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$solve_time))## First solve time is 30.889156#Solve second time
value(gamma) <- 2
result <- solve(prob, solver = "OSQP", warm_start = TRUE)
cat(sprintf("Second solve time is %f\n", result$solve_time))## Second solve time is 1.611814As you can see, there is a very significant speed up with warm starts than without.
Session Info
sessionInfo()## R version 4.4.2 (2024-10-31)
## Platform: x86_64-apple-darwin20
## Running under: macOS Sequoia 15.1
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## time zone: America/Los_Angeles
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices datasets utils methods base
##
## other attached packages:
## [1] Matrix_1.7-1 CVXR_1.0-15
##
## loaded via a namespace (and not attached):
## [1] slam_0.1-54 cli_3.6.3 knitr_1.48 rlang_1.1.4
## [5] xfun_0.49 clarabel_0.9.0.1 gurobi_11.0-0 Rglpk_0.6-5.1
## [9] cccp_0.3-1 assertthat_0.2.1 jsonlite_1.8.9 bit_4.5.0
## [13] Rcplex_0.3-6 Rmosek_10.2.0 htmltools_0.5.8.1 rcbc_0.1.0.9001
## [17] sass_0.4.9 gmp_0.7-5 rmarkdown_2.29 grid_4.4.2
## [21] evaluate_1.0.1 jquerylib_0.1.4 fastmap_1.2.0 yaml_2.3.10
## [25] lifecycle_1.0.4 bookdown_0.41 compiler_4.4.2 codetools_0.2-20
## [29] Rcpp_1.0.13-1 osqp_0.6.3.3 blogdown_1.19 lattice_0.22-6
## [33] digest_0.6.37 R6_2.5.1 bslib_0.8.0 tools_4.4.2
## [37] bit64_4.5.2 Rmpfr_0.9-5 cachem_1.1.0