LIA (Local improvement algorithm)

Last updated on Apr 25, 2023

Local improvement algorithms (LIAs) are a class of optimization algorithms that aim to find the best solution to a given problem within a local search space. LIAs are often used in problems where the search space is too large or too complex to explore exhaustively, and where a global optimization algorithm may be too slow or impractical.

In this article, we will discuss the basics of LIA, its applications, and some common types of local search algorithms.

Basics of Local Improvement Algorithms (LIAs)

LIAs start with an initial solution and iteratively improve it by making local modifications. The goal is to find a better solution by exploring the neighboring solutions in the search space.

The search space is defined by the problem constraints, and the neighboring solutions are defined by a set of operators that can modify the current solution. The operators are designed to change the solution in a way that preserves the constraints of the problem. For example, in a scheduling problem, the operators could be swapping two tasks or shifting the start time of a task by a certain amount.

The quality of a solution is measured by a cost or objective function that assigns a score to each solution. The goal of the algorithm is to minimize or maximize this score, depending on the problem.

The algorithm starts with an initial solution and iteratively applies the operators to generate new solutions. If a new solution improves the score, it becomes the current solution, and the search continues. Otherwise, the algorithm tries another operator or accepts the current solution and starts a new iteration.

The algorithm terminates when a stopping condition is met, such as reaching a maximum number of iterations or finding a solution that meets a certain quality threshold.

Applications of Local Improvement Algorithms (LIAs)

LIAs have been applied to a wide range of optimization problems in various domains, including scheduling, routing, clustering, and packing problems. Some common applications of LIAs are:

Traveling salesman problem: Given a set of cities and the distances between them, find the shortest possible route that visits each city exactly once and returns to the starting city.
Vehicle routing problem: Given a set of vehicles and a set of customers with delivery locations, find the optimal routes for the vehicles to deliver the goods to the customers.
Job shop scheduling problem: Given a set of jobs and a set of machines with processing times, find the optimal schedule that minimizes the total processing time or makespan.
Bin packing problem: Given a set of items with sizes and a set of bins with capacities, find the optimal way to pack the items into the bins to minimize the number of bins used.

Local Search Algorithms

There are several types of local search algorithms, including hill climbing, simulated annealing, tabu search, and genetic algorithms. Each algorithm has its strengths and weaknesses and is suitable for different types of problems.

Hill Climbing

Hill climbing is a simple and intuitive algorithm that starts with an initial solution and iteratively improves it by making small modifications. The algorithm selects the best neighbor and moves to that neighbor, repeating the process until no better neighbor can be found.

The main drawback of hill climbing is that it may get stuck in a local minimum or maximum and fail to find the global optimal solution. To overcome this problem, variants of hill climbing have been proposed, such as random restarts, which restart the algorithm with a new random initial solution when it gets stuck, and steepest ascent, which explores all neighbors and selects the best one.

Simulated Annealing

Simulated annealing is a probabilistic algorithm that starts with an initial solution and iteratively explores the neighboring solutions. The algorithm accepts a worse neighbor with a certain probability, which decreases over time, to escape from local minima.

The name of the algorithm is inspired by the process of annealing in metallurgy, where a material is slowly cooled to obtain a desired crystal structure. The idea is that the algorithm starts with a high temperature that allows it to accept worse neighbors and gradually decreases the temperature, reducing the probability of accepting worse solutions.

Simulated annealing is a powerful algorithm that can escape from local minima and find the global optimal solution, but it requires careful tuning of the temperature schedule and acceptance probability.

Tabu Search

Tabu search is an algorithm that uses a memory mechanism to avoid revisiting the same solutions. The algorithm starts with an initial solution and iteratively explores the neighboring solutions, but it keeps a list of the solutions visited in the past and prohibits revisiting them.

The memory mechanism is called a tabu list, and it can be used to prevent cycling, promote diversity, or enforce problem-specific constraints. The algorithm accepts a worse neighbor if it has not been visited before or if it violates a taboo rule with a certain penalty.

Tabu search is a powerful algorithm that can escape from local minima and enforce constraints, but it requires careful tuning of the tabu rules and penalties.

Genetic Algorithms

Genetic algorithms are a class of metaheuristic algorithms that mimic the process of natural selection and evolution. The algorithm starts with a population of solutions and iteratively generates new solutions by combining and mutating the existing ones.

The solutions are represented as chromosomes, which are encoded as strings of genes. The genes represent the components of the solution, and the chromosomes represent the complete solution.

The algorithm evaluates the fitness of each chromosome, which is a measure of how well it solves the problem. The fittest chromosomes are selected for reproduction, and their genes are combined to form new chromosomes. The new chromosomes are then mutated by randomly changing some of their genes.

Genetic algorithms are a powerful algorithm that can explore the search space and converge to good solutions, but they can be slow and require a large population size and a long running time.

Conclusion

Local improvement algorithms are a powerful class of optimization algorithms that can find good solutions to a wide range of problems. They start with an initial solution and iteratively improve it by exploring the neighboring solutions in the search space.

LIAs have been applied to various domains, including scheduling, routing, clustering, and packing problems, and they have been shown to produce high-quality solutions in practice. LIAs are particularly useful when the search space is too large or too complex to explore exhaustively, and when a global optimization algorithm may be too slow or impractical.