Rmosek: Primal vs Dual with \(l_1\) regularization

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

Number of iterations

Time per iteration

Session information

sessionInfo()

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