R中的递归optim()函数会导致错误

Question

我正在尝试使用R中的optim()函数来通过矩阵运算最小化值。在这种情况下，我试图将一组股票的波动性降到最低，这些股票的个人回报率彼此相同。被最小化的目标函数是calculate_portfolio_variance。

library(quantmod)
filter_and_sort_symbols <- function(symbols)
{
# Name: filter_and_sort_symbols
# Purpose: Convert to uppercase if not
# and remove any non valid symbols
# Input: symbols = vector of stock tickers
# Output: filtered_symbols = filtered symbols

# convert symbols to uppercase
symbols <- toupper(symbols)

# Validate the symbol names
valid <- regexpr("^[A-Z]{2,4}$", symbols)

# Return only the valid ones
return(sort(symbols[valid == 1]))
}
# Create the list of stock tickers and check that they are valid symbols
tickers <- filter_and_sort_symbols(c("AAPL", "NVDA", "MLM", "AA"))
benchmark <- "SPY"
# Set the start and end dates
start_date <- "2007-01-01"
end_date <- "2019-01-01"
# Gather the stock data using quantmod library
getSymbols(Symbols=tickers, from=start_date, to=end_date, auto.assign = TRUE)
getSymbols(benchmark, from=start_date, to=end_date, auto.assign = TRUE)
# Create a matrix of only the adj. prices
price_matrix <- NULL
for(ticker in tickers){price_matrix <- cbind(price_matrix, get(ticker)[,6])}
# Set the column names for the price matrix
colnames(price_matrix) <- tickers
benchmark_price_matrix <- NULL
benchmark_price_matrix <- cbind(benchmark_price_matrix, get(benchmark)[,6])
# Compute log returns
returns_matrix <- NULL
for(ticker in tickers){returns_matrix <- cbind(returns_matrix, annualReturn(get(ticker)))}
returns_covar <- cov(returns_matrix)
colnames(returns_covar) <- tickers
rownames(returns_covar) <- tickers
# get average returns for tickers and benchmark
ticker_avg <- NULL
for(ticker in tickers){ticker_avg <- cbind(ticker_avg, colMeans(annualReturn(get(ticker))))}
colnames(ticker_avg) <- tickers
benchmark_avg <- colMeans(annualReturn(get(benchmark)))
# create the objective function
calculate_portfolio_variance <- function(allocations, returns_covar, ticker_avg, benchmark_avg)
{
# Name: calculate_portfolio_variance
# Purpose: Computes expected portfolio variance, to be used as the minimization objective function
# Input: allocations = vector of allocations to be adjusted for optimality; returns_covar = covariance matrix of stock returns
#        ticker_avg = vector of average returns for all tickers, benchmark_avg = benchmark avg. return
# Output: Expected portfolio variance

# get benchmark volatility 
benchmark_variance <- (sd(annualReturn(get(benchmark))))^2
# scale allocations for 100% investment
allocations <- as.matrix(allocations/sum(allocations))
# get the naive allocations
naive_allocations <- rep(c(1/ncol(ticker_avg)), times=ncol(ticker_avg))
portfolio_return <-  sum(t(allocations)*ticker_avg)
portfolio_variance <- t(allocations)%*%returns_covar%*%allocations

# constraints = portfolio expected return must be greater than benchmark avg. return and
#               portfolio variance must be less than benchmark variance (i.e. a better reward at less risk)
if(portfolio_return < benchmark_avg | portfolio_variance > benchmark_variance)
{
allocations <- naive_allocations
}

portfolio_variance <- t(allocations)%*%returns_covar%*%allocations
return(portfolio_variance)
}

# Specify lower and upper bounds for the allocation percentages
lower <- rep(0, ncol(returns_matrix))
upper <- rep(1, ncol(returns_matrix))
# Initialize the allocations by evenly distributing among all tickers
set.seed(1234)
allocations <- rep(1/length(tickers), times=length(tickers))

当我手动调用目标函数时，它会按预期返回一个值：

> calculate_portfolio_variance(allocations, returns_covar, ticker_avg, benchmark_avg)
[,1]
[1,] 0.1713439

然而，当我使用optim()函数时，它会返回错误：

> optim_result <- optim(par=allocations, fn=calculate_portfolio_variance(allocations, ticker_avg, benchmark_avg), lower=lower, upper=upper, method="L-BFGS-B")
Error in t(allocations) %*% returns_covar : non-conformable arguments

我不确定原因，但可能是optim()递归使用allocations变量的方式。我能做些什么来解决这个问题？

编辑：FWIW，其他优化策略有效(差分进化，模拟退火(，但我更喜欢使用梯度下降，因为它比快得多

Answer 1

如果第一个参数被重命名为par，并且您切换将t((应用于侧翼矩阵乘法运算中使用的参数向量的顺序，则不会发生错误：

cpv <- function(par, returns_covar=returns_covar, ticker_avg, benchmark_avg)
{
# Name: calculate_portfolio_variance
# Purpose: Computes expected portfolio variance, to be used as the minimization objective function
# Input: allocations = vector of allocations to be adjusted for optimality; returns_covar = covariance matrix of stock returns
#        ticker_avg = vector of average returns for all tickers, benchmark_avg = benchmark avg. return
# Output: Expected portfolio variance

# get benchmark volatility 
benchmark_variance <- (sd(annualReturn(get(benchmark))))^2
# scale allocations for 100% investment
par <- as.matrix(par/sum(par))
# get the naive allocations
naive_allocations <- rep(c(1/ncol(ticker_avg)), times=ncol(ticker_avg))
portfolio_return <-  sum(t(par)*ticker_avg);print(par)
portfolio_variance <- t(par)%*%returns_covar%*%par

# constraints = portfolio expected return must be greater than benchmark avg. return and
#               portfolio variance must be less than benchmark variance (i.e. a better reward at less risk)
if(portfolio_return < benchmark_avg | portfolio_variance > benchmark_variance)
{
par <- naive_allocations
}

portfolio_variance <- t(par)%*%returns_covar%*%par
return(portfolio_variance)
}

我在代码中留下了par的调试打印，并显示了运行的顶部结果

optim_result <- optim(par=allocations, fn=cpv, lower=lower, upper=upper, returns_covar=returns_covar, ticker_avg=ticker_avg, benchmark_avg=benchmark_avg, method="L-BFGS-B")
[,1]
[1,] 0.25
[2,] 0.25
[3,] 0.25
[4,] 0.25
[,1]
[1,] 0.2507493
[2,] 0.2497502
[3,] 0.2497502
[4,] 0.2497502
[,1]
[1,] 0.2492492
[2,] 0.2502503
[3,] 0.2502503
[4,] 0.2502503
#--- snipped output of six more iterations.

结果：

> optim_result 
$par
[1] 0.25 0.25 0.25 0.25
$value
[1] 0.1713439
$counts
function gradient 
1        1 
$convergence
[1] 0
$message
[1] "CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL"

正如我在对一个无关问题的评论中所说，optim函数首先尝试提高然后降低par中的第一个元素，然后尝试对第二、第三和第四个元素进行同样的操作。在没有发现任何改进的情况下；决定"；它收敛于局部最小值，并声明收敛。

我应该指出，optim的代码相当旧，原始算法的作者Nash博士已经以optimx包的形式在CRAN上放置了更新版本。他说optim在当时很好，但他认为如果不成功，应该尝试其他程序。