Warm Starts
A new feature in 1.0 is warm starts for one of our solvers in
OSQP. Having warm starts allows the user to retain data of a
particular problem and change parameters of the problem without
repeating any of the calculations.
Lasso Example
We will demonstrate this in a simple lasso problem:
\[ \begin{array}{ll} \underset{x, \lambda}{\mbox{minimize}} & \frac{1}{2}||Ax - b||_2^2 + \lambda ||x||_1\\ \mbox{subject to} & \lambda > 0 \end{array} \]
#Problem data
set.seed(1)
m <- 2000
n <- 1000
A <- Matrix(rnorm(m*n), nrow = m, ncol = n)
b <- rnorm(m)
#Construct problem
gamma <- Parameter(pos = TRUE)
x <- Variable(n)
obj <- Minimize(.5 * sum_squares(A%*%x - b) + gamma * norm1(x))
constraint <- list(x >= 0)
prob <- Problem(obj, constraint)
#Solve first time
value(gamma) <- 1
result <- solve(prob, solver = "OSQP", warm_start = TRUE) #warm_start = T is not necessary for the first time
cat(sprintf("First solve time is %f\n", result$solve_time))## First solve time is 52.909901#Solve second time
value(gamma) <- 2
result <- solve(prob, solver = "OSQP", warm_start = TRUE)
cat(sprintf("Second solve time is %f\n", result$solve_time))## Second solve time is 2.843947As you can see, there is a very significant speed up with warm starts than without.
Session Info
sessionInfo()## R version 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin21.6.0 (64-bit)
## Running under: macOS Ventura 13.0
##
## Matrix products: default
## BLAS: /usr/local/Cellar/openblas/0.3.21/lib/libopenblasp-r0.3.21.dylib
## LAPACK: /usr/local/Cellar/r/4.2.1_4/lib/R/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 datasets utils methods base
##
## other attached packages:
## [1] Matrix_1.5-1 CVXR_1.0-11
##
## loaded via a namespace (and not attached):
## [1] gmp_0.6-6 Rcpp_1.0.9 bslib_0.4.0 compiler_4.2.1
## [5] jquerylib_0.1.4 tools_4.2.1 digest_0.6.30 bit_4.0.4
## [9] jsonlite_1.8.3 evaluate_0.17 lattice_0.20-45 rlang_1.0.6
## [13] gurobi_9.5-2 cli_3.4.1 Rglpk_0.6-4 yaml_2.3.6
## [17] blogdown_1.13 xfun_0.34 osqp_0.6.0.5 cccp_0.2-9
## [21] fastmap_1.1.0 Rmpfr_0.8-9 stringr_1.4.1 knitr_1.40
## [25] sass_0.4.2 bit64_4.0.5 grid_4.2.1 R6_2.5.1
## [29] rmarkdown_2.17 bookdown_0.29 magrittr_2.0.3 rcbc_0.1.0.9001
## [33] codetools_0.2-18 htmltools_0.5.3 assertthat_0.2.1 Rcplex_0.3-5
## [37] stringi_1.7.8 Rmosek_10.0.25 cachem_1.0.6 slam_0.1-50