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<title>Section 8.5.3: Analytic center of a set of linear inequalities</title>
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<h1>Section 8.5.3: Analytic center of a set of linear inequalities</h1>
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<span class="comment">% Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Jo&Atilde;&laquo;lle Skaf - 04/29/08</span>
<span class="comment">%</span>
<span class="comment">% The analytic center of a set of linear inequalities and equalities:</span>
<span class="comment">% a_i^Tx &lt;= b_i i=1,...,m,</span>
<span class="comment">% Fx = g,</span>
<span class="comment">% is the solution of the unconstrained minimization problem</span>
<span class="comment">% minimize -sum_{i=1}^m log(b_i-a_i^Tx).</span>
<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>, 0);
rand(<span class="string">'state'</span>, 0);
n = 10;
m = 50;
p = 5;
tmp = randn(n,1);
A = randn(m,n);
b = A*tmp + 10*rand(m,1);
F = randn(p,n);
g = F*tmp;
<span class="comment">% Analytic center</span>
cvx_begin
variable <span class="string">x(n)</span>
minimize <span class="string">-sum(log(b-A*x))</span>
F*x == g
cvx_end
disp([<span class="string">'The analytic center of the set of linear inequalities and '</span> <span class="keyword">...</span>
<span class="string">'equalities is: '</span>]);
disp(x);
</pre>
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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: 160 variables, 105 equality constraints
50 exponentials add 400 variables, 250 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
50/ 50 | 2.435e+00 3.573e-01 0.000e+00 | Solved
50/ 50 | 1.982e-01 3.069e-03 0.000e+00 | Solved
49/ 49 | 2.535e-02 4.842e-05 0.000e+00 | Solved
11/ 46 | 1.342e-03 1.284e-07 0.000e+00 | Solved
0/ 6 | 1.582e-04 1.762e-09 0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -64.8504
The analytic center of the set of linear inequalities and equalities is:
-0.3618
-1.5333
0.1387
0.2491
-1.1163
1.3142
1.2303
-0.0511
0.4031
0.1248
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