Update algorithm.md

docs/src/algorithm.md

@@ -6,66 +6,63 @@ The strength of the DIRECT algorithm lies in its ability to systematically explo
 
 Additionally, by not requiring the Lipschitz constant, the DIRECT algorithm is adaptable to various optimization problems, including those where the smoothness of the objective function is not well understood.
 
-- The figure below shows the DIRECT method after 16 iterations on the Branin function. The cells are much denser around the minima of the Branin function because the DIRECT method is designed to increase its resolution in promising regions.
+The figure below shows the DIRECT method after 16 iterations on the Branin function. The cells are much denser around the minima of the Branin function because the DIRECT method is designed to increase its resolution in promising regions.
 
 ![page_11](https://github.com/user-attachments/assets/b833bedd-41aa-40c5-a27f-26188a171797)
 
 ## Key Concepts of the DIRECT Algorithm
 
 ### **Search Space**:
 
-   - The algorithm minimizes an objective function f(x) over a hyper-rectangular search space.
-   - The search space is normalized to the unit hypercube to avoid oversensitivity to dimensions with larger domains. If minimizing $f(x)$ in the interval between lower and upper ranges $a$ and $b$, DIRECT will instead minimize:
+The algorithm minimizes an objective function f(x) over a hyper-rectangular search space. The search space is then normalized to the unit hypercube to avoid oversensitivity to dimensions with larger domains. If given an objective function $f(x)$ in the interval between lower and upper domains $a$ and $b$, DIRECT will instead minimize:
 
   ```math
    g(\mathbf{x}) = f(\mathbf{x} \odot (\mathbf{b} - \mathbf{a}) + \mathbf{a})
 ```
 
-After finding the minimum $x^*$ of $g$, The minimizer of $f$ is:
+After finding the minimum $x^*$ of $g$, the minimizer of $f$ is:
 
 ```math
 \mathbf{x}^* \odot (\mathbf{b} - \mathbf{a}) + \mathbf{a}
 ```
 
 ### **Function Evaluation**:
-   - DIRECT partitions its search space into hyperrectangular intervals.
-   - The objective function is evaluated at the center of each hyper-rectangle.
-   - Each interval has a center $c^{(i)}$, an associated objective function value $f(c^{(i)})$, and a radius $r^{(i)}$. The radius is the distance from the center to a vertex.
+DIRECT partitions its search space into hyperrectangular intervals. The objective function is evaluated at the center of each hyperrectangle. Each interval has a center $c^{(i)}$, an associated objective function value $f(c^{(i)})$, and a radius $r^{(i)}$. The radius is the distance from the center to a vertex.
 
 ### **Selection of Potentially Optimal Rectangles**:
-   - In each iteration, the algorithm identifies potentially optimal rectangles. A rectangle is considered potentially optimal if it could contain the global minimum based on the evaluations performed so far.
+In each iteration, the algorithm identifies potentially optimal intervals. An interval is considered potentially optimal if there exists a Lipschitz constant such that said interval could contain the global minimum based on the evaluations performed so far.
 
 ## Lipschitz Lower Bound:
 
-- The Lipschitz lower bound for an interval is a circular cone extending downward from its center $c^{(i)}$.
+The Lipschitz lower bound for an interval is a circular cone extending downward from its center $c^{(i)}$:
 
  ```math
   f(\mathbf{x}) \geq f(\mathbf{c}^{(i)}) - \ell \|\mathbf{x} - \mathbf{c}^{(i)}\|_2
 ``` 
-   - This lower bound is constrained by the extents of the interval, and its lowest value is achieved at the vertices, which are all a distance $r^{(i)}$ from the center.
+This lower bound is constrained by the extents of the interval, and its lowest value is achieved at the vertices, which are all a distance $r^{(i)}$ from the center.
 
 ```math
 f(c^{(i)}) - \ell r^{(i)}
 ```
-- The left plot shows the intervals for the DIRECT method after 5 iterations on the Branin function. The right plot shows the interval objective function values versus their radii, which is useful for identifying intervals to split with further evaluations.
+The left plot below shows the intervals for the DIRECT method after 5 iterations on the Branin function. The right plot shows the interval objective function values versus their radii, which is useful for identifying intervals to split with further evaluations.
 
