Package {EMNA}

Type:	Package
Title:	Estimation of Multivariate Normal Algorithm
Description:	Implements the Estimation of Multivariate Normal Algorithm (EMNA), an Estimation of Distribution Algorithm (EDA) for continuous optimization problems. The method iteratively selects the best individuals from a population, estimates the multivariate normal distribution of these selected individuals, and generates new candidate solutions by sampling from the estimated distribution. This process allows the probability model to adapt toward promising regions of the search space. The implementation includes an option to slow down convergence, along with an optional explicit exploration strategy for population initialization. This explicit exploration strategy is described in Salinas-Gutierrez and Muñoz-Zavala (2023) <doi:10.1016/j.asoc.2023.110230>.
Version:	0.1.0
License:	GPL-3
Encoding:	UTF-8
Imports:	EEEA, mvtnorm
Config/roxygen2/version:	8.1.1
NeedsCompilation:	no
Packaged:	2026-10-29 04:02:38 UTC; rogel
Author:	Rogelio Salinas Gutiérrez [aut, cre, cph], Manuel Alonso Segoviano Baltazar [aut, cph], Jhon Daniel Aguilar Payares [aut, cph], Juan Alberto Dávila del Alto [aut, cph], Jonathan Michell Jauregui Carranza [aut, cph], Hiram Efraim Macias Ruelas [aut, cph], Byron Axel Morales Gutiérrez [aut, cph], María Fernanda Nieto Guerrero [aut, cph], Pedro Abraham Montoya Calzada [aut, cph]
Maintainer:	Rogelio Salinas Gutiérrez <rogelio.salinas@edu.uaa.mx>
Repository:	CRAN
Date/Publication:	2026-11-13 22:50:15 UTC

Estimation of Multivariate Normal Algorithm

Description

An Estimation of Distribution Algorithm (EDA) that models a population using a multivariate normal distribution.

Usage

EMNA(fn, lower, upper, n, d, iter=100, eeea = TRUE,alpha=NULL,epsilon=NULL)

Arguments

Value

Note

If both alpha and epsilon are assigned a value, the function employs the slow-convergence method that utilizes alpha.

Author(s)

Rogelio Salinas Gutiérrez, Manuel Alonso Segoviano Baltazar, Jhon Daniel Aguilar Payares, Juan Alberto Dávila del Alto, Jonathan Michell Jauregui CarranzA, Hiram Efraim Macias Ruelas, Byron Axel Morales Gutiérrez, María Fernanda Nieto Guerrero, Pedro Abraham Montoya Calzada.

References

Larranaga, P. and Lozano, J. A. (2002). Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers.

See Also

https://CRAN.R-project.org/package=EEEA

Examples

# Sphere objective function

sphere <- function(x) {
return(sum(x^2))
}

# Run EMNA using

set.seed(123)
result <- EMNA(
fn = sphere,
lower = -50,
upper = 50,
n = 20,
d = 2,
iter = 100,
eeea = FALSE,
alpha=NULL,
epsilon=2.5)

# Best individual
result$par
# Best objective function value
result$value

Kullback-Leibler Divergence Between Two Multivariate Normal Distributions

Description

Computes the Kullback-Leibler divergence between two multivariate normal distributions with given mean vectors and covariance matrices.

Usage

kldivergence(mu1, sigma1, mu2, sigma2)

Arguments

Details

Given two multivariate normal distributions

Q \sim N(\mu_1, \Sigma_1), \qquad P \sim N(\mu_2, \Sigma_2)

of dimension d, the Kullback-Leibler divergence of Q from P is given by

D_{KL}(Q \| P) = \frac{1}{2}\left[\mathrm{tr}(\Sigma_2^{-1}\Sigma_1) + (\mu_2 - \mu_1)^\top \Sigma_2^{-1}(\mu_2 - \mu_1) - d + \ln\frac{\det \Sigma_2}{\det \Sigma_1}\right].

The function evaluates this expression directly from the provided means and covariances. The parameter d is taken as the length of mu1.

The KL divergence is non-negative and equals zero if and only if \mu_1 = \mu_2 and \Sigma_1 = \Sigma_2. It is not symmetric, i.e., D_{KL}(Q \| P) \neq D_{KL}(P \| Q) in general.

For the formula to be valid, both sigma1 and sigma2 must be symmetric and positive-definite, so that their inverses and determinants are well-defined.

Value

A numeric scalar giving the Kullback-Leibler divergence D_{KL}(Q \| P).

Author(s)

Rogelio Salinas Gutiérrez, Manuel Alonso Segoviano Baltazar, Jhon Daniel Aguilar Payares, Juan Alberto Dávila del Alto, Jonathan Michell Jauregui CarranzA, Hiram Efraim Macias Ruelas, Byron Axel Morales Gutiérrez, María Fernanda Nieto Guerrero, Pedro Abraham Montoya Calzada.

References

Cover, T. M., & Thomas, J. A. (1991). Elements of Information Theory. John Wiley & Sons.

Examples

# Bivariate case

mu1 <- c(0, 0)
mu2 <- c(1, -1)
sigma1 <- matrix(c(1, 0.2, 0.2, 1), nrow = 2)
sigma2 <- matrix(c(2, 0.1, 0.1, 1.5), nrow = 2)

kldivergence(mu1, sigma1, mu2, sigma2)

# Multivariate case

mu1<-c(1,-1,2,0)
mu2<-c(0,0,1,2)
sigma1 <- matrix(c(3,0.5,0.2,0.1,0.5,2,0.3,0.2,0.2,0.3,4,0.5,0.1,0.2,0.5,1.5),4,4,byrow = TRUE)
sigma2 <- matrix(c(2,0.1,0.1,0,0.1,1.5,0.2,0.1,0.1,0.2,2.5,0.3,0,0.1,0.3,2),4,4,byrow = TRUE)

# kldivergence = 0

kldivergence(mu1,sigma1,mu1,sigma1)

# kldivergence is not symetric

kldivergence(mu1,sigma1,mu2,sigma2)
kldivergence(mu2,sigma2,mu1,sigma1)