# basic finance
library(xts) # to manipulate time series of stock data
library(pob) # book package with financial data
# plotting
library(ggplot2) # for nice plots
library(reshape2) # to reshape data
library(dplyr) # for data frame manipulation (e.g., mutate)
library(patchwork) # for combining plots
library(scales) # for axes control
# optimization
library(alabama)
library(nloptr)
library(quadprog)
library(riskParityPortfolio) # for risk parity portfolios
library(microbenchmark) # for cpu computationR Code for Portfolio Optimization
Chapter 11 – Risk Parity Portfolios
Daniel P. Palomar (2025). Portfolio Optimization: Theory and Application. Cambridge University Press.
Last update: March 14, 2025
Loading packages
The following packages are used in the examples:
From dollar to risk diversification
Portfolio allocation and risk allocation for the \(1/N\) portfolio and risk parity portfolio:
library(pob)
library(riskParityPortfolio)
stock_prices <- SP500_2015to2020$stocks["2020::2020-09", c("AAPL", "NFLX", "TSCO", "MGM", "MSFT", "FB", "AMZN", "GOOGL")]
T <- nrow(stock_prices)
T_trn <- round(0.70*T)
X <- diff(log(stock_prices[1:T_trn, ]))[-1]
Sigma <- cov(X)
N <- ncol(X)
# portfolios
w_EWP <- rep(1/N, N)
w_RPP <- riskParityPortfolio(Sigma)$w
# barplots
df_w <- data.frame(
"stocks" = names(stock_prices),
"1/N" = w_EWP,
"RPP" = w_RPP,
check.names = FALSE)
# Using package riskParityPortfolio:
# barplotPortfolioRisk(as.matrix(df_w[-1]), Sigma)
# We define our own barplot function for completeness:
barplot_w_and_RCC <- function(df_w, Sigma) {
df_w$stocks <- factor(df_w$stocks, levels = unique(df_w$stocks))
# get relative risk contribution
df_rrc <- df_w
df_rrc[-1] <- lapply(as.list(df_w[-1]), function(w) {
rc <- as.vector(w * (Sigma %*% w))
rc/sum(rc)
})
# plots
p_w <- df_w |>
melt(id.vars = "stocks") |>
ggplot(aes(x = stocks, y = value, fill = variable)) +
geom_bar(stat = "identity", position = "dodge", color = "black", width = 0.8) +
labs(fill = "portfolios") +
labs(title = "Portfolio weights", x = NULL, y = "weight")
p_rrc <- df_rrc |>
melt(id.vars = "stocks") |>
ggplot(aes(x = stocks, y = value, fill = variable)) +
geom_bar(stat = "identity", position = "dodge", color = "black", width = 0.8) +
labs(fill = "portfolios") +
labs(title = "Relative risk contribution", y = "risk")
p_w / p_rrc + plot_layout(guides = "collect")
}
barplot_w_and_RCC(df_w, Sigma) 
Naive diagonal formulation
Portfolio allocation and risk contribution of the \(1/N\) portfolio and naive RPP:
library(pob)
library(riskParityPortfolio)
stock_prices <- SP500_2015to2020$stocks["2020::2020-09", c("AAPL", "NFLX", "TSCO", "MGM", "MSFT", "FB", "AMZN", "GOOGL")]
T <- nrow(stock_prices)
T_trn <- round(0.70*T)
X <- diff(log(stock_prices[1:T_trn, ]))[-1]
Sigma <- cov(X)
N <- ncol(X)
# portfolios
w_EWP <- rep(1/N, N)
#w_naive_RPP <- riskParityPortfolio(Sigma, formulation = "diag")$w
w_naive_RPP <- 1/sqrt(diag(Sigma))
w_naive_RPP <- w_naive_RPP/sum(w_naive_RPP)
# barplots
df_w <- data.frame(
"stocks" = names(stock_prices),
"1/N" = w_EWP,
