Introduction

Suppose you are planning a very large event and realize that you need to figure out the most efficient way to distribute the workload among the team members. You try a number of approaches but you get stuck and can’t move forward. This is where local search algorithms come in handy. Hill climbing and simulated annealing are some of these tricks that will help you escape these repetitive problems and come up with better solutions.

In this article, we’ll discuss LS algorithms, where they’re used in AI, and how they can make you a better problem solver, whether you’re working on task scheduling or function optimization.

Learning outcomes

• Understand the core principles of local search algorithms.
• Identify common types of local search algorithms and their possible applications.
• Learn how to implement and apply these algorithms in practical scenarios.
• Gain insight into optimizing local search processes and addressing potential challenges.

Core Principles of Local Search Algorithms

Local search algorithms are designed to solve optimization problems by moving from one solution to another in the neighborhood. Simply put, it consists of taking an initial solution and making incremental changes to optimize it.

• Initial solution: Start with an initial guess or solution.
• Generation neighbors: Generate adjacent solutions by making small changes to the current solution.
• Evaluation: Evaluate the quality of the adjacent solutions using a predefined objective function.
• Selection: Choose the best neighbor as the new current solution.
• Termination: Repeat the process until a stopping criterion is met (for example, a maximum number of iterations or no improvement).

Common Types of Local Search Algorithms

• Hill climbing: A simple algorithm that continuously moves to the highest valued neighboring solution. It is intuitive, but can get stuck in local optima.
• Simulated annealing: An extension of hill climbing that allows incidental moves to worse solutions to escape local optima. It uses a temperature parameter that gradually decreases over time.
• Genetic algorithms: Although many researchers categorize GA as belonging to the category of evolutionary algorithms, these algorithms also exploit features of local search through processes such as mutation and crossover to search the solution space.
• Tabu Search: Tabu search is a more advanced method than the basic Hill Climbing algorithm, as it incorporates specific memory structures that prevent the solutions from returning to previous states and thus avoiding the local optima.
• Particle Swarm Optimization (PSO): Another technique, Particle-Swarm Optimization (PSO), tries to find a solution in the domain of a function; during this, particles study their positions and adjust them based on their best individual position and the best position of the whole swarm. This method helps to find the best solutions by optimizing multi-variable functions in a special way.

Practical implementation

To effectively implement local search algorithms, follow these steps:

• Define the problem: Provide a clear description of the optimization problem, including the objective function and constraints.
• Choose an algorithm: Select a local search algorithm that suits the characteristics of the problem.
• Implement the algorithm: Write code to initialize the solution, generate neighbors, evaluate them, and handle termination.
• Tuning parameters off: Adjust algorithm parameters (e.g. simulated annealing temperature) to balance exploration and exploitation.
• Validate results: Test the algorithm at different points in the problem to make sure it performs well.

Examples of local search algorithms

Let’s take a closer look at some local search algorithms below.

Hill climbing

Hill Climbing is a simple approach that goes to the neighboring solution with the highest value. Although intuitive, it can get stuck in local optima.

Example

def hill_climbing(initial_solution, objective_function):
    current_solution = initial_solution
    current_score = objective_function(current_solution)

    while True:
        neighbors = generate_neighbors(current_solution)
        best_neighbor = None
        best_neighbor_score = current_score

        for neighbor in neighbors:
            score = objective_function(neighbor)
            if score > best_neighbor_score:
                best_neighbor = neighbor
                best_neighbor_score = score

        if best_neighbor is None:
            break

        current_solution = best_neighbor
        current_score = best_neighbor_score

    return current_solution, current_score

def generate_neighbors(solution):
    # Example neighbor generation for a simple case
    return (solution + 1, solution - 1)

def objective_function(x):
    return -x**2  # Example: maximization problem

initial_solution = 0
best_solution, best_score = hill_climbing(initial_solution, objective_function)
print(f"Best solution: {best_solution} with score: {best_score}")

Output:

Best solution: 0 with score: 0

Simulated Glow

The basis of the Simulated Annealing algorithm is the annealing process which refers to metallurgy, in which the metal is gradually cooled to eliminate the presence of defects in the structure. It initializes the temperature as high so that the algorithm can traverse more space of the solution, and then comes up with low temperatures to shorten the time of accepting the solution, which is worse.

