Homec4science
Diffusion EMPoWER (master) Browse

R1252/simulation-code/cvx/examples/cvxbook/Ch06_approx_fitting/html/preference_regions.html
de096106547amaster

Last Change
Raw File

preference_regions.html
View Options

<!DOCTYPE HTML>
<html>
<head>
<meta charset="UTF-8">
<title>Example 6.9: Bounding consumer preference</title>
<link rel="canonical" href="http://cvxr.com/cvx/examples/cvxbook/Ch06_approx_fitting/html/preference_regions.html">
<link rel="stylesheet" href="../../../examples.css" type="text/css">
</head>
<body>
<div id="header">
<h1>Example 6.9: Bounding consumer preference</h1>
Jump to:&nbsp;&nbsp;&nbsp;&nbsp;
<a href="#source">Source code</a>&nbsp;&nbsp;&nbsp;&nbsp;
<a href="#output">Text output</a>
&nbsp;&nbsp;&nbsp;&nbsp;
<a href="#plots">Plots</a>
&nbsp;&nbsp;&nbsp;&nbsp;<a href="../../../index.html">Library index</a>
</div>
<div id="content">
<a id="source"></a>
<pre class="codeinput">
<span class="comment">% Section 6.5.5, Figures 6.25-6.26</span>
<span class="comment">% Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX Argyris Zymnis - 11/30/2005</span>
<span class="comment">%</span>
<span class="comment">% We are given a set of consumer preference data for bundles</span>
<span class="comment">% of two goods x_1 and x_2. These points are generated by</span>
<span class="comment">% taking 40 random points and comparing them using the</span>
<span class="comment">% utility function: u(x_1,x_2) = (1.1*sqrt(x_1)+0.8*sqrt(x_2))/1.9</span>
<span class="comment">% Then, if we have u(i) &gt;= u(j) we say that (i,j) is in Pweak.</span>
<span class="comment">%</span>
<span class="comment">% Given this, we wish to compare the point (0.5,0.5) to each</span>
<span class="comment">% of the bundles in the given dataset. I.e. for each point k in the</span>
<span class="comment">% dataset, we wish to decide wether u(k) &gt;= u(0) or u(k) &lt;= u(0),</span>
<span class="comment">% or both, in which case we cannot make any conclusions about</span>
<span class="comment">% consumer preferences.</span>
<span class="comment">%</span>
<span class="comment">% To do this, we have to solve two LPs for each point:</span>
<span class="comment">% minimize u(k) - u(0)</span>
<span class="comment">% subject to g_i &gt;= 0</span>
<span class="comment">% u(j) &lt;= u(i) + g_i^T(a_j - a_i), for all i,j</span>
<span class="comment">% u(i) &gt;= u(j), for all (i,j) in Pweak</span>
<span class="comment">%</span>
<span class="comment">% and:</span>
<span class="comment">% maximize u(k) - u(0)</span>
<span class="comment">% subject to g_i &gt;= 0</span>
<span class="comment">% u(j) &lt;= u(i) + g_i^T(a_j - a_i), for all i,j</span>
<span class="comment">% u(i) &gt;= u(j), for all (i,j) in Pweak</span>
<span class="comment">%</span>
<span class="comment">% If the second LP has a strictly negative solution, we can deduce that</span>
<span class="comment">% u(k) &lt; u(0). If on the other hand the second LP has a nonnegative</span>
<span class="comment">% solution and the first LP has a strictly positive solution, we can</span>
<span class="comment">% deduce that u(k) &gt; u(0). Finally if none of the two previous cases</span>
<span class="comment">% holds, we cannot make a decision between the two bundles.</span>
<span class="comment">%</span>
<span class="comment">% NOTE: This file requires the auxilliary function utilfun.m to run.</span>
