li>Syntax : options= list("MaxIter", [---], "CpuTime", [---]);</li>
<li>MaxIter : a Scalar, containing the Maximum Number of Iteration that the solver should take.</li>
<li>CpuTime : a Scalar, containing the Maximum amount of CPU Time that the solver should take.</li>
<li>Default Values : options = list("MaxIter", [3000], "CpuTime", [600]);</li></ul></p>
<p class="para">The exitflag allows to know the status of the optimization which is given back by Ipopt.
<ul class="itemizedlist"><li>exitflag=0 : Optimal Solution Found</li>
<li>exitflag=1 : Maximum Number of Iterations Exceeded. Output may not be optimal.</li>
<li>exitflag=2 : Maximum CPU Time exceeded. Output may not be optimal.</li>
<li>exitflag=3 : Stop at Tiny Step.</li>
<li>exitflag=4 : Solved To Acceptable Level.</li>
<li>exitflag=5 : Converged to a point of local infeasibility.</li></ul></p>
<p class="para">For more details on exitflag see the ipopt documentation, go to http://www.coin-or.org/Ipopt/documentation/</p>
<p class="para">The output data structure contains detailed informations about the optimization process.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>output.iterations: The number of iterations performed during the search</li>
<li>output.constrviolation: The max-norm of the constraint violation.</li></ul></p>
<p class="para">The lambda data structure contains the Lagrange multipliers at the end
of optimization. In the current version the values are returned only when the the solution is optimal.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>lambda.lower: The Lagrange multipliers for the lower bound constraints.</li>
<li>lambda.upper: The Lagrange multipliers for the upper bound constraints.</li>
<li>lambda.eqlin: The Lagrange multipliers for the linear equality constraints.</li>
<li>lambda.ineqlin: The Lagrange multipliers for the linear inequality constraints.</li></ul></p>
<p class="para"></p></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//A simple linear least square example</span>
<span class="scilabid">C</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span> <span class="scilabnumber">2</span> <span class="scilabnumber">0</span><span class="scilabdefault">;</span>
<span class="scilaboperator">-</span><span class="scilabnumber">1</span> <span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilabnumber">0</span> <span class="scilabnumber">2</span><span class="scilabopenclose">]</span>
<span class="scilabid">d</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span>
<span class="scilabnumber">0</span>
<span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">10</span> <span class="scilaboperator">-</span><span class="scilabnumber">2</span><span class="scilabdefault">;</span>
<span class="scilaboperator">-</span><span class="scilabnumber">2</span> <span class="scilabnumber">10</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">4</span>
<span class="scilaboperator">-</span><span class="scilabnumber">4</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">resnorm</span><span class="scilabdefault">,</span><span class="scilabid">residual</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">lsqlin</span><span class="scilabopenclose">(</span><span class="scilabid">C</span><span class="scilabdefault">,</span><span class="scilabid">d</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//A basic example for equality, inequality constraints and variable bounds</span>
<span class="scilabid">C</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span> <span class="scilabnumber">1</span> <span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilabnumber">1</span> <span class="scilabnumber">1</span> <span class="scilabnumber">0</span><span class="scilabdefault">;</span>
<span class="scilabnumber">0</span> <span class="scilabnumber">1</span> <span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilabnumber">1</span> <span class="scilabnumber">0</span> <span class="scilabnumber">0</span><span class="scilabdefault">;</span>
<span class="scilabnumber">0</span> <span class="scilabnumber">0</span> <span class="scilabnumber">1</span><span class="scilabopenclose">]</span>
<span class="scilabid">d</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">89</span><span class="scilabdefault">;</span>
<span class="scilabnumber">67</span><span class="scilabdefault">;</span>
<span class="scilabnumber">53</span><span class="scilabdefault">;</span>
<span class="scilabnumber">35</span><span class="scilabdefault">;</span>
<span class="scilabnumber">20</span><span class="scilabdefault">;</span><span class="scilabopenclose">]</span>
<span class="scilabid">A</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">3</span> <span class="scilabnumber">2</span> <span class="scilabnumber">1</span><span class="scilabdefault">;</span>
<span class="scilabnumber">2</span> <span class="scilabnumber">3</span> <span class="scilabnumber">4</span><span class="scilabdefault">;</span>
<span class="scilabnumber">1</span> <span class="scilabnumber">2</span> <span class="scilabnumber">3</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">191</span>
<span class="scilabnumber">209</span>
<span class="scilabnumber">162</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">Aeq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span> <span class="scilabnumber">2</span> <span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span> <span class="scilaboperator">=</span> <span class="scilabnumber">10</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span> <span class="scilaboperator">=</span> <a class="scilabmacro" href="scilab://repmat">repmat</a><span class="scilabopenclose">(</span><span class="scilabnumber">0.1</span><span class="scilabdefault">,</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span> <span class="scilaboperator">=</span> <a class="scilabmacro" href="scilab://repmat">repmat</a><span class="scilabopenclose">(</span><span class="scilabnumber">4</span><span class="scilabdefault">,</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">resnorm</span><span class="scilabdefault">,</span><span class="scilabid">residual</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">lsqlin</span><span class="scilabopenclose">(</span><span class="scilabid">C</span><span class="scilabdefault">,</span><span class="scilabid">d</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabopenclose">)</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Authors</h3>
<ul class="itemizedlist"><li class="member">Harpreet Singh</li></ul></div>
<br />
<div class="manualnavbar">
<table width="100%">
<tr><td colspan="3" class="next"><a href="http://bugzilla.scilab.org/enter_bug.cgi?product=Scilab%20software&component=Documentation%20pages" class="ulink">Report an issue</a></td></tr>
<tr>
<td width="30%">
<span class="previous"><a href="linprog.html"><< linprog</a></span>
</td>
<td width="40%" class="center">
<span class="top"><a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a></span>
</td>
<td width="30%" class="next">
<span class="next"><a href="lsqnonlin.html">lsqnonlin >></a></span>
</td>
</tr></table>
<hr />
</div>
</body>
</html>
|