 ![page_12](https://github.com/user-attachments/assets/34da1f5e-c983-45cc-8b6c-531184d4b756)
 
-- The left plot shows the Lipschitz lowerbounds constructed for the DIRECT intervals using the Lipschitz constant = 200, and highlights one interval. The right plot shows how the minimum value for the lowerbound within the highlighted interval is the same as the x-intercept for a line of slope passing through that interval’s $(r, f(c))$ point.
+The left plot below shows the Lipschitz lowerbounds constructed for the DIRECT intervals using the Lipschitz constant = 200, and highlights one interval. The right plot shows how the minimum value for the lowerbound within the highlighted interval is the same as the x-intercept for a line of slope passing through that interval’s $(r, f(c))$ point.
 
 ![page_13](https://github.com/user-attachments/assets/f023e6b0-ee8a-48fb-a8b6-c1b68d819377)
 
-- The left plot continues to show the Lipschitz lowerbounds constructed for the DIRECT intervals using the Lipschitz constant = 200, but now highlights the interval containing the lowest value. The right plot shows how the lowest lowerbound for a given Lipschitz constant is the one with the lowest x-intercept in the right-hand plot.
+The left plot below continues to show the Lipschitz lowerbounds constructed for the DIRECT intervals using the Lipschitz constant = 200, but now highlights the interval containing the lowest value. The right plot shows how the lowest lowerbound for a given Lipschitz constant is the one with the lowest x-intercept in the right-hand plot.
 
 ![page_14](https://github.com/user-attachments/assets/7df39f70-2ef2-4eaa-a5de-54f93d21e653)
 
 ### **Recursive Division**:
-   - The selected rectangles are further divided, splitting into thirds along the axis directions. The order in which dimensions are split matters; lower function evaluations receive larger sub-rectangles. The process continues recursively, refining the search by focusing on the most promising regions.
+The selected intervals are further divided, splitting into thirds along the axis directions. The order in which dimensions are split matters; lower function evaluations receive larger sub-rectangles. The process continues recursively, refining the search by focusing on the most promising regions.
 
 ### **Convex Hull**:
-   - The DIRECT method selects all intervals for which a Lipschitz constant exists such that their lower bounds have minimal value. These intervals form a piecewise-linear boundary along the lower-right of the $(r, f(c))$ space.
+The DIRECT method selects all intervals for which a Lipschitz constant exists such that their lower bounds have minimal value. These intervals form a piecewise-linear boundary along the lower-right of the $(r, f(c))$ space.
 
-- The left plot shows the split intervals identified for this iteration of DIRECT on the Branin function. The right plot shows the lower-right convex hull formed by these split intervals in $(r, f(c))$ space.
+The left plot shows the split intervals identified for this iteration of DIRECT on the Branin function. The right plot shows the lower-right convex hull formed by these split intervals in $(r, f(c))$ space.
 
 ![page_15](https://github.com/user-attachments/assets/5142788b-814e-4221-b50f-1746672278df)
 
@@ -77,11 +74,12 @@ that has been split $d$ times will have a width of $3^{−d}$. If we have $n$ di
 ```math
 r = \left\|\left[ \frac{1}{2 \cdot 3^{-d_1}}, \dots, \frac{1}{2 \cdot 3^{-d_n}} \right]\right\|_2
 ```
-- **Interval splitting in multiple dimensions for DIRECT** requires choosing an ordering for the split dimensions:
+The figure below shows how interval splitting in for DIRECT requires choosing an ordering for the split dimensions:
 
   ![page_16](https://github.com/user-attachments/assets/962993d9-372a-4733-9d1f-0260cdacdff1)
 
-- **DIRECT will only split the longest dimensions of intervals**. The algorithm only divides intervals larger than a minimum radius. This minimum radius prevents inefficient function evaluations very close to existing points:
+DIRECT will only split the longest dimensions of intervals:
 
   ![page_17](https://github.com/user-attachments/assets/99caea66-02b5-4371-90e2-69305c035ddf)
 
+Finally, DIRECT only splits intervals larger than a minimum radius. This minimum radius prevents inefficient function evaluations very close to existing points.