data= [<span class="keyword">...</span>
4.5e-01 9.6e-01
2.1e-01 3.4e-01
9.6e-01 3.0e-02
8.0e-02 9.2e-01
2.0e-02 2.2e-01
0.0e+00 3.9e-01
2.6e-01 6.4e-01
3.5e-01 9.7e-01
9.1e-01 7.8e-01
1.2e-01 1.4e-01
5.8e-01 8.4e-01
4.9e-01 2.7e-01
7.0e-02 8.0e-01
9.3e-01 8.7e-01
4.4e-01 8.6e-01
3.3e-01 4.2e-01
8.9e-01 9.0e-01
4.9e-01 7.0e-02
9.5e-01 3.3e-01
6.6e-01 2.6e-01
9.5e-01 7.3e-01
4.2e-01 9.1e-01
6.8e-01 2.0e-01
5.2e-01 6.2e-01
7.7e-01 6.3e-01
2.0e-02 2.9e-01
9.8e-01 2.0e-02
5.0e-02 7.9e-01
7.9e-01 1.9e-01
6.2e-01 6.0e-02
2.8e-01 8.7e-01
6.9e-01 1.0e-01
6.9e-01 3.7e-01
0.0e+00 7.2e-01
8.7e-01 1.7e-01
6.3e-01 4.0e-02
3.2e-01 7.3e-01
4.0e-02 4.6e-01
3.6e-01 9.5e-01
8.2e-01 6.7e-01 ];
<span class="comment">% objective point</span>
obj=[0.5,0.5];
figure(1);
<span class="comment">% display the utility function's level sets on some data points.</span>
plot(data(:,1),data(:,2),<span class="string">'o'</span>);
hold <span class="string">on</span>;
[X,Y] = meshgrid(0:.01:1,0:.01:1);
Z=(1.1*X.^(1/2)+0.8*Y.^(1/2))/1.9;
[C,h] = contour(X,Y,Z,[.1,.2,.3,.4,.5,.6,.7,.8,.9],<span class="string">'--'</span>);
clear <span class="string">X</span> <span class="string">Y</span> <span class="string">Z</span> <span class="string">C</span>
hold <span class="string">off</span>;
xlabel(<span class="string">'x_1'</span>);
ylabel(<span class="string">'x_2'</span>);
hold <span class="string">off</span>
m = size(data,1); <span class="comment">% number of baskets, including 0,1</span>
<span class="comment">% add preference data</span>
Pweak = zeros(m+1,m+1);
<span class="keyword">for</span> i=1:m,
<span class="keyword">for</span> j=1:m
<span class="keyword">if</span> (i~=j) &amp; (1.1*data(i,1).^(1/2)+0.8*data(i,2).^(1/2))/1.9 &gt;= <span class="keyword">...</span>
(1.1*data(j,1).^(1/2)+0.8*data(j,2).^(1/2))/1.9,
Pweak(i,j) = 1;
<span class="keyword">end</span>;
<span class="keyword">end</span>;
<span class="keyword">end</span>;
<span class="comment">% Find consumer preferences</span>
data = [data; 0.5 0.5];
bounds = zeros(m,2);
<span class="keyword">for</span> k = 1:m
fprintf(1,<span class="string">'Deciding on bundle %d of %d: '</span>,k,m);
<span class="comment">% Check for u(k) &gt;= u(0.5,0.5)</span>
cvx_begin <span class="string">quiet</span>
variables <span class="string">u(m+1)</span> <span class="string">g_x(m+1)</span> <span class="string">g_y(m+1)</span>
minimize(u(k)-u(m+1))
subject <span class="string">to</span>
g_x &gt;= 0;
g_y &gt;= 0;
ones(m+1,1)*u' &lt;= u*ones(1,m+1)+(g_x*ones(1,m+1)).*<span class="keyword">...</span>
(ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+<span class="keyword">...</span>
(g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
(u*ones(1,m+1)).*Pweak &gt;= (ones(m+1,1)*u').*Pweak;
cvx_end
bounds(k,1) = cvx_optval;
fprintf( 1,<span class="string">'%g'</span>, round(cvx_optval) );
<span class="comment">% Check for u(0.5,0.5) &gt;= u(k)</span>
cvx_begin <span class="string">quiet</span>
variables <span class="string">u(m+1)</span> <span class="string">g_x(m+1)</span> <span class="string">g_y(m+1)</span>
maximize(u(k)-u(m+1))
subject <span class="string">to</span>
g_x &gt;= 0;
g_y &gt;= 0;
ones(m+1,1)*u' &lt;= u*ones(1,m+1) + (g_x*ones(1,m+1)).*<span class="keyword">...</span>
(ones(m+1,1)*data(:,1)'-data(:,1)*ones(1,m+1))+<span class="keyword">...</span>
