R中整数变量的非线性优化/编程

Question

我想知道是否有人能够建议一些软件包来解决非线性优化问题，该问题可以为最佳解决方案提供整数变量？问题是最小化具有等式约束的函数，该函数受一些下边界和上边界约束。

我在 R 中使用了 'nloptr' 包来解决非线性优化问题，该问题效果很好，但现在想扩展该方法以将某些变量作为整数。从我到目前为止对 nloptr 的使用和理解来看，它只能返回连续变量，而不是整数变量以获得最佳解决方案。

我相信这类问题需要使用混合整数非线性规划来解决。

nloptr 形式的问题的一个示例：

min f(x) (x-y)^2/y + (p-q)^2/q
so that (x-y)^2/y + (p-q)^2/q = 10.2

where 
x and p are positive integers not equal to 0 
and 
y and q may or may not be positive integers not equal to 0

R 中的 nloptr 代码如下所示

library('nloptr')

x1 <- c(50,25,20,15)

fn <- function(x) {
  (((x[1] - x[2])^2)/x[2]) + (((x[3] - x[4])^2)/x[4])
  }

heq <- function(x) {
  fn(x)-10.2
}

lower_limit <- c(0,0,0,0)
upper_limit <- c(67.314, 78, 76.11, 86)


slsqp(x1, fn, lower = lower_limit, upper = upper_limit,  hin = NULL, heq = heq, control = list(xtol_rel = 1e-8, check_derivatives = FALSE))

这将输出以下内容：

$par
[1] 46.74823 29.72770 18.93794 16.22137

$value
[1] 10.2

$iter
[1] 6

$convergence
[1] 4

$message
[1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."

这是我正在寻找的那种结果，但如上所述，我需要 x 和 p 作为整数。

我看过https://cran.r-project.org/web/views/Optimization.html，它有一个非常好的混合整数非线性编程包列表，但只是想知道是否有人有经验他们中的任何一个以及他们认为最适合解决上述问题的方法。

大约 7 年前在这里发布了一个类似的问题，但它最终提供了一个指向 cran 页面的链接，因此认为值得再问一次。

非常感谢您的意见。

干杯，

安德鲁

Answer 1

这是一个示例，说明它在没有更简单的目标函数的情况下如何不适用于 CVXR。我怀疑问题不是凸的，即使有约束，所以需要一个替代选项。

#base example from https://cvxr.rbind.io/cvxr_examples/cvxr_gentle-intro/
#install.packages("CVXR")
library(CVXR)

#modified for Stackoverflow integer MIQP ####
#Solves, but terms not normalised by y and q respectively

# Variables minimized over
x <- Variable(1, integer=TRUE)
y <- Variable(1)
p <- Variable(1, integer=TRUE)
q <- Variable(1)

# Problem definition (terms not normalised by y and q respectively)
objective <- Minimize((x - y)^2 + (p - q)^2)
constraints <- list(x >= 0, y >= 0, p >= 0, q >= 0, 
                    x <= 67.314, y <= 78, p <= 76.11, q <= 86)
prob2.1 <- Problem(objective, constraints)

# Problem solution
solution2.1 <- solve(prob2.1)
solution2.1$status
solution2.1$value
solution2.1$getValue(x)
solution2.1$getValue(y)
solution2.1$getValue(p)
solution2.1$getValue(q)


#modified for Stackoverflow integer NLP (not integer) ####
#Does not solve, not convex?

# Variables minimized over
x <- Variable(1)
y <- Variable(1)
p <- Variable(1)
q <- Variable(1)

# Problem definition
objective <- Minimize((x - y)^2/y + (p - q)^2/q)
constraints <- list(x >= 0, y >= 0, p >= 0, q >= 0, 
                    x <= 67.314, y <= 78, p <= 76.11, q <= 86)
prob2.1 <- Problem(objective, constraints)

# Problem solution
solution2.1 <- solve(prob2.1, gp = TRUE)
solution2.1 <- solve(prob2.1, gp = FALSE)
# solution2.1$status
# solution2.1$value
# solution2.1$getValue(x)
# solution2.1$getValue(y)
# solution2.1$getValue(p)
# solution2.1$getValue(q)


#modified for Stackoverflow integer MINLP ####
#Does not solve

# Variables minimized over
x <- Variable(1, integer=TRUE)
y <- Variable(1)
p <- Variable(1, integer=TRUE)
q <- Variable(1)

# Problem definition
objective <- Minimize((x - y)^2/y + (p - q)^2/q)
constraints <- list(x >= 0, y >= 0, p >= 0, q >= 0, 
                    x <= 67.314, y <= 78, p <= 76.11, q <= 86)
prob2.1 <- Problem(objective, constraints)

# Problem solution
solution2.1 <- solve(prob2.1, gp = TRUE)
solution2.1 <- solve(prob2.1, gp = FALSE)
# solution2.1$status
# solution2.1$value
# solution2.1$getValue(x)
# solution2.1$getValue(y)
# solution2.1$getValue(p)
# solution2.1$getValue(q)


#modified for Stackoverflow integer NLP (not integer) ####
#objective multiplied by y*q, Does not solve, not convex?

# Variables minimized over
x <- Variable(1)
y <- Variable(1)
p <- Variable(1)
q <- Variable(1)

# Problem definition
objective <- Minimize((x - y)^2*q + (p - q)^2*y)
constraints <- list(x >= 0, y >= 0, p >= 0, q >= 0, 
                    x <= 67.314, y <= 78, p <= 76.11, q <= 86)
prob2.1 <- Problem(objective, constraints)

# Problem solution
solution2.1 <- solve(prob2.1, gp = TRUE)
solution2.1 <- solve(prob2.1, gp = FALSE)
# solution2.1$status
# solution2.1$value
# solution2.1$getValue(x)
# solution2.1$getValue(y)
# solution2.1$getValue(p)
# solution2.1$getValue(q)