在R: Rsolnp或Auglag中具有线性约束的二次目标最大化

Question

我正在尝试使用auglag或Rsolnp找到以下优化问题的解决方案。

Max t(w1 - w2) * Kf * Sf * t(Kf) * (w1 - w2)
subject to Kc * w1 = Kc * w2
and sum(w1) = 1 and sum(w2) = 1 and w1,w2 >= 0
Sc and Sf are variance covariance matrices at the coarse and fine level respectively.
Kc and Kf are exposure matrices as the coarse and fine level respectively.
Nc and Nf are nodes at which exposure nodes at the coarse and fine level.

这是有效地试图找到两个投资组合w1和w2的wts，在更精细的暴露水平下，在wts = 1和所有wts> 0的总和下，使TEV最大化。还有另一个等式约束(这实际上意味着两个投资组合在粗水平上的风险敞口是相同的)。Rsolnp不能最大化并给出了一个解决方案，其中目标函数为0，并且auglag完全爆炸并且不满足带有一堆警告的约束。

谁能帮我理解我错在哪里?

    seqFineNodes <- c(1, 2, 3, 4, 5, 6)
Nc <- c(2, 3, 5)
Kc <- matrix(c(0.2481316799436,0.495478766935844,0,0,0,0,0,0,0.743360061619584,0.497321712603124,0,0,0,0,0,0.497321712603124,1.23913608908603,1.48240730986596), nrow=length(seqFineNodes), ncol=length(Nc))
dimnames(Kc) <- list(as.character(seqFineNodes), as.character(Nc))
Sc <- matrix(c(619.806079280659,627.832850585004,549.805085990891,627.832850585004,668.726833059322,624.524848194842,549.805085990891,624.524848194842,696.498483673357), nrow=length(Nc), ncol=length(Nc))
dimnames(Sc) <- list(as.character(Nc), as.character(Nc))
Nf <- c(2, 3, 4, 5)
Kf <- matrix(c(0.2481316799436,0.495478766935844,0,0,0,0,0,0,0.743360061619584,0,0,0,0,0,0,0.994643425206249,0,0,0,0,0,0,1.23913608908603,1.48240730986596), nrow=length(seqFineNodes), ncol=length(Nf))
dimnames(Kf) <- list(as.character(seqFineNodes), as.character(Nf))
Sf <- matrix(c(619.806079280659,627.832850585004,602.504944834256,549.805085990891,627.832850585004,668.726833059322,666.196728425214,624.524848194842,602.504944834256,666.196728425214,696.688027074344,681.064062606848,549.805085990891,624.524848194842,681.064062606848,696.498483673357), nrow=length(Nf), ncol=length(Nf))
dimnames(Sf) <- list(as.character(Nf), as.character(Nf))
KRD_fine <- Kf
KRD_coarse <- Kc
VC_fine <- Sf
VC_coarse <- Sc
countw <- length(seqFineNodes)

t1 <- diag(x = 1, nrow = countw, ncol = countw)
t2 <- diag(x = -1, nrow = countw, ncol = countw)
tr <- cbind(t1,t2)
D_fine <- t(tr) %*% KRD_fine %*% VC_fine %*% t(KRD_fine) %*% tr
#round(eigen(Dmat)$values, 4)
D_fine <- as.matrix(nearPD(D_fine)$mat)
#round(eigen(Dmat)$values, 4)
eq_coarse_krd_A <- t(KRD_coarse) %*% tr
eq_coarse_krd_b <- rep(0, nrow(VC_coarse))
# Equality constraints
eq_A1 <- c(rep(1, countw), rep(0,countw))
eq_A2 <- c(rep(0, countw), rep(1,countw))
eq_b <- c(1 , 1)
# Constraint wts greater than zero
ineq_A <- diag(x = 1, nrow = 2 * countw, ncol = 2 * countw)
ineq_b <- rep(0, 2 * countw)
# Combine constraints
heq <- rbind(eq_coarse_krd_A, eq_A1, eq_A2)
beq <- c(eq_coarse_krd_b, eq_b)
hin <- ineq_A
theta <- c(1, rep(0, countw - 1), 1, rep(0, countw - 1))
krdsol <- solnp(par = theta, 
                fun = function(x) -c(t(x) %*% D_fine %*% x), 
                ineqfun = function(x) c(hin %*% x),
                ineqLB = rep(0, 2 * countw),
                ineqUB = rep(1, 2 * countw),
                eqfun = function(x) c(heq %*% x),
                eqB = beq)

krdFine <- auglag(par = theta, 
                  fn = function(x) c(t(x) %*% D_fine %*% x), 
                  hin = function(x) c(hin %*% x),
                  heq = function(x) c(heq %*% x) - beq,
                  control.outer = list(method = "nlminb"),
                  control.optim=list(fnscale=-1))

Answer 1

我解决了你关于solnp的问题。?solnp表示fun, ineqfun和eqfun返回vector，但您的返回matrix。所以我添加了c(...)。

library(Rsolnp)
krdsol <- solnp(par = theta, 
                fun = function(x) c(-t(x) %*% D_fine %*% x), 
                ineqfun = function(x) c(hin %*% x),
                ineqLB = rep(0, 2 * countw),
                ineqUB = rep(1, 2 * countw),
                eqfun = function(x) c(heq %*% x),
                eqB = beq)

(编辑)

auglag(control.optim=list(...))作为参数的元素列在?nlminb()中(参见?auglag())