"Naive diagonal RPP" = w_naive_RPP,
check.names = FALSE)
barplot_w_and_RCC(df_w, Sigma)
Vanilla convex formulations
Portfolio allocation and risk contribution of the vanilla convex RPP compared to benchmarks:
library(riskParityPortfolio)
stock_prices <- SP500_2015to2020$stocks["2020::2020-09", c("AAPL", "NFLX", "TSCO", "MGM", "MSFT", "FB", "AMZN", "GOOGL")]
T <- nrow(stock_prices)
T_trn <- round(0.70*T)
X <- diff(log(stock_prices[1:T_trn, ]))[-1]
Sigma <- cov(X)
N <- ncol(X)
# portfolios
w_EWP <- rep(1/N, N)
w_naive_RPP <- riskParityPortfolio(Sigma, formulation = "diag")$w
w_convex_RPP <- riskParityPortfolio(Sigma)$w
# barplots
df_w <- data.frame(
"stocks" = names(stock_prices),
"1/N" = w_EWP,
"Naive diagonal RPP" = w_naive_RPP,
"Vanilla convex RPP" = w_convex_RPP,
check.names = FALSE)
barplot_w_and_RCC(df_w, Sigma)
Algorithms
General-purpose nonlinear solver
We can solve this problem with the general-purpose nonlinear solver optim() in R:
# initial point
w <- rep(1/N, N)
x0 <- w/as.numeric(sqrt(w %*% Sigma %*% w))
# function definition
fn_convex <- function(x, Sigma) {
N <- nrow(Sigma)
return(0.5 * t(x) %*% Sigma %*% x - (1/N)*sum(log(x)))
}
# optimize with general-purpose solver
result <- optim(par = x0, fn = fn_convex, Sigma = Sigma, method = "BFGS")
x_general_solver <- result$par
w_general_solver_RPP <- x_general_solver/sum(x_general_solver)
# sanity check of the solution
b <- rep(1/N, N)
Sigma %*% x_general_solver - b/x_general_solver [,1]
AAPL -7.391053e-06
NFLX -1.951736e-05
TSCO -9.158081e-06
MGM 9.466781e-06
MSFT 8.160602e-06
FB -1.373207e-05
AMZN -3.604614e-05
GOOGL -3.071543e-05Newton’s method
Newton’s method for Spinu’s RPP formulation:
library(microbenchmark)
library(dplyr)
b <- rep(1/N, N)
x_opt <- w_convex_RPP / as.vector(sqrt(w_convex_RPP %*% Sigma %*% w_convex_RPP))
opt_value <- 0.5 * x_opt %*% Sigma %*% x_opt - b %*% log(x_opt)
#
# Newton Spinu algorithm (1/N initial point)
#
num_iter <- 5L
w <- rep(1/N, N)
x <- w/as.numeric(sqrt(w %*% Sigma %*% w))
df <- data.frame("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
check.names = FALSE)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
grad <- Sigma %*% x - b/x
Hessian <- Sigma + diag(b/x^2)
x_new <- x - solve(Hessian, grad)
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value))
}
df <- df |> mutate("CPU time [ms]" = cumsum(`cpu time k`)/1e6)
p_gap_vs_iter <- df |>
ggplot(aes(x = k, y = gap)) +
geom_line(linewidth = 1.2, col = "blue") +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
scale_x_continuous(limits = c(0, num_iter), breaks = seq(0, num_iter, by = 1), minor_breaks = NULL) +
coord_cartesian(ylim = c(1e-10, 1e1)) +
ggtitle("Optimality gap versus iterations")
p_gap_vs_cpu_time <- df |>
ggplot(aes(x = `CPU time [ms]`, y = gap, color = method)) +
geom_line(linewidth = 1.2, col = "blue") +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
coord_cartesian(xlim = c(0, 0.25), ylim = c(1e-10, 1e1)) +
ggtitle("Optimality gap versus CPU time")
p_gap_vs_iter / p_gap_vs_cpu_time + plot_layout(guides = "collect")
