Constraint Optimization with constrOptim

In optimization problems, the objective is to find the values of variables that maximize or minimize a given function, subject to certain constraints. One such method for solving these types of problems is constraint optimization using the constrOptim function in R.

Introduction to Production Function and Constraint Function

The production function represents the relationship between the inputs used to produce a good and the output produced. In this case, we have two inputs: labor (L) and capital (K). The production function can be represented as:

production_function <- function(L, K) { L * K + L }

The constraint function represents the total cost of production and is given by:

constraint_function <- function(L, K) { 2 * L + 1.4 * K }

Maximizing Production with a Constraint on Total Cost

In this problem, we want to maximize the production function while ensuring that the total cost is equal to $100.

result <-
  constrOptim(theta = c(30, 30), 
             ui = rbind(c(2, 1.4), c(0, 1), c(1, 0)), 
             ci = c(100, 0, 0), 
             f = production_function, 
             grad = NULL)

However, the constrOptim function throws an error because it expects a single parameter vector for the function to optimize.

Correcting the Production Function

The first issue is that we need to define our functions with only one parameter. We can do this by modifying the production function and constraint function as follows:

production_function <- function(theta) {
  theta[2] * theta[1] + theta[1]
}

constraint_function <- function(theta) {
  2 * theta[1] + 1.4 * theta[2]
}

Applying the Corrected Production Function to constrOptim

Now that we have corrected our functions, we can apply them to constrOptim:

result <-
  constrOptim(theta = c(30, 30), 
             ui = rbind(c(2, 1.4), c(0, 1), c(1, 0)), 
             ci = c(100, 0, 0), 
             f = production_function, 
             grad = NULL)

Understanding the Constraints

The constraints are represented in constrOptim as:

ui %*% theta - ci >= 0

This means that we have three constraints:

• 2 * L + 1.4 * K = 100
• L > 0
• K > 0

Note that these constraints are not different from the original inequality constraint. The difference is in how they are represented.

Using constrOptim with Multiple Constraints

The example above illustrates using constrOptim to find the values of variables that satisfy multiple constraints. To demonstrate this, let’s use the corrected production function and add two additional constraints:

# Define the functions
production_function <- function(theta) {
  theta[2] * theta[1] + theta[1]
}

constraint_function <- function(theta) {
  2 * theta[1] + 1.4 * theta[2]
}

# Define the constraints
constraints <- list(
  constraint_function, 
  L = -30, # L must be less than or equal to 30
  K = -30  # K must be less than or equal to 30
)

# Run constrOptim with multiple constraints
result <-
  constrOptim(theta = c(30, 30), 
             ui = rbind(c(2, 1.4), c(0, 1), c(1, 0)), 
             ci = c(100, 0, 0), 
             f = production_function, 
             grad = NULL, 
             method = "L-BFGS-B", 
             S = constraints)

In this example, we define the constraints list with the original constraint function and two additional constraints. We then pass this list to constrOptim, specifying the method as "L-BFGS-B" for large-scale optimization.

Optimization Results

When you run constrOptim with multiple constraints, it returns a result that includes the optimal values of the variables, the corresponding constraint values, and an error message if any constraints are violated.

In our case, we expect to get results where:

• The production is maximized
• The total cost equals $100
• Labor (L) is greater than 0
• Capital (K) is greater than 0

The optimal solution should be close to the initial guess c(30, 30).

# Print the result
print(result$par)

This code prints the optimal values of the variables that solve the problem. It will return a vector with two elements: [L, K].

Conclusion

Constraint optimization using constrOptim is an efficient method for solving production functions subject to multiple constraints. By applying the corrected production function and defining the constraints properly, we can leverage constrOptim to find optimal solutions that maximize production while satisfying the total cost constraint.

Example Code

Below is a self-contained code snippet illustrating how to use constrOptim with multiple constraints:

# Load necessary libraries
library(ggplot2)

# Define the functions
production_function <- function(theta) {
  theta[2] * theta[1] + theta[1]
}

constraint_function <- function(theta) {
  2 * theta[1] + 1.4 * theta[2]
}

# Define the constraints
constraints <- list(
  constraint_function, 
  L = -30, # L must be less than or equal to 30
  K = -30  # K must be less than or equal to 30
)

# Run constrOptim with multiple constraints
result <-
  optim(c(30, 30), production_function, 
       method="L-BFGS-B", S=constraints$constraint_function,
       constraint.etype.c("eq","ineq"), lower = c(-Inf,-Inf), upper=c(Inf, Inf))

# Print the result
print(result$x)

# Plot the results
plot(theta ~ L, data.frame(L = 1:10))

Note that optimization algorithms may vary and different methods are available in optim() function as well.

Last modified on 2024-08-03