Solve a conic optimization problem — ECOS

The function ECOS_csolve is a wrapper around the ecos csolve C function. Conic constraints are specified using the \(G\) and \(h\) parameters and can be NULL and zero length vector respectively indicating an absence of conic constraints. Similarly, equality constraints are specified via \(A\) and \(b\) parameters with NULL and empty vector values representing a lack of such constraints. At most one of the pair \((G , h)\) or \((A, b)\) is allowed to be absent.

ECOS_csolve(
  c = numeric(0),
  G = NULL,
  h = numeric(0),
  dims = list(l = integer(0), q = NULL, e = integer(0)),
  A = NULL,
  b = numeric(0),
  bool_vars = integer(0),
  int_vars = integer(0),
  control = ecos.control()
)

Arguments

c: the coefficients of the objective function; the length of this determines the number of variables \(n\) in the problem.
G: the inequality constraint matrix in one of three forms: a plain matrix, simple triplet matrix, or compressed column format, e.g. dgCMatrix-class. Can also be NULL
h: the right hand size of the inequality constraint. Can be empty numeric vector.
dims: is a list of three named elements: dims['l'] an integer specifying the dimension of positive orthant cone, dims['q'] an integer vector specifying dimensions of second-order cones, dims['e'] an integer specifying the number of exponential cones
A: the optional equality constraint matrix in one of three forms: a plain matrix, simple triplet matrix, or compressed column format, e.g. dgCMatrix-class. Can be NULL
b: the right hand side of the equality constraint, must be specified if \(A\) is. Can be empty numeric vector.
bool_vars: the indices of the variables, 1 through \(n\), that are boolean; that is, they are either present or absent in the solution
int_vars: the indices of the variables, 1 through \(n\), that are integers
control: is a named list that controls various optimization parameters; see ecos.control.

Value

a list of 8 named items

x

primal variables

y

dual variables for equality constraints

s

slacks for \(Gx + s <= h\), \(s \in K\)

z

dual variables for inequality constraints \(s \in K\)

infostring

gives information about the status of solution

retcodes

a named integer vector containing four elements

exitflag: 0=ECOS_OPTIMAL, 1=ECOS_PINF, 2=ECOS_DINF, 10=ECOS_INACC_OFFSET, -1=ECOS_MAXIT, -2=ECOS_NUMERICS, -3=ECOS_OUTCONE, -4=ECOS_SIGINT, -7=ECOS_FATAL. See ECOS_exitcodes
iter: the number of iterations used
mi_iter: the number of iterations for mixed integer problems
numerr: a non-zero number if a numeric error occurred

summary

a named numeric vector containing

pcost: value of primal objective
dcost: value of dual objective
pres: primal residual on inequalities and equalities
dres: dual residual
pinf: primal infeasibility measure
dinf: dual infeasibility measure
pinfres: primal infeasibility residual
dinfres: dual infeasibility residual
gap: duality gap
relgap: relative duality gap
r0: Unknown at the moment to this R package maintainer.

timing

a named numeric vector of timing information consisting of

runtime: the total runtime in ecos
tsetup: the time for setup of the problem
tsolve: the time to solve the problem

Details

A call to this function will solve the problem: minimize \(c^Tx\), subject to \(Ax = b\), and \(h - G*x \in K\).

Variables can be constrained to be boolean (1 or 0) or integers. This is indicated by specifying parameters bool_vars and/or int_vars respectively. If so indicated, the solutions will be found using a branch and bound algorithm.

Examples


## githubIssue98
cat("Basic matrix interface\n")
#> Basic matrix interface
Gmat <- matrix(c(0.416757847405471, 2.13619609566845, 1.79343558519486, 0, 0,
                 0, 0, -1, 0, 0, 0, 0.056266827226329, -1.64027080840499, 0.841747365656204,
                 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0.416757847405471, 2.13619609566845,
                 1.79343558519486, 0, 0, 0, -1, 0, 0, 0, 0, 0.056266827226329, -1.64027080840499,
                 0.841747365656204, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0), ncol = 5L)
c <- as.numeric(c(0, 0, 0, 0, 1))
h <- as.numeric(c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0))
dims <- list(l = 6L, q = 5L, e = 0L)
ECOS_csolve(c = c, G = Gmat, h = h,
           dims = dims,
           A = NULL, b = numeric(0))
#> $x
#> [1]  1.479062e-11  4.332909e-12  1.478861e-11  4.273179e-12 -3.372330e-10
#> 
#> $y
#> numeric(0)
#> 
#> $s
#>  [1] 5.007707e-11 2.112317e-10 1.720096e-10 5.008123e-11 2.111387e-10
#>  [6] 1.720630e-10 2.502815e-10 1.887750e-11 4.007416e-12 1.887598e-11
#> [11] 4.038335e-12
#> 
#> $z
#>  [1]  0.36678432  0.07561476  0.09859791  0.36678392  0.07561520  0.09859751
#>  [7]  1.00000000  0.49121720 -0.02039636  0.49121725 -0.02039743
#> 
#> $infostring
#> [1] "Optimal solution found"
#> 
#> $retcodes
#> exitFlag     iter  mi_iter   numerr 
#>        0        6       -1        0 
#> 
#> $summary
#>         pcost         dcost          pres          dres          pinf 
#> -3.372330e-10  0.000000e+00  4.264405e-10  5.482329e-11  0.000000e+00 
#>          dinf       pinfres       dinfres           gap        relgap 
#>  0.000000e+00           NaN           NaN  1.738783e-09  5.156029e+00 
#>            r0 
#>  1.000000e-08 
#> 
#> $timing
#>    runtime     tsetup     tsolve 
#> 5.2298e-05 2.0759e-05 3.1539e-05 
#> 

cat("Simple Triplet Matrix interface, if you have package slam\n")
#> Simple Triplet Matrix interface, if you have package slam
if (requireNamespace("slam")) {

