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<title>Entropy maximization</title>
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<h1>Entropy maximization</h1>
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<pre class="codeinput">
<span class="comment">% Jo&euml;lle Skaf - 04/24/08</span>
<span class="comment">%</span>
<span class="comment">% Consider the linear inequality constrained entroy maximization problem</span>
<span class="comment">% maximize -sum_{i=1}^n x_i*log(x_i)</span>
<span class="comment">% subject to sum(x) = 1</span>
<span class="comment">% Fx &lt;= g</span>
<span class="comment">% where the variable is x \in \reals^{n}</span>
<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>, 0);
rand(<span class="string">'state'</span>, 0);
n = 20;
m = 10;
p = 5;
tmp = rand(n,1);
A = randn(m,n);
b = A*tmp;
F = randn(p,n);
g = F*tmp + rand(p,1);
<span class="comment">% Entropy maximization</span>
cvx_begin
variable <span class="string">x(n)</span>
maximize <span class="string">sum(entr(x))</span>
A*x == b
F*x &lt;= g
cvx_end
<span class="comment">% Results</span>
display(<span class="string">'The optimal solution is:'</span> );
disp(x);
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<pre class="codeoutput">
Successive approximation method to be employed.
SDPT3 will be called several times to refine the solution.
Original size: 65 variables, 35 equality constraints
20 exponentials add 160 variables, 100 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
20/ 20 | 1.734e+00 1.965e-01 0.000e+00 | Solved
19/ 20 | 1.190e-01 1.021e-03 0.000e+00 | Solved
18/ 20 | 4.647e-03 1.546e-06 0.000e+00 | Solved
1/ 12 | 5.714e-04 2.291e-08 0.000e+00 | Solved
0/ 0 | 0.000e+00 0.000e+00 0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.70335
The optimal solution is:
0.3445
0.3181
0.7539
0.8020
0.6418
0.3517
0.1981
0.2578
0.6373
0.3357
0.7109
0.8998
0.6085
0.6444
0.3061
0.4522
0.8886
0.7801
0.3106
0.6131
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