Question

我想知道是否有人能够提出一些方案来解决非线性优化问题，而非线性优化问题可以为优化解决方案提供整数变量？问题在于最小化具有相等约束的函数，该函数受某些上下边界约束的约束。

我已经在R中使用了'nloptr'软件包来解决非线性优化问题，该问题效果很好，但是现在想扩展该方法以使某些变量为整数。到目前为止，从我对nloptr的使用和理解来看，它只能返回连续的值，而不能返回整数变量以获得最佳解决方案。

我认为这类问题需要使用混合整数非线性编程来解决。

以nloptr形式出现的问题的一个示例：

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."

这是我要寻找的结果，但如上所述，我需要x和p为整数。

我看过https://cran.r-project.org/web/views/Optimization.html，其中有一个非常好的混合整数非线性编程软件包列表，但我想知道是否有人对它们进行了体验，以及他们认为最合适的软件包是什么解决上述问题。

大约7年前在这里发布了与此类似的问题，但最终链接到cran页面，因此认为值得再次询问。

非常感谢您的输入。

干杯

安德鲁

Answer 1

这是一个示例，说明在没有更简单的目标函数的情况下，它不能使用CVXR的情况。我怀疑即使有约束条件，问题也不是凸面的，因此需要替代方法。

#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)

Answer 2

我已经使用您的示例尝试了以下代码，以尝试在原始问题中复制nloptr示例：

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

#modified for Stackoverflow integer MINLP (MIQP) ####
#Solves

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

# Problem definition (terms not normalised by y and q respectively)
objective <- Minimize((x - y)^2 + (p - q)^2 -z)
constraints <- list(x <= 67.314, y <= 78, p <= 76.11, q <= 86, z == 10.2)
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)
solution2.1$getValue(z)

但是，我得到的值为-10.19989，当它应该为0时。

> solution2.1$status
[1] "optimal"
> solution2.1$value
[1] -10.19989
> solution2.1$getValue(x)
[1] -1060371
> solution2.1$getValue(y)
[1] -1060371
> solution2.1$getValue(p)
[1] -1517
> solution2.1$getValue(q)
[1] -1517.002
> solution2.1$getValue(z)
[1] 10.2

我无法弄清上面要做的事情，才能使其像nloptr示例一样工作，但要确保x和p是整数值！

干杯，安德鲁

Answer 3

ROI是解决MINLP问题的一种选择。我相信它可以使用一些合适的求解器。它允许访问neos（在对您的问题的另一个答案中描述）。

如果您有兴趣了解ROI优化的外观，请参阅LP（线性编程示例：

#ROI example https://epub.wu.ac.at/5858/1/ROI_StatReport.pdf
#install.packages("ROI")
library(ROI)
ROI_available_solvers()

ROI_registered_solvers() #has one solver loaded by default

## Get and load "lpsolve" solver
#install.packages("ROI.plugin.lpsolve", repos=c("https://r-forge.r-project.org/src/contrib",
#                                   "http://cran.at.r-project.org"),dependencies=TRUE)
library(ROI.plugin.lpsolve)
ROI_registered_solvers() #Now it is available to use

## Describe model
A <- rbind(c(5, 7, 2), c(3, 2, -9), c(1, 3, 1))
dir <- c("<=", "<=", "<=")
rhs <- c(61, 35, 31)
lp <- OP(objective = L_objective(c(3, 7, -12)),
         constraints = L_constraint(A, dir = dir, rhs = rhs),
         bounds = V_bound(li = 3, ui = 3, lb = -10, ub = 10, nobj = 3),
         maximum = TRUE)

## When you have a model, you can find out which solvers can solve it
ROI_available_solvers(lp)[, c("Package", "Repository")]

## Solve model
lp_sol <- ROI_solve(lp, solver = "lpsolve")

Answer 4

由于此问题属于难以解决的类型，因此不能保证任何通用算法都可以很好地解决此确切问题（请参阅免费午餐定理）。实际上，对于一个难题，许多算法甚至不可能收敛于全局最优。有趣的是，至少对问题空间的随机搜索最终会收敛，因为最终它会搜索整个空间！

tl / dr 尝试枚举问题空间。例如，如果您的四个变量是0到80之间的整数，则只能循环通过〜80 ^ 4 =〜4,000万个组合。中间选项可能是（如果只有两个变量是整数），可以通过优化方法针对剩余的两个变量（给定两个整数的值）来解决问题（也许现在是凸问题？）并遍历整数值。

Answer 5

rneos是一个软件包，可让您访问neos，这是一种免费的解决方案，具有多种算法，其中包括一些适用于MINLP问题的算法（例如BONMIN和Couenne，请参见列表here）。不幸的是，该问题需要格式化为GAMS或AMPL模型。对您来说，这可能意味着学习一些基本的GAMS，在这种情况下，也许您可以只使用GAMS软件see here？免费版本可能足以满足您的目的。它可以作为命令行运行，因此您可以根据需要从R调用它。

如果您有兴趣了解neos优化的外观，请参阅LP（线性编程示例：

#rneos example
#from p11 of https://www.pfaffikus.de/talks/rif/files/rif2011.pdf

#install.packages("rneos")
library(rneos)
#library(devtools)
#install_github("duncantl/XMLRPC")
library(XMLRPC)
## NEOS: ping
Nping()
## NEOS: listCategories
NlistCategories()
## NEOS: listSolversInCategory
NlistSolversInCategory(category = "lp")
## NEOS: getSolverTemplate
template <- NgetSolverTemplate(category = "lp", solvername = "MOSEK", inputMethod = "GAMS")
template
#gams file below sourced from https://github.com/cran/rneos/blob/master/inst/ExGAMS/TwoStageStochastic.gms
modc <- paste(paste(readLines("TwoStageStochastic.gms"), collapse = "\n"), "\n")
cat(modc)
argslist <- list(model = modc, options = "", wantlog = "", comments = "")
xmls <- CreateXmlString(neosxml = template, cdatalist = argslist)
## NEOS: printQueue
NprintQueue()
## NEOS: submitJob
(test <- NsubmitJob(xmlstring = xmls, user = "rneos", interface = "", id = 0))
## NEOS: getJobStatus
NgetJobStatus(obj = test)
## NEOS: getFinalResults
NgetFinalResults(obj = test)

Answer 6

R 不是解决该问题的工具。它需要非线性编程的高级功能。我将注意力转向 Julia 语言。我使用了 JuMP 包以及 Juniper 和 Ipopt 算法。它对我的 MINLP 问题很有效。

R中具有整数变量的非线性优化/编程

6 个答案: