Local Search in Continuous Spaces

Local search algorithms can also be used in continuous spaces, where the solution space is represented by a set of continuous variables. In this case, the search process involves moving from one solution to another by making small changes to the values of the variables.

One common local search algorithm for continuous spaces is gradient descent, where the search process follows the direction of the steepest descent of a function. In gradient descent, the search starts from an initial solution and iteratively moves towards the minimum of a cost function by updating the values of the variables in the direction of the negative gradient.

Another variant of gradient descent is stochastic gradient descent, where the gradient is estimated from a random subset of the training data. This method is often used in machine learning, where the cost function is based on a training set of examples.

Simulated annealing is another local search algorithm that can be used in continuous spaces. In simulated annealing, the search process starts with an initial solution and iteratively generates new candidate solutions by making small changes to the values of the variables. The algorithm accepts a candidate solution if it improves the objective function, but may also accept worse solutions with a certain probability, which decreases over time according to a cooling schedule.

Particle swarm optimization is a metaheuristic algorithm inspired by the social behavior of bird flocking and fish schooling. In particle swarm optimization, a population of candidate solutions, represented as particles, move around the solution space guided by their own experience and the experience of other particles in the swarm.

Local search algorithms for continuous spaces can be used for a wide range of optimization problems, including function optimization, parameter tuning, and machine learning. However, these algorithms may get stuck in local optima or saddle points, which can limit their ability to find the global optimum of a problem. To address this issue, variants of these algorithms, such as simulated annealing with multiple restarts or particle swarm optimization with diversity maintenance, can be used.