Example

Let us focus on the formal problem, such as a traveling salesman problem in which a salesman has to travel through several cities and return to the starting point in the minimum time. One technique to quickly find a constraint-optimal route is to use simulated annealing. This method sometimes accepts a longer route in the hope of discovering a better overall route.

   import random
   import math

   def objective_function(route):
       # Example function: the total distance of the route
       return sum(math.sqrt((route(i) - route(i-1))**2) for i in range(len(route)))

   def simulated_annealing(initial_route, temperature, cooling_rate):
       current_route = initial_route
       current_score = objective_function(current_route)
       best_route = current_route
       best_score = current_score

       while temperature > 0.1:
           new_route = current_route(:)
           i, j = random.sample(range(len(route)), 2)
           new_route(i), new_route(j) = new_route(j), new_route(i)
           new_score = objective_function(new_route)

           if new_score 

Output:

Best route: (0, 1, 2, 3, 4) with score: 8.0

Tabu Search

Tabu Search uses memory structures to keep track of recently visited solutions, preventing the algorithm from revisiting them. This helps avoid cycles and encourages exploration of new areas of the solution space.

Example

You can use taboo searches for job scheduling issues to assign jobs to different machines and minimize overall completion time by avoiding recently attempted job assignments.

   import random

   def objective_function(schedule):
       # Example function: total completion time
       return sum(job('duration') for job in schedule)

   def tabu_search(initial_schedule, iterations, tabu_tenure):
       current_schedule = initial_schedule
       best_schedule = current_schedule
       best_score = objective_function(current_schedule)
       tabu_list = ()

       for _ in range(iterations):
           neighbors = generate_neighbors(current_schedule)
           best_neighbor = None
           best_neighbor_score = float('inf')

           for neighbor in neighbors:
               if neighbor not in tabu_list:
                   score = objective_function(neighbor)
                   if score  tabu_tenure:
                   tabu_list.pop(0)

               if best_neighbor_score 

Output:

Best schedule: ({'job': 'A', 'duration': 3}, {'job': 'B', 'duration': 2}, {'job': 'C', 'duration': 1}) with score: 6

Greedy algorithms

Many organizations use GA build up solutions piecemeal, often choosing the piece that will deliver the most short-term benefits. While they are not always the best solutions, they can be powerful for a variety of problems.

Example

In the knapsack problem, if you need to capture as much value as possible within the allowed weight of the bag, you can tackle this by using a greedy algorithm. This approach sorts items based on their value-to-weight ratio.

   def knapsack_greedy(items, capacity):
       items = sorted(items, key=lambda x: x('value') / x('weight'), reverse=True)
       total_value = 0
       total_weight = 0

       for item in items:
           if total_weight + item('weight') 

Output:

Maximum value in backpack: 160

Particle swarm optimization

PSO is based on the imitation of the activity of birds and fish. Agents (or particles) roam around the search space of the problems while adjusting their positions based on their own learning experiences and the learning experiences of their neighbors.

Example

You can apply PSO to function optimization problems, where particles explore the domain of the function and update their positions based on their individual and collective best solutions.

   import numpy as np

   def objective_function(x):
       return sum(x**2)

   def particle_swarm_optimization(num_particles, dimensions, iterations):
       particles = np.random.rand(num_particles, dimensions)
       velocities = np.random.rand(num_particles, dimensions)
       personal_best = particles.copy()
       global_best = particles(np.argmin((objective_function(p) for p in particles)))

       for _ in range(iterations):
           for i in range(num_particles):
               r1, r2 = np.random.rand(dimensions), np.random.rand(dimensions)
               velocities(i) = 0.5 * velocities(i) + 2 * r1 * (personal_best(i) - particles(i)) + 2 * r2 * (global_best - particles(i))
               particles(i) += velocities(i)
               if objective_function(particles(i)) 

Output:

Best position: ( 3.35110987e-07  6.94381793e-07 -1.03625781e-06  2.22941746e-06
 -9.73259302e-07) with value: 7.585831600413816e-12

Conclusion

The local search algorithms are efficient decision-making tools for solving the optimization problems, taking into account the improvement of the given neighborhood solutions. Therefore, the introduction of indices, even from the point of view of local search, is instrumental in reaching cognitive preliminary theorems, regardless of the tasks you are likely to encounter – schema determination, routing or varieties of design problems. If you make a good choice of the algorithm, tune its parameters correctly and check the results, you can deal with complex solutions of the space and obtain a good or almost the best solution to solve the problem at hand.

Frequently Asked Questions