Introduction

The Jaya package provides a gradient-free optimization algorithm that can solve single-objective and multi-objective optimization problems. This vignette demonstrates how to use the Jaya package for optimization tasks.

Key Features

Single-objective optimization
Multi-objective Pareto optimization
Adaptive population adjustment
Early stopping based on improvement thresholds

Installation

Ensure the Jaya package is installed and loaded:

library(Jaya)

Single-Objective Optimization

To solve a single-objective optimization problem using jaya, define an objective function and specify the bounds of the variables.

Example: Minimize Sphere Function

# Define the sphere function
sphere_function <- function(x) sum(x^2)

# Set optimization parameters
lower_bounds <- c(-5, -5, -5)
upper_bounds <- c(5, 5, 5)

# Run optimization
single_result <- jaya(
  fun = sphere_function,
  lower = lower_bounds,
  upper = upper_bounds,
  popSize = 20,
  maxiter = 50,
  n_var = length(lower_bounds),
  opt = "minimize"
)

# Summary of results
summary(single_result)
#> Jaya Algorithm
#> Population Size      = 20 
#> Number of iterations = 50 
#> Number of variables  = 3 
#> 
#> Objective: minimize 
#> Objective Function:
#> [[1]]
#> function (x) 
#> sum(x^2)
#> <bytecode: 0x556e56132980>
#> 
#> 
#> Limits:
#> x1 = [-5, 5]
#> x2 = [-5, 5]
#> x3 = [-5, 5]
#> 
#> Best Result:
#>            Best.x1      Best.x2       Best.x3   Best.f.x.
#> Best -0.0004232535 4.714157e-05 -0.0008202799 8.54225e-07

# Plot the best value over iterations
plot(single_result)

Multi-Objective Optimization

To solve a multi-objective optimization problem, provide a list of objectives and specify the bounds of the variables.

Example: Optimize Two Sphere Functions

# Define multiple objectives
objective1 <- function(x) sum(x^2)
objective2 <- function(x) sum((x - 2)^2)

# Combine objectives
objectives <- list(objective1, objective2)

# Run optimization
multi_result <- jaya_multi(
  objectives = objectives,
  lower = lower_bounds,
  upper = upper_bounds,
  popSize = 30,
  maxiter = 100,
  n_var = length(lower_bounds)
)

# Summary of results
summary(multi_result)
#>              Length Class      Mode
#> Pareto_Front 5      data.frame list
#> Solutions    5      data.frame list


# Pairwise plots for multi-objective Pareto front
plot_jaya_multi_pairwise(multi_result)

Advanced Features

Early Stopping

To stop optimization early if the improvement falls below a threshold, use the early_stopping parameter:

early_stopping_result <- jaya(
  fun = sphere_function,
  lower = lower_bounds,
  upper = upper_bounds,
  popSize = 20,
  maxiter = 50,
  n_var = length(lower_bounds),
  early_stopping = TRUE,
  tolerance = 1e-6,
  patience = 5
)

summary(early_stopping_result)
#> Jaya Algorithm
#> Population Size      = 20 
#> Number of iterations = 50 
#> Number of variables  = 3 
#> 
#> Objective: minimize 
#> Objective Function:
#> [[1]]
#> function (x) 
#> sum(x^2)
#> <bytecode: 0x556e56132980>
#> 
#> 
#> Limits:
#> x1 = [-5, 5]
#> x2 = [-5, 5]
#> x3 = [-5, 5]
#> 
#> Best Result:
#>            Best.x1      Best.x2       Best.x3    Best.f.x.
#> Best -6.414401e-05 0.0001618335 -0.0002268213 8.175243e-08

Conclusion

The Jaya package offers a versatile and easy-to-use optimization framework for a variety of problems. Customize the algorithm parameters to fit your specific needs and leverage its advanced features for improved performance.

References

R. V. Rao (2016). Jaya: A Simple and New Optimization Algorithm for Solving Constrained and Unconstrained Optimization Problems. International Journal of Industrial Engineering Computations. doi:10.5267/j.ijiec.2015.8.004

Optimizing with Jaya: A Vignette