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
|
<!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: Nonlinear Programming</title>
<meta name="description" content="GNU Octave: Nonlinear Programming">
<meta name="keywords" content="GNU Octave: Nonlinear 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="Linear-Least-Squares.html#Linear-Least-Squares" rel="next" title="Linear Least Squares">
<link href="Quadratic-Programming.html#Quadratic-Programming" rel="prev" title="Quadratic 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="Nonlinear-Programming"></a>
<div class="header">
<p>
Next: <a href="Linear-Least-Squares.html#Linear-Least-Squares" accesskey="n" rel="next">Linear Least Squares</a>, Previous: <a href="Quadratic-Programming.html#Quadratic-Programming" accesskey="p" rel="prev">Quadratic 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="Nonlinear-Programming-1"></a>
<h3 class="section">25.3 Nonlinear Programming</h3>
<p>Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.
</p>
<a name="XREFsqp"></a><dl>
<dt><a name="index-sqp"></a>Function File: <em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>iter</var>, <var>nf</var>, <var>lambda</var>] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>)</em></dt>
<dt><a name="index-sqp-1"></a>Function File: <em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>)</em></dt>
<dt><a name="index-sqp-2"></a>Function File: <em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>)</em></dt>
<dt><a name="index-sqp-3"></a>Function File: <em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>)</em></dt>
<dt><a name="index-sqp-4"></a>Function File: <em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>, <var>maxiter</var>)</em></dt>
<dt><a name="index-sqp-5"></a>Function File: <em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>, <var>maxiter</var>, <var>tol</var>)</em></dt>
<dd><p>Solve the nonlinear program
</p>
<div class="example">
<pre class="example">min phi (x)
x
</pre></div>
<p>subject to
</p>
<div class="example">
<pre class="example">g(x) = 0
h(x) >= 0
lb <= x <= ub
</pre></div>
<p>using a sequential quadratic programming method.
</p>
<p>The first argument is the initial guess for the vector <var>x0</var>.
</p>
<p>The second argument is a function handle pointing to the objective
function <var>phi</var>. The objective function must accept one vector
argument and return a scalar.
</p>
<p>The second argument may also be a 2- or 3-element cell array of
function handles. The first element should point to the objective
function, the second should point to a function that computes the
gradient of the objective function, and the third should point to a
function that computes the Hessian of the objective function. If the
gradient function is not supplied, the gradient is computed by finite
differences. If the Hessian function is not supplied, a BFGS update
formula is used to approximate the Hessian.
</p>
<p>When supplied, the gradient function <code><var>phi</var>{2}</code> must accept
one vector argument and return a vector. When supplied, the Hessian
function <code><var>phi</var>{3}</code> must accept one vector argument and
return a matrix.
</p>
<p>The third and fourth arguments <var>g</var> and <var>h</var> are function
handles pointing to functions that compute the equality constraints
and the inequality constraints, respectively. If the problem does
not have equality (or inequality) constraints, then use an empty
matrix ([]) for <var>g</var> (or <var>h</var>). When supplied, these equality
and inequality constraint functions must accept one vector argument
and return a vector.
</p>
<p>The third and fourth arguments may also be 2-element cell arrays of
function handles. The first element should point to the constraint
function and the second should point to a function that computes the
gradient of the constraint function:
</p>
<div class="example">
<pre class="example"> [ d f(x) d f(x) d f(x) ]
transpose ( [ ------ ----- ... ------ ] )
[ dx_1 dx_2 dx_N ]
</pre></div>
<p>The fifth and sixth arguments, <var>lb</var> and <var>ub</var>, contain lower
and upper bounds on <var>x</var>. These must be consistent with the
equality and inequality constraints <var>g</var> and <var>h</var>. If the
arguments are vectors then <var>x</var>(i) is bound by <var>lb</var>(i) and
<var>ub</var>(i). A bound can also be a scalar in which case all elements
of <var>x</var> will share the same bound. If only one bound (lb, ub) is
specified then the other will default to (-<var>realmax</var>,
+<var>realmax</var>).
</p>
<p>The seventh argument <var>maxiter</var> specifies the maximum number of
iterations. The default value is 100.
</p>
<p>The eighth argument <var>tol</var> specifies the tolerance for the
stopping criteria. The default value is <code>sqrt (eps)</code>.
</p>
<p>The value returned in <var>info</var> may be one of the following:
</p>
<dl compact="compact">
<dt>101</dt>
<dd><p>The algorithm terminated normally.
Either all constraints meet the requested tolerance, or the stepsize,
delta <var>x</var>,
is less than <code><var>tol</var> * norm (x)</code>.
</p>
</dd>
<dt>102</dt>
<dd><p>The BFGS update failed.
</p>
</dd>
<dt>103</dt>
<dd><p>The maximum number of iterations was reached.
</p></dd>
</dl>
<p>An example of calling <code>sqp</code>:
</p>
<div class="example">
<pre class="example">function r = g (x)
r = [ sumsq(x)-10;
x(2)*x(3)-5*x(4)*x(5);
x(1)^3+x(2)^3+1 ];
endfunction
function obj = phi (x)
obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
endfunction
x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
[x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, [])
x =
-1.71714
1.59571
1.82725
-0.76364
-0.76364
obj = 0.053950
info = 101
iter = 8
nf = 10
lambda =
-0.0401627
0.0379578
-0.0052227
</pre></div>
<p><strong>See also:</strong> <a href="Quadratic-Programming.html#XREFqp">qp</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Linear-Least-Squares.html#Linear-Least-Squares" accesskey="n" rel="next">Linear Least Squares</a>, Previous: <a href="Quadratic-Programming.html#Quadratic-Programming" accesskey="p" rel="prev">Quadratic 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>
|
