SOCP 1

Description

maximize \ \ x + y

subject \ to \ \ x^2 + y^2 \leq 1

x \geq 0, y \geq 0

Examples

library(ROI)
obj <- L_objective(c(1, 1))
## NOTE: chol(diag(2)) == diag(2)
con <- C_constraint(L = rbind(0, -diag(2)), 
                    cones = K_soc(3), 
                    rhs = c(1, 0, 0))
op <- OP(obj, con, maximum=TRUE)
x <- ROI_solve(op, solver="ecos")
x
## Optimal solution found.
## The objective value is: 1.414214e+00
solution(x)
## [1] 0.7071068 0.7071068

SOCP 2

Description

The following example is also known as Problem 10 from the Hock-Schittkowski-Collection Hock and Schittkowski (1981).

minimize \ \ x - y \\

subject \ to \ \ -3 x^2 + 2 x y + 1 \geq 0

References

W. Hock, K. Schittkowski (1981): Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer

Examples

library(ROI)
obj <- L_objective(c(1, -1))
L <- chol(rbind(c(3, -1), c(-1, 1)))
con <- C_constraint(L = rbind(0, -L), 
                    cones = K_soc(3), 
                    rhs = c(1, 0, 0))
op <- OP(objective = obj, constraints = con, 
         bounds = V_bound(li=1:2, lb=rep(-Inf, 2)))
x <- ROI_solve(op, solver="ecos")
x
## Optimal solution found.
## The objective value is: -1.000000e+00
solution(x)
## [1] 1.996387e-10 1.000000e+00

SOCP 3

Description

The following example is originally from the CVXOPT (http://cvxopt.org/userguide/coneprog.html) homepage.

minimize \ \ -2x_1 + x_2 + 5 x_3

subject to

\left\| \begin{array}{c} -13 x_1 + 3 x_2 + 5 x_3 - 3 \\ -12 x_1 + 12 x_2 - 6 x_3 - 2 \end{array} \right\|_2 \leq -12 x_1 - 6 x_2 + 5 x_3 - 12

\left\| \begin{array}{c} -3 x_1 + 6 x_2 + 2 x_3 \\ x_1 + 9 x_2 + 2 x_3 + 3 \\ - x_1 - 19 x_2 + 3 x_3 - 42 \end{array} \right\|_2 \leq -3 x_1 + 6 x_2 - 10 x_3 + 27

References

[CVXOPT] Andersen, Martin S and Dahl, Joachim and Vandenberghe, Lieven (2016) CVXOPT: A Python package for convex optimization, version 1.1.8, http://cvxopt.org/

Examples

library(ROI)
lo <- L_objective(c(-2, 1, 5))
lc1 <- rbind(c(12, 6, -5), c(13, -3, -5), c(12, -12, 6))
lc2 <- rbind(c(3, -6, 10), c(3, -6, -2), c(-1, -9, -2), c(1, 19, -3))
lc <- C_constraint(L = rbind(lc1, lc2), cones = K_soc(c(3, 4)), 
                   rhs=c(c(-12, -3, -2), c(27, 0, 3, -42)))
vb <- V_bound(li=1:3, lb=rep(-Inf, 3))
op <- OP(objective = lo, constraints = lc, bounds = vb)
x <- ROI_solve(op, solver="ecos")
x
## Optimal solution found.
## The objective value is: -3.834637e+01
solution(x)
## [1] -5.014767 -5.766924 -8.521796