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
|
<html><head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>intfminbnd</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="fminunc.html"><< fminunc</a></span>
</td>
<td width="40%" class="center">
<span class="top"><a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a></span>
</td>
<td width="30%" class="next">
<span class="next"><a href="intfmincon.html">intfmincon >></a></span>
</td>
</tr></table>
<hr />
</div>
<span class="path"><a href="index.html">FOSSEE Optimization Toolbox</a> >> <a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a> > intfminbnd</span>
<br /><br />
<div class="refnamediv"><h1 class="refname">intfminbnd</h1>
<p class="refpurpose">Solves a multi-variable optimization problem on a bounded interval</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">intfminbnd</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">intcon</span><span class="default">,</span><span class="default">x1</span><span class="default">,</span><span class="default">x2</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">intfminbnd</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">intcon</span><span class="default">,</span><span class="default">x1</span><span class="default">,</span><span class="default">x2</span><span class="default">,</span><span class="default">options</span><span class="default">)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">] = </span><span class="functionid">intfminbnd</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">]= </span><span class="functionid">intfminbnd</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</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="functionid">intfminbnd</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">gradient</span><span class="default">,</span><span class="default">hessian</span><span class="default">]=</span><span class="functionid">intfminbnd</span><span class="default">(.....)</span></pre></div></div>
<div class="refsection"><h3 class="title">Input Parameters</h3>
<dl><dt><span class="term">f :</span>
<dd><p class="para">A function, representing the objective function of the problem.</p></dd></dt>
<dt><span class="term"><span><img src='./_LaTeX_intfminbnd.xml_1.png' style='position:relative;top:5px;width:19px;height:14px'/></span> :</span>
<dd><p class="para">A vector, containing the lower bound of the variables of size (1 X n) or (n X 1) where n is number of variables. If it is empty it means that the lower bound is <span><img src='./_LaTeX_intfminbnd.xml_2.png' style='position:relative;top:4px;width:33px;height:16px'/></span>.</p></dd></dt>
<dt><span class="term"><span><img src='./_LaTeX_intfminbnd.xml_3.png' style='position:relative;top:5px;width:19px;height:14px'/></span> :</span>
<dd><p class="para">A vector, containing the upper bound of the variables of size (1 X n) or (n X 1) or (0 X 0) where n is the number of variables. If it is empty it means that the upper bound is <span><img src='./_LaTeX_intfminbnd.xml_4.png' style='position:relative;top:2px;width:20px;height:11px'/></span>.</p></dd></dt>
<dt><span class="term">intcon :</span>
<dd><p class="para">A vector of integers, representing the variables that are constrained to be integers.</p></dd></dt>
<dt><span class="term">options :</span>
<dd><p class="para">A list, containing the options for user to specify. See below for details.</p></dd></dt></dl></div>
<div class="refsection"><h3 class="title">Outputs</h3>
<dl><dt><span class="term">xopt :</span>
<dd><p class="para">A vector of doubles, containing the computed solution of the optimization problem.</p></dd></dt>
<dt><span class="term">fopt :</span>
<dd><p class="para">A double, containing the the function value at x.</p></dd></dt>
<dt><span class="term">exitflag :</span>
<dd><p class="para">An integer, containing the flag which denotes the reason for termination of algorithm. See below for details.</p></dd></dt>
<dt><span class="term">gradient :</span>
<dd><p class="para">A vector of doubles, containing the objective's gradient of the solution.</p></dd></dt>
<dt><span class="term">hessian :</span>
<dd><p class="para">A matrix of doubles, containing the Lagrangian's hessian of the solution.</p></dd></dt></dl></div>
<div class="refsection"><h3 class="title">Description</h3>
<p class="para">Search the minimum of a multi-variable function on bounded interval specified by :
Find the minimum of f(x) such that</p>
<p class="para"><span><img src='./_LaTeX_intfminbnd.xml_5.png' style='position:relative;top:29px;width:179px;height:66px'/></span></p>
<p class="para">intfminbnd calls Bonmin, which is an optimization library written in C++, to solve the bound optimization problem.</p>
<p class="para"><h3 class="title">Options</h3>
The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:</p>
<p class="para">options= list("IntegerTolerance", [---], "MaxNodes",[---], "MaxIter", [---], "AllowableGap",[---] "CpuTime", [---],"gradobj", "off", "hessian", "off" );
<ul class="itemizedlist"><li>IntegerTolerance : A Scalar, a number with that value of an integer is considered integer.</li>
<li>MaxNodes : A Scalar, containing the maximum number of nodes that the solver should search.</li>
<li>CpuTime : A scalar, specifying the maximum amount of CPU Time in seconds that the solver should take.</li>
<li>AllowableGap : A scalar, that specifies the gap between the computed solution and the the objective value of the best known solution stop, at which the tree search can be stopped.</li>
<li>MaxIter : A scalar, specifying the maximum number of iterations that the solver should take.</li>
<li>gradobj : A string, to turn on or off the user supplied objective gradient.</li>
<li>hessian : A scalar, to turn on or off the user supplied objective hessian.</li></ul>
The default values for the various items are given as:</p>
<p class="para">options = list('integertolerance',1d-06,'maxnodes',2147483647,'cputime',1d10,'allowablegap',0,'maxiter',2147483647,'gradobj',"off",'hessian',"off")</p>
<p class="para">The exitflag allows the user to know the status of the optimization which is returned by Bonmin. The values it can take and what they indicate is described below:
<ul class="itemizedlist"><li>0 : Optimal Solution Found</li>
<li>1 : Maximum Number of Iterations Exceeded. Output may not be optimal.</li>
<li>2 : Maximum amount of CPU Time exceeded. Output may not be optimal.</li>
<li>3 : Stop at Tiny Step.</li>
<li>4 : Solved To Acceptable Level.</li>
<li>5 : Converged to a point of local infeasibility.</li></ul></p>
<p class="para">For more details on exitflag, see the Bonmin documentation which can be found on http://www.coin-or.org/Bonmin</p>
<p class="para"></p></div>
