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我正在尝试用 R 中的 CVXR 解决混合整数问题。以下代码用于解决它：

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)

这给出了以下错误：

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

当我将代码更改为以下代码时：

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

它有效，但这些不是我想要的限制。

有谁知道如何解决这个问题？

4

1 回答 1

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我们可以通过引入一个布尔矩阵来凸化这个问题，如果第 i,j 个不等式大于y，则为 1，否则为 0。同样，如果第 i,j 个不等式大于，则为 1，否则为 0。因此，我们添加了 2*(n-1)*n 个布尔变量。也设置为 9 并为避免数值困难设置为 0.1。有关更多信息，请参阅：https ://math.stackexchange.com/questions/37075/how-can-not-equals-be-expressed-as-an-inequality-for-a-linear-programming-model/1517850y[i,j]diff(beta)yy[i,j]diff(t(beta))Mepsilon

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
于 2021-01-06T16:12:14.937 回答