Clarabel solves linear programs (LPs), quadratic programs (QPs), second-order cone programs (SOCPs) and semidefinite programs (SDPs). It also solves problems with exponential and power cone constraints. The specific problem solved is:

Minimize $$\frac{1}{2}x^TPx + q^Tx$$ subject to $$Ax + s = b$$ $$s \in K$$ where $$x \in R^n$$, $$s \in R^m$$, $$P = P^T$$ and nonnegative-definite, $$q \in R^n$$, $$A \in R^{m\times n}$$, and $$b \in R^m$$. The set $$K$$ is a composition of convex cones.

clarabel(A, b, q, P = NULL, cones, control = list(), strict_cone_order = TRUE)

## Arguments

A

a matrix of constraint coefficients.

b

a numeric vector giving the primal constraints

q

a numeric vector giving the primal objective

P

a symmetric positive semidefinite matrix, default NULL

cones

a named list giving the cone sizes, see “Cone Parameters” below for specification

control

a list giving specific control parameters to use in place of default values, with an empty list indicating the default control parameters. Specified parameters should be correctly named and typed to avoid Rust system panics as no sanitization is done for efficiency reasons

strict_cone_order

a logical flag, default TRUE for forcing order of cones described below. If FALSE cones can be specified in any order and even repeated and directly passed to the solver without type and length checks

## Value

named list of solution vectors x, y, s and information about run

## Details

The order of the rows in matrix $$A$$ has to correspond to the order given in the table “Cone Parameters”, which means means rows corresponding to primal zero cones should be first, rows corresponding to non-negative cones second, rows corresponding to second-order cone third, rows corresponding to positive semidefinite cones fourth, rows corresponding to exponential cones fifth and rows corresponding to power cones at last.

When the parameter strict_cone_order is FALSE, one can specify the cones in any order and even repeat them in the order they appear in the A matrix. See below.

### Clarabel can solve

1. linear programs (LPs)

2. second-order cone programs (SOCPs)

3. exponential cone programs (ECPs)

4. power cone programs (PCPs)

5. problems with any combination of cones, defined by the parameters listed in “Cone Parameters” below

### Cone Parameters

The table below shows the cone parameter specifications. Mathematical definitions are in the vignette.

 Parameter Type Length Description z integer $$1$$ number of primal zero cones (dual free cones), which corresponds to the primal equality constraints l integer $$1$$ number of linear cones (non-negative cones) q integer $$\ge 1$$ vector of second-order cone sizes s integer $$\ge 1$$ vector of positive semidefinite cone sizes ep integer $$1$$ number of primal exponential cones p numeric $$\ge 1$$ vector of primal power cone parameters gp list $$\ge 1$$ list of named lists of two items, a : a numeric vector of at least 2 exponent terms in the product summing to 1, and n : an integer dimension of generalized power cone parameters

When the parameter strict_cone_order is FALSE, one can specify the cones in the order they appear in the A matrix. The cones argument in such a case should be a named list with names matching ^z* indicating primal zero cones, ^l* indicating linear cones, and so on. For example, either of the following would be valid: list(z = 2L, l = 2L, q = 2L, z = 3L, q = 3L), or, list(z1 = 2L, l1 = 2L, q1 = 2L, zb = 3L, qx = 3L), indicating a zero cone of size 2, followed by a linear cone of size 2, followed by a second-order cone of size 2, followed by a zero cone of size 3, and finally a second-order cone of size 3. Generalized power cones parameters have to specified as named lists, e.g., list(z = 2L, gp1 = list(a = c(0.3, 0.7), n = 3L), gp2 = list(a = c(0.5, 0.5), n = 1L)).

Note that when strict_cone_order = FALSE, types of cone parameters such as integers, reals have to be explicit since the parameters are directly passed to the Rust interface with no sanity checks.!

clarabel_control()

## Examples

A <- matrix(c(1, 1), ncol = 1)
b <- c(1, 1)
obj <- 1
cone <- list(z = 2L)
control <- clarabel_control(tol_gap_rel = 1e-7, tol_gap_abs = 1e-7, max_iter = 100)
clarabel(A = A, b = b, q = obj, cones = cone, control = control)
#> $x #> [1] 1 #> #>$z
#> [1] -0.5 -0.5
#>
#> $s #> [1] 0 0 #> #>$obj_val
#> [1] 1
#>
#> $obj_val_dual #> [1] 1 #> #>$status
#> [1] 2
#>
#> $solve_time #> [1] 0.000111267 #> #>$iterations
#> [1] 0
#>
#> $r_prim #> [1] 0 #> #>$r_dual
#> [1] 0
#>
#> $info #>$info$μ #> [1] 1 #> #>$info$sigma #> [1] 1 #> #>$info$step_length #> [1] 0 #> #>$info$cost_primal #> [1] 1 #> #>$info$cost_dual #> [1] 1 #> #>$info$res_primal #> [1] 0 #> #>$info$res_dual #> [1] 0 #> #>$info$res_primal_inf #> [1] 1 #> #>$info$res_dual_inf #> [1] 1.414214 #> #>$info$gap_abs #> [1] 0 #> #>$info$gap_rel #> [1] 0 #> #>$info$ktratio #> [1] 1 #> #>$info$solve_time #> [1] 0.000111267 #> #>$info$iterations #> [1] 0 #> #>$info\$status
#> [1] 2
#>
#>