Last updated: 2018-05-12

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    rmd b9c4ac9 LSun 2017-06-23 wflow_publish(“rmosek_primal_dual_2.rmd”)

Introduction

Following previous simulation, we are adding \(l_1\) regularization to the primal form such that

\[ \begin{array}{rl} \min\limits_{f \in \mathbb{R}^m, \ \ g \in \mathbb{R}^n} & -\sum\limits_{i = 1}^n\log\left(g_i\right) + \sum\limits_{j = 1}^m\lambda_j\left|f_j\right| \\ \text{s.t.} & Af + a = g\\ & g \geq 0 \ . \end{array} \]

Its dual form is

\[ \begin{array}{rl} \min\limits_{\nu \in \mathbb{R}^n} & a^T\nu-\sum\limits_{i = 1}^n\log\left(\nu_i\right) \\ \text{s.t.} & \left|A^T\nu\right| \leq \lambda\\ & \nu\geq0 \ . \end{array} \]

Right now we haven’t figured out how to program the \(l_1\) regularized primal form in Rmosek, so here we are only comparing the dual form with or without regularization.

Simulation

Let \(\lambda\) be

\[ \lambda_i = \begin{cases} 0 & i \text{ odd ;}\\ a / \rho^{i/2} & i \text{ even .}\\ \end{cases} \] with \(a = 10\), \(\rho = 0.5\). \(n = 10^4\), \(m = 10\), \(A\) and \(a\) are generated in the same way.

The dual optimization in all \(1000\) simulation trials reaches the optimal solution both with and without regularization.

Total time cost

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Number of iterations

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Time per iteration

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Session information

sessionInfo()
R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.4

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] workflowr_1.0.1   Rcpp_0.12.16      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.6     whisker_0.3-2     R.oo_1.21.0      
[13] R.utils_2.6.0     rmarkdown_1.9     tools_3.4.3      
[16] stringr_1.3.0     yaml_2.1.18       compiler_3.4.3   
[19] htmltools_0.3.6   knitr_1.20

This reproducible R Markdown analysis was created with workflowr 1.0.1