<p class="para">A few examples displaying the various functionalities of intfminbnd have been provided below. You will find a series of problems and the appropriate code snippets to solve them.</p>
<div class="refsection"><h3 class="title">Example</h3>
<p class="para">We start with a simple objective function. Find x in R^6 such that it minimizes:</p>
<p class="para"><span><img src='./_LaTeX_intfminbnd.xml_6.png' style='position:relative;top:12px;width:506px;height:128px'/></span></p>
<p class="para"></p>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 1:</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabnumber">0</span>
<span class="scilabskeyword">for</span> <span class="scilabid">i</span> <span class="scilaboperator">=</span><span class="scilabnumber">1</span><span class="scilabspecial">:</span><span class="scilabnumber">6</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabinputoutputargs">y</span><span class="scilaboperator">+</span><a class="scilabcommand" href="scilab://sin">sin</a><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabid">i</span><span class="scilabopenclose">)</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabskeyword">end</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Variable bounds</span>
<span class="scilabid">x1</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</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="scilabdefault">,</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="scilabdefault">,</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="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">x2</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">intcon</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span> <span class="scilabnumber">3</span> <span class="scilabnumber">4</span><span class="scilabopenclose">]</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">intfminbnd</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span> <span class="scilabdefault">,</span><span class="scilabid">intcon</span><span class="scilabdefault">,</span> <span class="scilabid">x1</span><span class="scilabdefault">,</span> <span class="scilabid">x2</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">Example</h3>
Here we solve a bounded objective function in R^6. We use this function to illustrate how we can further enhance the functionality of fminbnd by setting input options. We can pre-define the gradient of the objective function and/or the hessian of the lagrange function and thereby improve the speed of computation. This is elaborated on in example 2. We also set solver parameters using the options.
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 2:</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabnumber">0</span>
<span class="scilabskeyword">for</span> <span class="scilabid">i</span> <span class="scilaboperator">=</span><span class="scilabnumber">1</span><span class="scilabspecial">:</span><span class="scilabnumber">6</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabinputoutputargs">y</span><span class="scilaboperator">+</span><a class="scilabcommand" href="scilab://sin">sin</a><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabid">i</span><span class="scilabopenclose">)</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabskeyword">end</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Variable bounds</span>
<span class="scilabid">x1</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</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="scilabdefault">,</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="scilabdefault">,</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="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">x2</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">intcon</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span> <span class="scilabnumber">3</span> <span class="scilabnumber">4</span><span class="scilabopenclose">]</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabnumber">1500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">100</span><span class="scilabopenclose">]</span><span class="scilabopenclose">)</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">intfminbnd</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span> <span class="scilabdefault">,</span><span class="scilabid">intcon</span><span class="scilabdefault">,</span> <span class="scilabid">x1</span><span class="scilabdefault">,</span> <span class="scilabid">x2</span><span class="scilabdefault">,</span> <span class="scilabid">options</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">Example</h3>
<p class="para">Unbounded Problems: Find x in R^2 such that it minimizes:</p>
<p class="para"><span><img src='./_LaTeX_intfminbnd.xml_7.png' style='position:relative;top:12px;width:237px;height:148px'/></span></p>
<p class="para"></p>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">///Example 3: Unbounded problem:</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilaboperator">-</span><span class="scilabopenclose">(</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">+</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Variable bounds</span>
<span class="scilabid">x1</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilaboperator">-</span><span class="scilabconstants">%inf</span> <span class="scilabdefault">,</span> <span class="scilaboperator">-</span><span class="scilabconstants">%inf</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">x2</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span> <span class="scilabconstants">%inf</span> <span class="scilabdefault">,</span> <span class="scilabconstants">%inf</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabnumber">1500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">100</span><span class="scilabopenclose">]</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabid">intcon</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span> <span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</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">intfminbnd</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span><span class="scilabdefault">,</span><span class="scilabid">intcon</span><span class="scilabdefault">,</span> <span class="scilabid">x1</span><span class="scilabdefault">,</span> <span class="scilabid">x2</span><span class="scilabdefault">,</span> <span class="scilabid">options</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="fminunc.html"><< fminunc</a></span>
</td>
<td width="40%" class="center">
<span class="top"><a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a></span>
</td>
<td width="30%" class="next">
<span class="next"><a href="intfmincon.html">intfmincon >></a></span>
</td>
</tr></table>
<hr />
</div>
</body>
</html>
|
