鉴于此处优化问题与您之前的问题的相似之处,我将直接从我对该问题的回答中借用一些语言。但是它们有很大的不同(之前的问题是线性规划问题,这是二次规划问题,并且约束不同),因此它们不是重复的。
扩展我们得到的优化目标Quw*ruw^2 - 2*Quw*ruw*yuw + Quw*yuw^2
。我们看到这是决策变量的二次函数yuw
,因此可以使用包的solve.QP
方法来解决优化问题。quadProg
为了稍微抽象一下,让我们假设R=20
并C=10
描述输入矩阵的维度。然后是R*C
决策变量,我们可以为它们分配 order y11, y21, ... yR1, y12, y22, ... yR2, ..., y1C, y2C, ..., yRC
,读取变量矩阵的列。
从中?solve.QP
,我们读到目标采用-d'b + 0.5b'Db
决策变量的形式b
。d
决策变量对应的元素yuw
有值2*Quw*ruw
,D
是一个对角矩阵,决策变量对应的元素yuw
取值2*Quw
。请注意,该solve.QP
函数要求D
矩阵是正定的,因此我们需要Quw > 0
每一u, w
对。
第一个R*(C-1)
约束对应于yuw >= yu,w+1
约束,下一个(R-1)*C
约束对应于yuw >= yu-1,w
约束。下一个2*R
约束对应于yuC = 0
约束(输入为yuC >= 0
and -yuC >= 0
),最后一个约束是-yR1 >= -100
(逻辑上等价于yR0 = 100
)。
quadProg
我们可以使用以下 R 命令将这个模型输入到包中,使用随机输入数据:
# Sample data
set.seed(144)
Quw <- matrix(runif(200), nrow=20)
ruw <- matrix(runif(200), nrow=20)
R <- nrow(Quw)
C <- ncol(Quw)
# Build constraint matrix
part1 <- matrix(0, nrow=R*(C-1), ncol=R*C)
part1[cbind(1:(R*C-R), 1:(R*C-R))] <- 1
part1[cbind(1:(R*C-R), (R+1):(R*C))] <- -1
part2 <- matrix(0, nrow=(R-1)*C, ncol=R*C)
pos2 <- as.vector(sapply(2:R, function(r) r+seq(0, R*(C-1), by=R)))
part2[cbind(1:nrow(part2), pos2)] <- 1
part2[cbind(1:nrow(part2), pos2-1)] <- -1
part3 <- matrix(0, nrow=2*R, ncol=R*C)
part3[cbind(1:R, (R*C-R+1):(R*C))] <- 1
part3[cbind((R+1):(2*R), (R*C-R+1):(R*C))] <- -1
part4 <- rep(0, R*C)
part4[R] <- -1
const.mat <- rbind(part1, part2, part3, part4)
library(quadProg)
mod <- solve.QP(Dmat = 2*diag(as.vector(Quw)),
dvec = 2*as.vector(ruw)*as.vector(Quw),
Amat = t(const.mat),
bvec = c(rep(0, nrow(const.mat)-1), -100))
我们现在可以访问模型解决方案:
# Objective (including the constant term):
mod$value + sum(Quw*ruw^2)
# [1] 9.14478
matrix(mod$solution, nrow=R)
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
# [1,] 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.3215992 0.1818095 0.1818095 0.1818095 0.000000e+00
# [2,] 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [3,] 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 2.775558e-17
# [4,] 0.5728478 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [5,] 0.5728478 0.5111456 0.5111456 0.4699046 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [6,] 0.5728478 0.5111456 0.5111456 0.4699046 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [7,] 0.5728478 0.5111456 0.5111456 0.4699046 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [8,] 0.5728478 0.5111456 0.5111456 0.5111456 0.4346995 0.4346995 0.4346995 0.2882339 0.2882339 0.000000e+00
# [9,] 0.5728478 0.5111456 0.5111456 0.5111456 0.4346995 0.4346995 0.4346995 0.4346995 0.2882339 1.110223e-16
# [10,] 0.5728478 0.5111456 0.5111456 0.5111456 0.4346995 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [11,] 0.6298100 0.5111456 0.5111456 0.5111456 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [12,] 0.6298100 0.5111456 0.5111456 0.5111456 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [13,] 0.6298100 0.5111456 0.5111456 0.5111456 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [14,] 0.6298100 0.5111456 0.5111456 0.5111456 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [15,] 0.6298100 0.6009718 0.5111456 0.5111456 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [16,] 0.6298100 0.6009718 0.6009718 0.6009718 0.4518205 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [17,] 0.6298100 0.6009718 0.6009718 0.6009718 0.6009718 0.4346995 0.4346995 0.4346995 0.4346995 0.000000e+00
# [18,] 0.6298100 0.6009718 0.6009718 0.6009718 0.6009718 0.6009718 0.4346995 0.4346995 0.4346995 0.000000e+00
# [19,] 0.6298100 0.6009718 0.6009718 0.6009718 0.6009718 0.6009718 0.4346995 0.4346995 0.4346995 0.000000e+00
# [20,] 0.6298100 0.6009718 0.6009718 0.6009718 0.6009718 0.6009718 0.5643033 0.5643033 0.5643033 0.000000e+00