Cyclical coordinate descent algorithm
Cyclical coordinate descent algorithm for Spinu’s RPP formulation:
#
# Cyclical Spinu coordinate descent algorithm (1/N initial point)
#
num_iter <- 10L
w <- rep(1/N, N)
x <- w/as.numeric(sqrt(w %*% Sigma %*% w))
df <- data.frame("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
check.names = FALSE)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
x_new <- x
#Sigma_xk <- as.numeric(Sigma %*% x_new)
for (i in 1:N) {
Sigma_xk_i <- as.numeric(x_new[-i] %*% Sigma[-i, i])
#Sigma_xk_i <- Sigma_xk[i] - x_new[i] * Sigma[i, i]
x_new[i] <- (- Sigma_xk_i + sqrt(Sigma_xk_i^2 + 4*Sigma[i, i]*b[i]))/(2*Sigma[i, i])
#x_diff <- x_new[i] - x[i]
#Sigma_xk <- Sigma_xk + Sigma[, i] * x_diff
}
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value))
}
df <- df |> mutate("CPU time [ms]" = cumsum(`cpu time k`)/1e6)
p_gap_vs_iter <- df |>
ggplot(aes(x = k, y = gap)) +
geom_line(linewidth = 1.2, col = "blue") +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
scale_x_continuous(limits = c(0, num_iter), breaks = seq(0, num_iter, by = 1), minor_breaks = NULL) +
coord_cartesian(ylim = c(1e-10, 1e1)) +
ggtitle("Optimality gap versus iterations")
p_gap_vs_cpu_time <- df |>
ggplot(aes(x = `CPU time [ms]`, y = gap, color = method)) +
geom_line(linewidth = 1.2, col = "blue") +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
coord_cartesian(xlim = c(0, 50), ylim = c(1e-10, 1e1)) +
ggtitle("Optimality gap versus CPU time")
p_gap_vs_iter / p_gap_vs_cpu_time + plot_layout(guides = "collect")
Numerical comparison
Convergence of Newton, MM, and SCA methods for Spinu’s RPP problem, comparing the original version with the improved one (i.e., using the correlation matrix instead of the covariance matrix and using the linear normalization step) (Palomar, 2025, Chapter 11). It seems that the best method is the improved SCA.
num_iter <- 25L
sigma <- sqrt(diag(Sigma))
C <- cov2cor(Sigma)
#
# Newton Spinu algorithm
#
x <- sqrt(b)/sqrt(rowSums(Sigma))
df <- data.frame("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "Newton",
check.names = FALSE)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
grad <- Sigma %*% x - b/x
Hessian <- Sigma + diag(b/x^2)
x_new <- x - solve(Hessian, grad)
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "Newton"))
}
#
# Newton Spinu algorithm (normalized and based on correlation matrix)
#
x <- sqrt(b)/sqrt(rowSums(C))
rowsum_C <- rowSums(C)
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "Newton - improved"))
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
grad <- C %*% x - b/x
Hessian <- C + diag(b/x^2)
x_new <- x - solve(Hessian, grad)
x_new <- x_new * sqrt(sum(b/x_new)/sum(rowsum_C * x_new))
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "Newton - improved"))
}
#
# Parallel Spinu MM
#
x <- sqrt(b)/sqrt(rowSums(Sigma))
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "MM"))
lmd_max <- eigen(Sigma)$values[1]
Sigma_lmd_max <- Sigma - lmd_max*diag(N)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
Sigma_lmd_max_xk <- Sigma_lmd_max %*% x