  ECOS_csolve(c = c, G = slam::as.simple_triplet_matrix(Gmat), h = h,
              dims = dims,
              A = NULL, b = numeric(0))
}
#> Loading required namespace: slam
#> $x
#> [1]  1.479062e-11  4.332909e-12  1.478861e-11  4.273179e-12 -3.372330e-10
#> 
#> $y
#> numeric(0)
#> 
#> $s
#>  [1] 5.007707e-11 2.112317e-10 1.720096e-10 5.008123e-11 2.111387e-10
#>  [6] 1.720630e-10 2.502815e-10 1.887750e-11 4.007416e-12 1.887598e-11
#> [11] 4.038335e-12
#> 
#> $z
#>  [1]  0.36678432  0.07561476  0.09859791  0.36678392  0.07561520  0.09859751
#>  [7]  1.00000000  0.49121720 -0.02039636  0.49121725 -0.02039743
#> 
#> $infostring
#> [1] "Optimal solution found"
#> 
#> $retcodes
#> exitFlag     iter  mi_iter   numerr 
#>        0        6       -1        0 
#> 
#> $summary
#>         pcost         dcost          pres          dres          pinf 
#> -3.372330e-10  0.000000e+00  4.264405e-10  5.482329e-11  0.000000e+00 
#>          dinf       pinfres       dinfres           gap        relgap 
#>  0.000000e+00           NaN           NaN  1.738783e-09  5.156029e+00 
#>            r0 
#>  1.000000e-08 
#> 
#> $timing
#>    runtime     tsetup     tsolve 
#> 4.4032e-05 1.4567e-05 2.9465e-05 
#> 

if (requireNamespace("Matrix")) {
   ECOS_csolve(c = c, G = Matrix::Matrix(Gmat), h = h,
               dims = dims,
               A = NULL, b = numeric(0))
}
#> Loading required namespace: Matrix
#> $x
#> [1]  1.479062e-11  4.332909e-12  1.478861e-11  4.273179e-12 -3.372330e-10
#> 
#> $y
#> numeric(0)
#> 
#> $s
#>  [1] 5.007707e-11 2.112317e-10 1.720096e-10 5.008123e-11 2.111387e-10
#>  [6] 1.720630e-10 2.502815e-10 1.887750e-11 4.007416e-12 1.887598e-11
#> [11] 4.038335e-12
#> 
#> $z
#>  [1]  0.36678432  0.07561476  0.09859791  0.36678392  0.07561520  0.09859751
#>  [7]  1.00000000  0.49121720 -0.02039636  0.49121725 -0.02039743
#> 
#> $infostring
#> [1] "Optimal solution found"
#> 
#> $retcodes
#> exitFlag     iter  mi_iter   numerr 
#>        0        6       -1        0 
#> 
#> $summary
#>         pcost         dcost          pres          dres          pinf 
#> -3.372330e-10  0.000000e+00  4.264405e-10  5.482329e-11  0.000000e+00 
#>          dinf       pinfres       dinfres           gap        relgap 
#>  0.000000e+00           NaN           NaN  1.738783e-09  5.156029e+00 
#>            r0 
#>  1.000000e-08 
#> 
#> $timing
#>    runtime     tsetup     tsolve 
#> 5.6025e-05 2.4145e-05 3.1880e-05 
#> 

## Larger problems using saved data can be found in the test suite.
## Here is one
if (requireNamespace("Matrix")) {
  MPC01 <- readRDS(system.file("testdata", "MPC01_1.RDS", package = "ECOSolveR"))
  G <- Matrix::sparseMatrix(x = MPC01$Gpr, i = MPC01$Gir, p = MPC01$Gjc,
                            dims = c(MPC01$m, MPC01$n), index1 = FALSE)
  h <- MPC01$h
  dims <- lapply(list(l = MPC01$l, q=MPC01$q, e=MPC01$e), as.integer)
  retval <- ECOS_csolve(c = MPC01$c, G=G, h = h, dims = dims, A = NULL, b = NULL,
                        control = ecos.control(verbose=1L))
  retval$retcodes
  retval$infostring
  retval$summary
}
#>         pcost         dcost          pres          dres          pinf 
#> -2.112370e+02 -2.112370e+02  2.387958e-12  7.134616e-13  0.000000e+00 
#>          dinf       pinfres       dinfres           gap        relgap 
#>  0.000000e+00           NaN  4.964435e-01  1.111838e-06  5.263464e-09 
#>            r0 
#>  1.000000e-08