(g_y*ones(1,m+1)).*(ones(m+1,1)*data(:,2)'-data(:,2)*ones(1,m+1));
(u*ones(1,m+1)).*Pweak &gt;= (ones(m+1,1)*u').*Pweak;
cvx_end
bounds(k,2) = cvx_optval;
fprintf( 1,<span class="string">' %g\n'</span>, round(cvx_optval) );
<span class="keyword">end</span>
figure(2);
hold <span class="string">off</span>
<span class="comment">% plot data pt and contour line through it</span>
val = 1.1*sqrt(0.5)+ 0.8*sqrt(.5); <span class="comment">% value at center</span>
t = linspace(((val-.8)/1.1)^2, 1, 1000);
y = ( (val - 1.1*(t.^(1/2)))/.8 ).^2;
plot(t,y,<span class="string">'--'</span>, [.5 .5], [0 1], <span class="string">':'</span>, [0 1], [.5 .5], <span class="string">':'</span>);
axis([0 1 0 1]);
hold <span class="string">on</span>
<span class="keyword">for</span> k=1:m
<span class="keyword">if</span> bounds(k,2) &lt; 1e-5, <span class="comment">% preferred over (.5,.5)</span>
dot = plot(data(k,1),data(k,2),<span class="string">'o'</span>);
<span class="comment">%'MarkerSize',8);</span>
<span class="keyword">elseif</span> bounds(k,1) &gt; -1e-5, <span class="comment">% rejected in favor of (.5,.5)</span>
dot = plot(data(k,1),data(k,2),<span class="string">'o'</span>,<span class="string">'MarkerFaceColor'</span>,[0 0 0]);
<span class="keyword">else</span> <span class="comment">% no conclusion</span>
dot = plot(data(k,1),data(k,2),<span class="string">'square'</span>, <span class="string">'LineWidth'</span>,1.0,<span class="keyword">...</span>
<span class="string">'MarkerSize'</span>,10);
<span class="keyword">end</span>;
<span class="keyword">end</span>;
xlabel(<span class="string">'x_1'</span>); ylabel(<span class="string">'x_2'</span>);
</pre>
<a id="output"></a>
<pre class="codeoutput">
Deciding on bundle 1 of 40: 0 Inf
Deciding on bundle 2 of 40: -Inf -0
Deciding on bundle 3 of 40: -Inf -0
Deciding on bundle 4 of 40: -Inf -0
Deciding on bundle 5 of 40: -Inf -0
Deciding on bundle 6 of 40: -Inf -0
Deciding on bundle 7 of 40: -Inf -0
Deciding on bundle 8 of 40: 0 Inf
Deciding on bundle 9 of 40: 0 Inf
Deciding on bundle 10 of 40: -Inf -0
Deciding on bundle 11 of 40: 0 Inf
Deciding on bundle 12 of 40: -Inf -0
Deciding on bundle 13 of 40: -Inf -0
Deciding on bundle 14 of 40: NaN Inf
Deciding on bundle 15 of 40: 0 Inf
Deciding on bundle 16 of 40: -Inf -0
Deciding on bundle 17 of 40: 0 Inf
Deciding on bundle 18 of 40: -Inf -0
Deciding on bundle 19 of 40: 0 Inf
Deciding on bundle 20 of 40: -Inf -0
Deciding on bundle 21 of 40: 0 Inf
Deciding on bundle 22 of 40: 0 Inf
Deciding on bundle 23 of 40: -Inf -0
Deciding on bundle 24 of 40: 0 Inf
Deciding on bundle 25 of 40: 0 Inf
Deciding on bundle 26 of 40: -Inf -0
Deciding on bundle 27 of 40: -Inf -0
Deciding on bundle 28 of 40: -Inf -0
Deciding on bundle 29 of 40: -Inf Inf
Deciding on bundle 30 of 40: -Inf -0
Deciding on bundle 31 of 40: -Inf Inf
Deciding on bundle 32 of 40: -Inf -0
Deciding on bundle 33 of 40: -Inf Inf
Deciding on bundle 34 of 40: -Inf -0
Deciding on bundle 35 of 40: -Inf Inf
Deciding on bundle 36 of 40: -Inf -0
Deciding on bundle 37 of 40: -Inf Inf
Deciding on bundle 38 of 40: -Inf -0
Deciding on bundle 39 of 40: 0 Inf
Deciding on bundle 40 of 40: 0 Inf
</pre>
<a id="plots"></a>
<div id="plotoutput">
<img src="preference_regions__01.png" alt=""> <img src="preference_regions__02.png" alt="">
</div>
</div>
</body>
</html>
c4science · Help