What is steepest ascent hill climbing?

Steepest ascent hill climbing, or hill climbing, is an iterative algorithm that finds a local maximum of a function by moving in the direction of increasing function values. It is a greedy algorithm, meaning that it always makes the choice that appears to be the best at the moment, without considering the potential long-term consequences. This can sometimes lead to the algorithm getting stuck in a local maximum, rather than finding the global maximum.

The steepest ascent hill climbing algorithm works as follows:

1. Start at an initial state.

2. Evaluate the function at the current state.

3. Move to the neighbor with the highest function value.

4. Repeat steps 2 and 3 until a local maximum is reached.

A local maximum is a point where the function value is higher than all the neighbor values. The hill climbing algorithm stops at a local maximum because there are no neighbors with a higher function value.

Here is an example of steepest ascent hill climbing. Suppose we want to find the maximum of the function f(x) = x^2. We can start at the initial state x = 0. The function value at x = 0 is 0. The neighbors of x = 0 are x = 1 and x = -1. The function values at x = 1 and x = -1 are 1 and 1, respectively. Since 1 is greater than 0, we move to the neighbor x = 1. The function value at x = 1 is 1. The neighbors of x = 1 are x = 2 and x = 0. The function values at x = 2 and x = 0 are 4 and 0, respectively. Since 4 is greater than 1, we move to the neighbor x = 2. The function value at x = 2 is 4. The neighbors of x = 2 are x = 3 and x = 1. The function values at x = 3 and x = 1 are 9 and 1, respectively. Since 9 is greater than 4, we move to the neighbor x = 3. The function value at x = 3 is 9. The neighbors of x = 3 are x = 4 and x = 2. The function values at x = 4 and x = 2 are 16 and 4, respectively. Since 16 is greater than 9, we move to the neighbor x = 4. The function value at x = 4 is 16. The neighbors of x = 4 are x = 5 and x = 3. The function values at x = 5 and x = 3 are 25 and 9, respectively. Since 25 is greater than 16, we move to the neighbor x = 5. The function value at x = 5 is 25. The neighbors of x = 5 are x = 6 and x = 4. The function values at x = 6 and x = 4 are 36 and 16, respectively. Since 36 is greater than 25, we move to the neighbor x = 6. The function value at x = 6 is 36. The neighbors of x = 6 are x = 7 and x = 5. The function values at x = 7 and x = 5 are 49 and 25, respectively. Since 49 is greater than 36, we move to the neighbor x = 7. The function value at x = 7 is 49. The neighbors of x = 7 are x = 8 and x = 6. The function values at x = 8 and x = 6 are 64 and 36, respectively. Since 64 is greater than 49, we move to the neighbor x = 8. The function value at x = 8 is 64. The neighbors of x = 8 are x