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
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Minimizers (GNU Octave (version 4.4.1))</title>
<meta name="description" content="Minimizers (GNU Octave (version 4.4.1))">
<meta name="keywords" content="Minimizers (GNU Octave (version 4.4.1))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<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="Nonlinear-Equations.html#Nonlinear-Equations" rel="up" title="Nonlinear Equations">
<link href="Diagonal-and-Permutation-Matrices.html#Diagonal-and-Permutation-Matrices" rel="next" title="Diagonal and Permutation Matrices">
<link href="Solvers.html#Solvers" rel="prev" title="Solvers">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
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.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
ul.no-bullet {list-style: none}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<a name="Minimizers"></a>
<div class="header">
<p>
Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</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="Minimizers-1"></a>
<h3 class="section">20.2 Minimizers</h3>
<a name="index-local-minimum"></a>
<a name="index-finding-minimums"></a>
<p>Often it is useful to find the minimum value of a function rather than just
the zeroes where it crosses the x-axis. <code>fminbnd</code> is designed for the
simpler, but very common, case of a univariate function where the interval
to search is bounded. For unbounded minimization of a function with
potentially many variables use <code>fminunc</code> or <code>fminsearch</code>. The two
functions use different internal algorithms and some knowledge of the objective
function is required. For functions which can be differentiated,
<code>fminunc</code> is appropriate. For functions with discontinuities, or for
which a gradient search would fail, use <code>fminsearch</code>.
See <a href="Optimization.html#Optimization">Optimization</a>, for minimization with the presence of constraint
functions. Note that searches can be made for maxima by simply inverting the
objective function
(<code>Fto_max = -Fto_min</code>).
</p>
<a name="XREFfminbnd"></a><dl>
<dt><a name="index-fminbnd"></a><em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>] =</em> <strong>fminbnd</strong> <em>(<var>fun</var>, <var>a</var>, <var>b</var>, <var>options</var>)</em></dt>
<dd><p>Find a minimum point of a univariate function.
</p>
<p><var>fun</var> should be a function handle or name. <var>a</var>, <var>b</var> specify a
starting interval. <var>options</var> is a structure specifying additional
options. Currently, <code>fminbnd</code> recognizes these options:
<code>"FunValCheck"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>,
<code>"MaxIter"</code>, <code>"MaxFunEvals"</code>. For a description of these
options, see <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p>
<p>On exit, the function returns <var>x</var>, the approximate minimum point and
<var>fval</var>, the function value thereof.
</p>
<p><var>info</var> is an exit flag that can have these values:
</p>
<ul>
<li> 1
The algorithm converged to a solution.
</li><li> 0
Maximum number of iterations or function evaluations has been exhausted.
</li><li> -1
The algorithm has been terminated from user output function.
</li></ul>
<p>Notes: The search for a minimum is restricted to be in the interval bound by
<var>a</var> and <var>b</var>. If you only have an initial point to begin searching
from you will need to use an unconstrained minimization algorithm such as
<code>fminunc</code> or <code>fminsearch</code>. <code>fminbnd</code> internally uses a
Golden Section search strategy.
</p>
<p><strong>See also:</strong> <a href="Solvers.html#XREFfzero">fzero</a>, <a href="#XREFfminunc">fminunc</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminunc"></a><dl>
<dt><a name="index-fminunc"></a><em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminunc-1"></a><em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminunc-2"></a><em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>, <var>grad</var>, <var>hess</var>] =</em> <strong>fminunc</strong> <em>(<var>fcn</var>, …)</em></dt>
<dd><p>Solve an unconstrained optimization problem defined by the function
<var>fcn</var>.
</p>
<p><var>fcn</var> should accept a vector (array) defining the unknown variables, and
return the objective function value, optionally with gradient.
<code>fminunc</code> attempts to determine a vector <var>x</var> such that
<code><var>fcn</var> (<var>x</var>)</code> is a local minimum.
</p>
<p><var>x0</var> determines a starting guess. The shape of <var>x0</var> is preserved in
all calls to <var>fcn</var>, but otherwise is treated as a column vector.
</p>
<p><var>options</var> is a structure specifying additional options. Currently,
<code>fminunc</code> recognizes these options:
<code>"FunValCheck"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>,
<code>"TolFun"</code>, <code>"MaxIter"</code>, <code>"MaxFunEvals"</code>,
<code>"GradObj"</code>, <code>"FinDiffType"</code>, <code>"TypicalX"</code>,
<code>"AutoScaling"</code>.
</p>
<p>If <code>"GradObj"</code> is <code>"on"</code>, it specifies that <var>fcn</var>, when
called with two output arguments, also returns the Jacobian matrix of
partial first derivatives at the requested point. <code>TolX</code> specifies
the termination tolerance for the unknown variables <var>x</var>, while
<code>TolFun</code> is a tolerance for the objective function value <var>fval</var>.
The default is <code>1e-7</code> for both options.
</p>
<p>For a description of the other options, see <code>optimset</code>.
</p>
<p>On return, <var>x</var> is the location of the minimum and <var>fval</var> contains
the value of the objective function at <var>x</var>.
</p>
<p><var>info</var> may be one of the following values:
</p>
<dl compact="compact">
<dt>1</dt>
<dd><p>Converged to a solution point. Relative gradient error is less than
specified by <code>TolFun</code>.
</p>
</dd>
<dt>2</dt>
<dd><p>Last relative step size was less than <code>TolX</code>.
</p>
</dd>
<dt>3</dt>
<dd><p>Last relative change in function value was less than <code>TolFun</code>.
