# box_qp_f: box constrained Quadratic Program (QP)

Description Usage Arguments Details Value Author(s) References Examples

### Description

box_qp_f solves the minimization problem

\mathrm{minimize}_{u}\;\; (b +u)' Q (b + u );\;\; \mathrm{subject\;\; to}\;\; \|u\|_∞ ≤q ρ

where Q_{m \times m} is symmetric PSD, u,b \in \Re^m. The algorithm used is one-at-a-time cyclical coordinate descent.

### Usage

 1 box_qp_f(Q, u, b, rho, Maxiter, tol = 10^-4) 

### Arguments

 Q (Required) is a symmetric PSD matrix of dimension m \times m. This is a problem parameter. u (Required) is the optimization variable, a vector of length m. The value of u serves as an initialization for the coordinate-wise algorithm. If a suitable starting point is unavailable, start with u = 0 b (Required) is a vector of length m, this is a problem parameter. rho (Required) is the degree of shrinkage. This is a non-negative scalar. Maxiter (Required) is an integer denoting the maximum number of iterations (full sweeps across all the m variables), to be performed by box_qp_f. tol is the convergence tolerance. It is a real positive number (defaults to 10^-4). box_qp_f converges if the relative difference of the objective values is less than tol.

### Details

This box QP function is a R wrapper to a Fortran code. This is primarily meant to be called from the R function dpglasso. One needs to be very careful (as in supplying the inputs of the progra properly) while using this as a stand alone program.

### Value

 u the optimal value of the argument u, upon convergence grad_vec the gradient of the objective function at u

### Author(s)

Rahul Mazumder and Trevor Hastie

### References

This algorithm is used as a part of the algorithm DPGLASSO described in our paper: “The Graphical Lasso: New Insights and Alternatives by Rahul Mazumder and Trevor Hastie" available at http://arxiv.org/abs/1111.5479

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 set.seed(2008) # create problem data m<-20; aa<-array(rnorm(m^2),dim=c(m,m)); Q<-aa Q<- Q + diag(rep(0.1,m)); b<-rnorm(m); soln<-box_qp_f(Q, u=rep(0,m), b, rho=.2 , Maxiter=1000, tol = 10^-4) 

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