x_new <- (-Sigma_lmd_max_xk + sqrt(Sigma_lmd_max_xk^2 + 4*lmd_max*b))/(2*lmd_max)
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "MM"))
}
#
# Parallel Spinu MM (normalized and based on correlation matrix)
#
x <- sqrt(b)/sqrt(rowSums(C))
rowsum_C <- rowSums(C)
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "MM - improved"))
lmd_max <- eigen(C)$values[1]
C_lmd_max <- C - lmd_max*diag(N)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
C_lmd_max_xk <- C_lmd_max %*% x
x_new <- (-C_lmd_max_xk + sqrt(C_lmd_max_xk^2 + 4*lmd_max*b))/(2*lmd_max)
x_new <- x_new * sqrt(sum(b/x_new)/sum(rowsum_C * x_new))
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "MM - improved"))
}
#
# Parallel Spinu SCA
#
x <- sqrt(b)/sqrt(rowSums(Sigma))
gamma <- 1 #(1, 0.1), (0.9, 0.01)
eps <- 0.1
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "SCA"))
Sigma_Diag_Sigma <- Sigma - diag(diag(Sigma))
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
Sigma_Diag_Sigma_xk <- Sigma_Diag_Sigma %*% x
x_hat <- (-Sigma_Diag_Sigma_xk + sqrt(Sigma_Diag_Sigma_xk^2 + 4*diag(Sigma)*b))/(2*diag(Sigma))
x_new <- gamma*x_hat + (1 - gamma)*x
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
gamma <- gamma * (1 - eps*gamma)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * x %*% Sigma %*% x - b %*% log(x),
"gap" = 0.5 * x %*% Sigma %*% x - b %*% log(x) - opt_value,
"method" = "SCA"))
}
#
# Parallel Spinu SCA (normalized and based on correlation matrix)
#
x <- sqrt(b)/sqrt(rowSums(C))
rowsum_C <- rowSums(C)
gamma <- 1 #(1, 0.1), (0.9, 0.01)
eps <- 0.1
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "SCA - improved"))
C_minus_I <- C - diag(N)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
C_minus_I_xk <- C_minus_I %*% x
x_hat <- (-C_minus_I_xk + sqrt(C_minus_I_xk^2 + 4*b))/2
#x_hat <- x_hat /sqrt(as.numeric(t(x_hat) %*% C %*% x_hat))
x_hat <- x_hat * sqrt(sum(b/x_hat)/sum(rowsum_C * x_hat)) # my version
x_new <- gamma*x_hat + (1 - gamma)*x
}, unit = "microseconds", times = 10L)$time |> median()
x <- as.numeric(x_new)
gamma <- gamma * (1 - eps*gamma)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"obj_value" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma),
"gap" = 0.5 * (x/sigma) %*% Sigma %*% (x/sigma) - b %*% log(x/sigma) - opt_value,
"method" = "SCA - improved"))
}
# process data frame
df$method <- factor(df$method, levels = unique(df$method))
df <- df |>
group_by(method) |>
mutate("CPU time [ms]" = cumsum(`cpu time k`)/1e6) |>
ungroup()
# plots
p_gap_vs_iter <- df |>
ggplot(aes(x = k, y = gap, color = method)) +
geom_line(linewidth = 1.2) +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
scale_x_continuous(limits = c(0, num_iter), breaks = seq(0, num_iter, by = 1), minor_breaks = NULL) +
coord_cartesian(ylim = c(1e-10, 1e-2)) +
labs(color = "method")
ggtitle("Optimality gap versus iterations")$title
[1] "Optimality gap versus iterations"
attr(,"class")
[1] "labels"p_gap_vs_cpu_time <- df |>
ggplot(aes(x = `CPU time [ms]`, y = gap, color = method)) +
