Solving optimization problem with CVXR in R, using several constraints

Question

I am trying to solve an mixed integer problem with CVXR in R. Following code is used to solve it:

n <- 6
beta <- Variable(n, n, integer = TRUE)
epsilon <- 0.1*10^-5
objective <- Minimize(1)
constraints <- list(beta >= 1,
                    beta <= 9,
                    abs(diff(beta)) >= epsilon,
                    abs(diff(t(beta))) >= epsilon)

prob <- Problem(objective, constraints)
CVXR_result <- solve(prob)

This gives the following error:

Error in construct_intermediate_chain(object, candidate_solvers, gp = gp) : 
  Problem does not follow DCP rules.

When I change the code into following code:

n <- 6
beta <- Variable(n, n, integer = TRUE)
epsilon <- 0.1*10^-5
objective <- Minimize(1)
constraints <- list(beta >= 1,
                    beta <= 9,
                    abs(diff(beta)) <= epsilon,
                    abs(diff(t(beta))) <= epsilon)

prob <- Problem(objective, constraints)
CVXR_result <- solve(prob)

CVXR_result$status

CVXR_result$value
cvxrBeta <- CVXR_result$getValue(beta)
cvxrBeta

It works, but these are not the constraints that I want.

Does anyone know how to solve this?

Answer 1

We can convexivy the problem by introducing a boolean matrix y such that y[i,j] is 1 if the i,jth inequality on diff(beta) is greater-than and 0 otherwise. Similarly yy[i,j] is 1 if the i,jth inequality on diff(t(beta)) is greater-than and 0 otherwise. Thus we have added 2*(n-1)*n boolean variables. Also set M to 9 and to avoid numerical difficulties set epsilon to 0.1. For more information see: https://math.stackexchange.com/questions/37075/how-can-not-equals-be-expressed-as-an-inequality-for-a-linear-programming-model/1517850

library(CVXR)

n <- 6
epsilon <- 0.1
M <- 9

beta <- Variable(n, n, integer = TRUE)
y <- Variable(n-1, n, boolean = TRUE)
yy <- Variable(n-1, n, boolean = TRUE)

objective <- Minimize(1)
constraints <- list(beta >= 1,
                    beta <= M,
                    diff(beta) <= -epsilon + 2*M*y,
                    diff(beta) >= epsilon - (1-y)*2*M,
                    diff(t(beta)) <= -epsilon + 2*M*yy,
                    diff(t(beta)) >= epsilon - (1-yy)*2*M)

prob <- Problem(objective, constraints)
CVXR_result <- solve(prob)

CVXR_result$status
## [1] "optimal"

CVXR_result$getValue(beta)
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    9    1    9    8    7
## [2,]    9    8    7    6    9    4
## [3,]    3    2    1    9    8    2
## [4,]    7    6    2    1    7    6
## [5,]    3    5    3    2    8    5
## [6,]    5    1    4    3    6    9