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
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Octave: Quadratic Programming</title>
<meta name="description" content="GNU Octave: Quadratic Programming">
<meta name="keywords" content="GNU Octave: Quadratic Programming">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Optimization.html#Optimization" rel="up" title="Optimization">
<link href="Nonlinear-Programming.html#Nonlinear-Programming" rel="next" title="Nonlinear Programming">
<link href="Linear-Programming.html#Linear-Programming" rel="prev" title="Linear Programming">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Quadratic-Programming"></a>
<div class="header">
<p>
Next: <a href="Nonlinear-Programming.html#Nonlinear-Programming" accesskey="n" rel="next">Nonlinear Programming</a>, Previous: <a href="Linear-Programming.html#Linear-Programming" accesskey="p" rel="prev">Linear Programming</a>, Up: <a href="Optimization.html#Optimization" accesskey="u" rel="up">Optimization</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Quadratic-Programming-1"></a>
<h3 class="section">25.2 Quadratic Programming</h3>
<p>Octave can also solve Quadratic Programming problems, this is
</p>
<div class="example">
<pre class="example">min 0.5 x'*H*x + x'*q
</pre></div>
<p>subject to
</p>
<div class="example">
<pre class="example"> A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre></div>
<a name="XREFqp"></a><dl>
<dt><a name="index-qp"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(<var>x0</var>, <var>H</var>)</em></dt>
<dt><a name="index-qp-1"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(<var>x0</var>, <var>H</var>, <var>q</var>)</em></dt>
<dt><a name="index-qp-2"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(<var>x0</var>, <var>H</var>, <var>q</var>, <var>A</var>, <var>b</var>)</em></dt>
<dt><a name="index-qp-3"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(<var>x0</var>, <var>H</var>, <var>q</var>, <var>A</var>, <var>b</var>, <var>lb</var>, <var>ub</var>)</em></dt>
<dt><a name="index-qp-4"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(<var>x0</var>, <var>H</var>, <var>q</var>, <var>A</var>, <var>b</var>, <var>lb</var>, <var>ub</var>, <var>A_lb</var>, <var>A_in</var>, <var>A_ub</var>)</em></dt>
<dt><a name="index-qp-5"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>lambda</var>] =</em> <strong>qp</strong> <em>(…, <var>options</var>)</em></dt>
<dd><p>Solve the quadratic program
</p>
<div class="example">
<pre class="example">min 0.5 x'*H*x + x'*q
x
</pre></div>
<p>subject to
</p>
<div class="example">
<pre class="example">A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
</pre></div>
<p>using a null-space active-set method.
</p>
<p>Any bound (<var>A</var>, <var>b</var>, <var>lb</var>, <var>ub</var>, <var>A_lb</var>,
<var>A_ub</var>) may be set to the empty matrix (<code>[]</code>) if not
present. If the initial guess is feasible the algorithm is faster.
</p>
<dl compact="compact">
<dt><var>options</var></dt>
<dd><p>An optional structure containing the following
parameter(s) used to define the behavior of the solver. Missing elements
in the structure take on default values, so you only need to set the
elements that you wish to change from the default.
</p>
<dl compact="compact">
<dt><code>MaxIter (default: 200)</code></dt>
<dd><p>Maximum number of iterations.
</p></dd>
</dl>
</dd>
</dl>
<dl compact="compact">
<dt><var>info</var></dt>
<dd><p>Structure containing run-time information about the algorithm. The
following fields are defined:
</p>
<dl compact="compact">
<dt><code>solveiter</code></dt>
<dd><p>The number of iterations required to find the solution.
</p>
</dd>
<dt><code>info</code></dt>
<dd><p>An integer indicating the status of the solution.
</p>
<dl compact="compact">
<dt>0</dt>
<dd><p>The problem is feasible and convex. Global solution found.
</p>
</dd>
<dt>1</dt>
<dd><p>The problem is not convex. Local solution found.
</p>
</dd>
<dt>2</dt>
<dd><p>The problem is not convex and unbounded.
</p>
</dd>
<dt>3</dt>
<dd><p>Maximum number of iterations reached.
</p>
</dd>
<dt>6</dt>
<dd><p>The problem is infeasible.
</p></dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<a name="XREFpqpnonneg"></a><dl>
<dt><a name="index-pqpnonneg"></a>Function File: <em><var>x</var> =</em> <strong>pqpnonneg</strong> <em>(<var>c</var>, <var>d</var>)</em></dt>
<dt><a name="index-pqpnonneg-1"></a>Function File: <em><var>x</var> =</em> <strong>pqpnonneg</strong> <em>(<var>c</var>, <var>d</var>, <var>x0</var>)</em></dt>
<dt><a name="index-pqpnonneg-2"></a>Function File: <em>[<var>x</var>, <var>minval</var>] =</em> <strong>pqpnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-pqpnonneg-3"></a>Function File: <em>[<var>x</var>, <var>minval</var>, <var>exitflag</var>] =</em> <strong>pqpnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-pqpnonneg-4"></a>Function File: <em>[<var>x</var>, <var>minval</var>, <var>exitflag</var>, <var>output</var>] =</em> <strong>pqpnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-pqpnonneg-5"></a>Function File: <em>[<var>x</var>, <var>minval</var>, <var>exitflag</var>, <var>output</var>, <var>lambda</var>] =</em> <strong>pqpnonneg</strong> <em>(…)</em></dt>
<dd><p>Minimize <code>1/2*x'*c*x + d'*x</code> subject to <code><var>x</var> >= 0</code>. <var>c</var>
and <var>d</var> must be real, and <var>c</var> must be symmetric and positive
definite. <var>x0</var> is an optional initial guess for <var>x</var>.
</p>
<p>Outputs:
</p>
<ul>
<li> minval
<p>The minimum attained model value, 1/2*xmin’*c*xmin + d’*xmin
</p>
</li><li> exitflag
<p>An indicator of convergence. 0 indicates that the iteration count
was exceeded, and therefore convergence was not reached; >0 indicates
that the algorithm converged. (The algorithm is stable and will
converge given enough iterations.)
</p>
</li><li> output
<p>A structure with two fields:
</p>
<ul>
<li> <code>"algorithm"</code>: The algorithm used (<code>"nnls"</code>)
</li><li> <code>"iterations"</code>: The number of iterations taken.
</li></ul>
</li><li> lambda
<p>Not implemented.
</p></li></ul>
<p><strong>See also:</strong> <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>, <a href="Linear-Least-Squares.html#XREFlsqnonneg">lsqnonneg</a>, <a href="#XREFqp">qp</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Nonlinear-Programming.html#Nonlinear-Programming" accesskey="n" rel="next">Nonlinear Programming</a>, Previous: <a href="Linear-Programming.html#Linear-Programming" accesskey="p" rel="prev">Linear Programming</a>, Up: <a href="Optimization.html#Optimization" accesskey="u" rel="up">Optimization</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