geom_line(linewidth = 1.2) +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
coord_cartesian(xlim = c(0, 0.25), ylim = c(1e-10, 1e-2)) +
labs(color = "method")
ggtitle("Optimality gap versus CPU time")$title
[1] "Optimality gap versus CPU time"
attr(,"class")
[1] "labels"p_gap_vs_iter / p_gap_vs_cpu_time + plot_layout(guides = "collect")
General nonconvex formulations
Portfolio allocation and risk contribution of general nonconvex RPP (with \(w_i \\leq 0.15\)) compared to benchmarks (\(1/N\) portfolio, naive diagonal RPP, and vanilla convex RPP):
library(riskParityPortfolio)
stock_prices <- SP500_2015to2020$stocks["2020::2020-09", c("AAPL", "NFLX", "TSCO", "MGM", "MSFT", "FB", "AMZN", "GOOGL")]
T <- nrow(stock_prices)
T_trn <- round(0.70*T)
X <- diff(log(stock_prices[1:T_trn, ]))[-1]
Sigma <- cov(X)
N <- ncol(X)
# portfolios
w_EWP <- rep(1/N, N)
w_naive_RPP <- riskParityPortfolio(Sigma, formulation = "diag")$w
w_convex_RPP <- riskParityPortfolio(Sigma)$w
w_nonconvex_RPP <- riskParityPortfolio(Sigma, w_ub = 0.15)$w
# barplots
df_w <- data.frame(
"stocks" = names(stock_prices),
"1/N" = w_EWP,
"Naive diagonal RPP" = w_naive_RPP,
"Vanilla convex RPP" = w_convex_RPP,
"General nonconvex RPP" = w_nonconvex_RPP,
check.names = FALSE)
barplot_w_and_RCC(df_w, Sigma)
Algorithms
Converge comparison for different algorithms to solve the nonconvex RPP formulation: \[ \begin{array}{ll} \underset{\w}{\textm{minimize}} & \begin{aligned}\sum_{i=1}^N \left(\frac{w_i\left(\bSigma\w\right)_i}{\w^\T\bSigma\w} - b_i\right)^2\end{aligned}\\ \textm{subject to} & \w \in \mathcal{W}. \end{array} \]
library(riskParityPortfolio)
library(microbenchmark)
library(dplyr)
library(alabama)
library(nloptr)
library(quadprog)
N <- 200
Sigma <- cov(diff(log(SP500_2015to2020$stocks[, 1:N]))[-1])
b <- rep(1/N, N)
w_ub <- 0.008 # RPP with upper bound!!
w_init <- rep(1/N, N)
# problem definition
R <- riskParityPortfolio:::R_rc_over_var_vs_b
R_grad <- riskParityPortfolio:::R_grad_rc_over_var_vs_b
num_iter <- 20L
num_times <- 10L # to compute the cpu time
#
# General nonlinear solver (alabama)
#
heq <- function(w, ...) sum(w) - 1 # sum(w) - 1 = 0
heq.jac <- function(w, ...) matrix(1, 1, N)
hin <- function(w, ...) return(c(w, w_ub - w)) # w >= 0 and w <= ub (ub - w >= 0)
hin.jac <- function(w, ...) return(rbind(diag(N), -diag(N)))
w <- w_init
df <- data.frame("k" = 0L,
"cpu time k" = NA,
"CPU time [ms]" = 0,
"obj_value" = R(w, Sigma, b),
"method" = "nonlinear solver (alabama)",
check.names = FALSE)
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
res <- alabama::constrOptim.nl(par = w_init,
fn = R, gr = R_grad,
hin = hin, hin.jac = hin.jac,
heq = heq, heq.jac = heq.jac,
control.outer = list(trace = FALSE, itmax = k),
Sigma = Sigma, b = b)
w <- res$par
}, unit = "microseconds", times = num_times)$time |> median()
df <- rbind(df, list("k" = res$outer.iterations,
"cpu time k" = NA,
"CPU time [ms]" = cpu_time/1e6,
"obj_value" = R(w, Sigma, b),
"method" = "nonlinear solver (alabama)"))
}
#
# General nonlinear solver (nloptr)
#
w <- w_init
df <- rbind(df, list("k" = 0L,
