Update README.md

README.md

@@ -10,18 +10,7 @@
 **DividedRectangles.jl** provides an implementation of the DIRECT (DIvided RECTangles) [algorithm for global optimization](https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Lipschitzian+optimization+without+the+Lipschitz+constant&btnG=). The DIRECT algorithm is particularly useful for optimizing functions where the Lipschitz constant is unknown. This package allows users to perform both univariate and multivariate optimization efficiently.
 
 ---
-
-Example Objective Function:
-As an example of an objective function, consider:
-
-```julia
-f(x) = dot(coeffs, x)
-```
-where:
-- `x`: represents the variables.
-- `coeffs`: represents the coefficients corresponding to each variable.
-
-This is just one possible objective function that could be optimized using the DIRECT algorithm. The algorithm works by dividing the search space into smaller rectangles and evaluating the function at specific points within these rectangles to find the optimal solution.
+![page_11](https://github.com/user-attachments/assets/b833bedd-41aa-40c5-a27f-26188a171797)
 
 ---
 
@@ -36,29 +25,31 @@ using Pkg
 Pkg.add(url="https://github.com/sisl/DividedRectangles.jl")
 
 ```
-# Usage 
+# Usage  
 
 To use the `DividedRectangles` module, start your code with:
 
 ```julia
 using DividedRectangles
 ```
+---
+
 ## Core Functions
 
 ### `optimize`
 The `optimize` function is the primary function of the `DividedRectangles` module. It implements the DIRECT algorithm to find the minimum of a given objective function within specified bounds.
 
-To use the `optimize` function with a custom mathematical function:
+To use the `optimize` function with a custom objective function::
 
 ```julia
 using DividedRectangles
 
 # Define the objective function
-f(x) = dot([1.0, 2.0, 3.0], x)  # Example objective function with coefficients
+f(x) = x[1]^2 + x[2]^2 + 3 * sin(5 * x[1]) + 2 * cos(3 * x[2])  # Multivariate example
 
 # Set the search bounds
-a = [0.0, 0.0, 0.0]
-b = [1.0, 1.0, 1.0]
+a = [-2.0, -2.0]
+b = [2.0, 2.0]
 
 # Call the optimization function
 result = optimize(f, a, b)
@@ -78,43 +69,6 @@ println("Best design found: ", result)
 - The best design 𝑥 found by DIRECT.
 ---
 
-### Example: Multivariate Optimization
-
-The following example shows how to optimize a multivariate function using the DIRECT algorithm:
-
-```julia
-using DividedRectangles
-
-# Define the objective function
-f(x) = x[1]^2 + x[2]^2 + 3 * sin(5 * x[1]) + 2 * cos(3 * x[2])
-
-# Set the search bounds
-a = [-2.0, -2.0]
-b = [2.0, 2.0]
-
-# Optimize
-result = DividedRectangles.optimize(f, a, b)
-
-println("Best design found: ", result)
-```
-
-### Parameters
-- `f`: This is the objective function to minimize. Should be an operation that accepts a vector of Float64 values.
-- `a`: Vector representing the lower bounds of the search space.
-- `b`: Vector representing the upper bounds of the search space.
-- `max_iterations`:  Maximum number of iterations for the optimization (default: 100).
-- `min_radius`: Minimum size of hyper-rectangles (`default: 1e-5`).
-
-## Advanced Usage
-### Fine-Tuning Optimization:
-The `optimize` function offers several parameters for fine-tuning the optimization process:
-
-- `max_iterations`: Sets the maximum number of iterations. Increasing this value can improve accuracy but requires more computational time.
-- `min_radius`: Specifies the minimum size of hyper-rectangles to control the granularity of the search.
-
-```julia
-result = DividedRectangles.optimize(f, a, b, max_iterations=500, min_radius=1e-6)
-```
 ## Credits
 
 Contributors to this package include Anshrin Srivastava, Mykel Kochenderfer, Dylan Asmar, and Tim Wheeler.