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<title>Builds a norm minimization tradeoff curve</title>
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<h1>Builds a norm minimization tradeoff curve</h1>
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<pre class="codeinput">
<span class="comment">% This script solves a family of problems of the form</span>
<span class="comment">% minimize || A*x-b ||_1 + gamma * || x ||_Inf</span>
<span class="comment">% for varying values of gamma. For gamma = 0, this is simply an</span>
<span class="comment">% unconstrained norm minimization; for gamma = Inf, x = 0 and</span>
<span class="comment">% || A x - b || = || b || are optimal. Varying gamma allows us</span>
<span class="comment">% to genreate a tradeoff curve between these extremes.</span>
n = 10;
A = randn(2*n,n);
b = randn(2*n,1);
gamma = logspace( -1, 3 );
nrms = zeros( size( gamma ) );
xnrms = zeros( size( gamma ) );
fprintf( 1, <span class="string">'Gamma: '</span> );
<span class="keyword">for</span> k = 1 : length( gamma ),
<span class="keyword">if</span> k &gt; 1 &amp; rem( k, 10 ) == 1, fprintf( 1, <span class="string">'\n '</span> ); <span class="keyword">end</span>
fprintf( 1, <span class="string">'%g '</span>, gamma( k ) );
cvx_begin <span class="string">quiet</span>
variable <span class="string">x(n)</span>
minimize( norm( A * x - b, 1 ) + gamma( k ) * norm( x, Inf ) )
cvx_end
nrms( k ) = norm( A * x - b, 1 );
xnrms( k ) = norm( x, Inf );
<span class="keyword">end</span>
fprintf( 1, <span class="string">'done.\n'</span> );
figure
semilogx( gamma, nrms );
xlabel( <span class="string">'\gamma'</span> );
ylabel( <span class="string">'|| A * x - b ||_1'</span> );
figure
plot( xnrms, nrms );
xlabel( <span class="string">'|| x ||_{\infty}'</span> );
ylabel( <span class="string">'|| A * x - b ||_1'</span> );
</pre>
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<pre class="codeoutput">
Gamma: 0.1 0.120679 0.145635 0.175751 0.212095 0.255955 0.308884 0.372759 0.449843 0.542868
0.655129 0.790604 0.954095 1.1514 1.3895 1.67683 2.02359 2.44205 2.94705 3.55648
4.29193 5.17947 6.25055 7.54312 9.10298 10.9854 13.2571 15.9986 19.307 23.2995
28.1177 33.9322 40.9492 49.4171 59.6362 71.9686 86.8511 104.811 126.486 152.642
184.207 222.3 268.27 323.746 390.694 471.487 568.987 686.649 828.643 1000 done.
</pre>
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<img src="regularized_norm_tradeoff__01.png" alt=""> <img src="regularized_norm_tradeoff__02.png" alt="">
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