Warm Starts

Version 1.0 features

Author

Anqi Fu, David Kang and Balasubramanian Narasimhan

Starting with version 0.11.1, CVXR supports Parameter objects that allow you to change problem data and re-solve without reconstructing the problem from scratch. This uses the EvalParams reduction under the hood to substitute parameter values before each solve.

Lasso Example

We demonstrate the parametric workflow in a simple lasso problem:

\[ \begin{array}{ll} \underset{x}{\mbox{minimize}} & \frac{1}{2}\|Ax - b\|_2^2 + \gamma \|x\|_1\\ \mbox{subject to} & x \geq 0 \end{array} \]

where \(\gamma > 0\) is a regularization parameter.

## Problem data
set.seed(1)
m <- 200
n <- 100
A <- matrix(rnorm(m*n), nrow = m, ncol = n)
b <- rnorm(m)

## Define the problem once using a Parameter
x <- Variable(n)
gamma <- Parameter(nonneg = TRUE)
obj <- Minimize(.5 * sum_squares(A %*% x - b) + gamma * norm1(x))
constraint <- list(x >= 0)
prob <- Problem(obj, constraint)

## Sweep over gamma values
gamma_vals <- c(0.1, 0.5, 1, 2, 5, 10)
timings <- numeric(length(gamma_vals))
opt_vals <- numeric(length(gamma_vals))

for (i in seq_along(gamma_vals)) {
    value(gamma) <- gamma_vals[i]
    timings[i] <- system.time(opt_vals[i] <- psolve(prob, solver = "OSQP"))["elapsed"]
    check_solver_status(prob)
}

results <- data.frame(
    gamma = gamma_vals,
    optimal_value = round(opt_vals, 4),
    elapsed = round(timings, 3)
)

The first solve includes one-time compilation overhead; subsequent re-solves skip this step entirely:

knitr::kable(results, col.names = c("$\\gamma$", "Optimal Value", "Elapsed (s)"))

\(\gamma\)	Optimal Value	Elapsed (s)
0.1	69.0649	0.343
0.5	70.3046	0.109
1.0	71.8543	0.022
2.0	74.9537	0.021
5.0	84.2518	0.021
10.0	99.7486	0.021

Session Info

R version 4.5.3 (2026-03-11)
Platform: aarch64-apple-darwin20
Running under: macOS Tahoe 26.3.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

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 utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.7-4 CVXR_1.8.1  

loaded via a namespace (and not attached):
 [1] slam_0.1-55       cli_3.6.5         knitr_1.51        ECOSolveR_0.6.1  
 [5] rlang_1.1.7       xfun_0.56         clarabel_0.11.2   otel_0.2.0       
 [9] gurobi_13.0-1     Rglpk_0.6-5.1     highs_1.12.0-3    cccp_0.3-3       
[13] scs_3.2.7         S7_0.2.1          jsonlite_2.0.0    backports_1.5.0  
[17] rprojroot_2.1.1   htmltools_0.5.9   Rmosek_11.1.1     gmp_0.7-5.1      
[21] piqp_0.6.2        rmarkdown_2.30    grid_4.5.3        evaluate_1.0.5   
[25] fastmap_1.2.0     yaml_2.3.12       compiler_4.5.3    codetools_0.2-20 
[29] htmlwidgets_1.6.4 Rcpp_1.1.1        here_1.0.2        osqp_1.0.0       
[33] lattice_0.22-9    digest_0.6.39     checkmate_2.3.4   tools_4.5.3

Lasso Example

Session Info

References