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<title>Rmosek: Primal vs Dual with l_1 regularization</title>

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<h1 class="title toc-ignore"><code>Rmosek</code>: Primal vs Dual with <span class="math inline">\(l_1\)</span> regularization</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-06-23</em></h4>

</div>


<!-- The file analysis/chunks.R contains chunks that define default settings
shared across the workflowr files. -->
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2017-11-07</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
 package git2r is installed -->
<p><strong>Code version:</strong> 2c05d59</p>
<!-- Add your analysis here -->
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Following <a href="rmosek_primal_dual.html">previous simulation</a>, we are adding <span class="math inline">\(l_1\)</span> regularization to the primal form such that</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{f \in \mathbb{R}^m, \ \ g \in \mathbb{R}^n} &amp; -\sum\limits_{i = 1}^n\log\left(g_i\right) + \sum\limits_{j = 1}^m\lambda_j\left|f_j\right| \\
\text{s.t.} &amp; Af + a = g\\
&amp; g \geq 0 \ .
\end{array}
\]</span></p>
<p>Its dual form is</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{\nu \in \mathbb{R}^n} &amp; a^T\nu-\sum\limits_{i = 1}^n\log\left(\nu_i\right) \\
\text{s.t.} &amp; \left|A^T\nu\right| \leq \lambda\\
&amp; \nu\geq0 \ .
\end{array}
\]</span></p>
<p><strong>Right now we haven’t figured out how to program the <span class="math inline">\(l_1\)</span> regularized primal form in <code>Rmosek</code>, so here we are only comparing the dual form with or without regularization.</strong></p>
</div>
<div id="simulation" class="section level2">
<h2>Simulation</h2>
<p>Let <span class="math inline">\(\lambda\)</span> be</p>
<p><span class="math display">\[
\lambda_i = \begin{cases}
0 &amp; i \text{ odd ;}\\
a / \rho^{i/2} &amp; i \text{ even .}\\
\end{cases}
\]</span> with <span class="math inline">\(a = 10\)</span>, <span class="math inline">\(\rho = 0.5\)</span>. <span class="math inline">\(n = 10^4\)</span>, <span class="math inline">\(m = 10\)</span>, <span class="math inline">\(A\)</span> and <span class="math inline">\(a\)</span> are generated in the <a href="rmosek_primal_dual.html">same</a> way.</p>
<p>The dual optimization in all <span class="math inline">\(1000\)</span> simulation trials reaches the optimal solution both with and without regularization.</p>
</div>
<div id="total-time-cost" class="section level2">
<h2>Total time cost</h2>
<p><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-3-2.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="number-of-iterations" class="section level2">
<h2>Number of iterations</h2>
<p><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="time-per-iteration" class="section level2">
<h2>Time per iteration</h2>
<p><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/rmosek_primal_dual_2.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.2 (2017-09-28)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Sierra 10.12.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] compiler_3.4.2  backports_1.1.1 magrittr_1.5    rprojroot_1.2  
 [5] tools_3.4.2     htmltools_0.3.6 yaml_2.1.14     Rcpp_0.12.13   
 [9] stringi_1.1.5   rmarkdown_1.6   knitr_1.17      git2r_0.19.0   
[13] stringr_1.2.0   digest_0.6.12   evaluate_0.10.1</code></pre>
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