"cpu time k" = NA,
"CPU time [ms]" = 0,
"obj_value" = R(w, Sigma, b),
"method" = "nonlinear solver (nloptr)"))
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
res <- nloptr::slsqp(x0 = w_init,
fn = R, gr = R_grad,
hin = hin, hinjac = hin.jac,
heq = heq, heqjac = heq.jac,
control = list(maxeval = k),
Sigma = Sigma, b = b)
w <- res$par
}, unit = "microseconds", times = num_times)$time |> median()
df <- rbind(df, list("k" = res$iter,
"cpu time k" = NA,
"CPU time [ms]" = cpu_time/1e6,
"obj_value" = R(w, Sigma, b),
"method" = "nonlinear solver (nloptr)"))
}
#
# SCA
#
gamma <- 0.9
zeta <- 1e-7
# constraints (A' * b >= b0)
meq <- 1
Amat <- t(rbind(1, -diag(N), diag(N)))
bvec <- c(1, -rep(w_ub, N), rep(0, N))
# intermediate variables
tau <- .05 * sum(diag(Sigma)) / (2*N)
tauI <- tau*diag(N)
g <- riskParityPortfolio:::g_rc_over_var_vs_b
A <- riskParityPortfolio:::A_rc_over_var_vs_b
wk <- w_init
df <- rbind(df, list("k" = 0L,
"cpu time k" = 0,
"CPU time [ms]" = NA,
"obj_value" = R(wk, Sigma, b),
"method" = "SCA"))
for (k in 1:num_iter) {
cpu_time <- microbenchmark({
# auxiliary quantities
Sigma_wk <- Sigma %*% wk
rk <- wk * Sigma_wk
Jk <- A(wk, Sigma, b, Sigma_w = Sigma_wk, r = rk)
gk <- g(wk, Sigma, b, r = rk)
Qk <- 2 * crossprod(Jk) + tauI
qk <- 2 * t(Jk) %*% gk - Qk %*% wk
w_hat <- quadprog::solve.QP(Dmat = Qk, dvec = -qk, Amat = Amat, bvec = bvec, meq = meq)$solution
w_new <- wk + gamma * (w_hat - wk)
}, unit = "microseconds", times = num_times)$time |> median()
wk <- w_new
gamma <- gamma * (1 - zeta*gamma)
df <- rbind(df, list("k" = k,
"cpu time k" = cpu_time,
"CPU time [ms]" = NA,
"obj_value" = R(wk, Sigma, b),
"method" = "SCA"))
}
# process data frame
df$method <- factor(df$method, levels = unique(df$method))
df <- df |>
group_by(method) |>
mutate("CPU time [ms]" = ifelse(is.na(`CPU time [ms]`), cumsum(`cpu time k`)/1e6, `CPU time [ms]`)) |>
ungroup()
# add optimality gap using the actual minimum value
df <- df |>
group_by(method) |>
mutate("gap" = obj_value - min(df$obj_value)) |>
ungroup()
# plot
p_gap_vs_iter <- df |>
ggplot(aes(x = k, y = gap, color = method)) +
geom_line(linewidth = 1.2) +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
scale_x_continuous(limits = c(0, num_iter), breaks = seq(0, num_iter, by = 1), minor_breaks = NULL) +
#coord_cartesian(ylim = c(1e-10, 1e-2)) +
labs(color = "method")
ggtitle("Optimality gap versus iterations") + theme(axis.title.x = element_text(face = "italic"))NULLp_gap_vs_cpu_time <- df |>
ggplot(aes(x = `CPU time [ms]`, y = gap, color = method)) +
geom_line(linewidth = 1.2) +
scale_y_log10(breaks = trans_breaks("log10", function(x) 10^x), labels = trans_format("log10", math_format(10^.x))) +
coord_cartesian(xlim = c(0, 250)) +
labs(color = "method")
ggtitle("Optimality gap versus CPU time")$title
[1] "Optimality gap versus CPU time"
attr(,"class")
[1] "labels"p_gap_vs_iter / p_gap_vs_cpu_time + plot_layout(guides = "collect")
References
Palomar, D. P. (2025). Portfolio optimization: Theory and application. Cambridge University Press.