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)a matrix of constraint coefficients.
a numeric vector giving the primal constraints
a numeric vector giving the primal objective
a symmetric positive semidefinite matrix, default
NULL
a named list giving the cone sizes, see “Cone Parameters” below for specification
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
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
named list of solution vectors x, y, s and information about run
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.
linear programs (LPs)
second-order cone programs (SOCPs)
exponential cone programs (ECPs)
power cone programs (PCPs)
problems with any combination of cones, defined by the parameters listed in “Cone Parameters” below
The table below shows the cone parameter specifications
| 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 | \(\geq1\) | vector of second-order cone sizes | |
s | integer | \(\geq1\) | vector of positive semidefinite cone sizes | |
ep | integer | \(1\) | number of primal exponential cones | |
p | numeric | \(\geq1\) | vector of primal 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.
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
#>
#> $status
#> [1] 2
#>
#> $solve_time
#> [1] 4.5391e-05
#>
#> $iterations
#> [1] 3
#>
#> $r_prim
#> [1] 0
#>
#> $r_dual
#> [1] 0
#>