</p>
</dd>
<dt>0</dt>
<dd><p>Iteration limit exceeded—either maximum number of algorithm iterations
<code>MaxIter</code> or maximum number of function evaluations <code>MaxFunEvals</code>.
</p>
</dd>
<dt>-1</dt>
<dd><p>Algorithm terminated by <code>OutputFcn</code>.
</p>
</dd>
<dt>-3</dt>
<dd><p>The trust region radius became excessively small.
</p></dd>
</dl>
<p>Optionally, <code>fminunc</code> can return a structure with convergence
statistics (<var>output</var>), the output gradient (<var>grad</var>) at the
solution <var>x</var>, and approximate Hessian (<var>hess</var>) at the solution
<var>x</var>.
</p>
<p>Application Notes: If the objective function is a single nonlinear equation
of one variable then using <code>fminbnd</code> is usually a better choice.
</p>
<p>The algorithm used by <code>fminunc</code> is a gradient search which depends
on the objective function being differentiable. If the function has
discontinuities it may be better to use a derivative-free algorithm such as
<code>fminsearch</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminsearch"></a><dl>
<dt><a name="index-fminsearch"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminsearch-1"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminsearch-2"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>, <var>options</var>, <var>fun_arg1</var>, <var>fun_arg2</var>, …)</em></dt>
<dt><a name="index-fminsearch-3"></a><em>[<var>x</var>, <var>fval</var>, <var>exitflag</var>, <var>output</var>] =</em> <strong>fminsearch</strong> <em>(…)</em></dt>
<dd>
<p>Find a value of <var>x</var> which minimizes the function <var>fun</var>.
</p>
<p>The search begins at the point <var>x0</var> and iterates using the
Nelder & Mead Simplex algorithm (a derivative-free method). This
algorithm is better-suited to functions which have discontinuities or for
which a gradient-based search such as <code>fminunc</code> fails.
</p>
<p>Options for the search are provided in the parameter <var>options</var> using the
function <code>optimset</code>. Currently, <code>fminsearch</code> accepts the options:
<code>"TolX"</code>, <code>"TolFun"</code>, <code>"MaxFunEvals"</code>, <code>"MaxIter"</code>,
<code>"Display"</code>, <code>"FunValCheck"</code>, and <code>"OutputFcn"</code>.
For a description of these options, see <code>optimset</code>.
</p>
<p>Additional inputs for the function <var>fun</var> can be passed as the fourth
and higher arguments. To pass function arguments while using the default
<var>options</var> values, use <code>[]</code> for <var>options</var>.
</p>
<p>On exit, the function returns <var>x</var>, the minimum point, and <var>fval</var>,
the function value at the minimum.
</p>
<p>The third return value <var>exitflag</var> is
</p>
<dl compact="compact">
<dt>1</dt>
<dd><p>if the algorithm converged
(size of the simplex is smaller than <code><var>options</var>.TolX</code> <strong>AND</strong>
the step in the function value between iterations is smaller than
<code><var>options</var>.TolFun</code>).
</p>
</dd>
<dt>0</dt>
<dd><p>if the maximum number of iterations or the maximum number of function
evaluations are exceeded.
</p>
</dd>
<dt>-1</dt>
<dd><p>if the iteration is stopped by the <code>"OutputFcn"</code>.
</p></dd>
</dl>
<p>The fourth return value is a structure <var>output</var> with the fields,
<code>funcCount</code> containing the number of function calls to <var>fun</var>,
<code>iterations</code> containing the number of iteration steps,
<code>algorithm</code> with the name of the search algorithm (always:
<code>"Nelder-Mead simplex direct search"</code>), and <code>message</code>
with the exit message.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
</pre></div>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminunc">fminunc</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<p>The function <code>humps</code> is a useful function for testing zero and
extrema finding functions.
</p>
<a name="XREFhumps"></a><dl>
<dt><a name="index-humps"></a><em><var>y</var> =</em> <strong>humps</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-humps-1"></a><em>[<var>x</var>, <var>y</var>] =</em> <strong>humps</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Evaluate a function with multiple minima, maxima, and zero crossings.
</p>
<p>The output <var>y</var> is the evaluation of the rational function:
</p>
<div class="example">
<pre class="example"> 1200*<var>x</var>^4 - 2880*<var>x</var>^3 + 2036*<var>x</var>^2 - 348*<var>x</var> - 88
<var>y</var> = - ---------------------------------------------
200*<var>x</var>^4 - 480*<var>x</var>^3 + 406*<var>x</var>^2 - 138*<var>x</var> + 17
</pre></div>
<p><var>x</var> may be a scalar, vector or array. If <var>x</var> is omitted, the
default range [0:0.05:1] is used.
</p>
<p>When called with two output arguments, [<var>x</var>, <var>y</var>], <var>x</var> will
contain the input values, and <var>y</var> will contain the output from
<code>humps</code>.
</p>
<p>Programming Notes: <code>humps</code> has two local maxima located near <var>x</var> =
0.300 and 0.893, a local minimum near <var>x</var> = 0.637, and zeros near
<var>x</var> = -0.132 and 1.300. <code>humps</code> is a useful function for testing
algorithms which find zeros or local minima and maxima.
</p>
<p>Try <code>demo humps</code> to see a plot of the <code>humps</code> function.
</p>
<p><strong>See also:</strong> <a href="Solvers.html#XREFfzero">fzero</a>, <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminunc">fminunc</a>, <a href="#XREFfminsearch">fminsearch</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</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>
|
