The Local Search algorithm is a method for finding the best solution within a vicinity of the current state. This technique is often used to refine approximate solutions by iteratively modifying individual components to discover better states.
How It Works
- Initialization: Begin with an initial state.
- Generate Neighbors: Generate neighboring states by changing a component of the current state.
- Evaluation: Evaluate the quality of neighboring states using an objective function.
- Select Best State: Choose the neighboring state with the best objective value.
- Repeat: Iterate through steps 2 to 4 until no better neighboring state can be found.
Example: Optimizing the Fibonacci Function
Consider the optimization problem of the Fibonacci function F(x) = F(x-1) + F(x-2) with F(0) = 0, F(1) = 1. We want to find the value of x for which F(x) is maximized. We can use the Local Search approach to iteratively explore farther from each step.
Code Example in C++
In this example, we utilize the Local Search method to optimize the Fibonacci function. We iterate through different values of x and calculate the Fibonacci value at each x. When a better value is found, we update the best value and its corresponding x.