**Iterated Local Search** is a stochastic global optimization algorithm.

It involves the repeated using of a local search algorithm to modified versions of a good solution found previously. In this way, it is like a clever version of the stochastic hill climbing with random restarts algorithm.

The intuition overdue the algorithm is that random restarts can help to locate many local optima in a problem and that largest local optima are often tropical to other local optima. Therefore modest perturbations to existing local optima may locate largest or plane weightier solutions to an optimization problem.

In this tutorial, you will discover how to implement the iterated local search algorithm from scratch.

After completing this tutorial, you will know:

- Iterated local search is a stochastic global search optimization algorithm that is a smarter version of stochastic hill climbing with random restarts.
- How to implement stochastic hill climbing with random restarts from scratch.
- How to implement and wield the iterated local search algorithm to a nonlinear objective function.

Let’s get started.

## Tutorial Overview

This tutorial is divided into five parts; they are:

- What Is Iterated Local Search
- Ackley Objective Function
- Stochastic Hill Climbing Algorithm
- Stochastic Hill Climbing With Random Restarts
- Iterated Local Search Algorithm

## What Is Iterated Local Search

Iterated Local Search, or ILS for short, is a stochastic global search optimization algorithm.

It is related to or an extension of stochastic hill climbing and stochastic hill climbing with random starts.

It’s substantially a increasingly clever version of Hill-Climbing with Random Restarts.

— Page 26, Essentials of Metaheuristics, 2011.

Stochastic hill climbing is a local search algorithm that involves making random modifications to an existing solution and unsuspicious the modification only if it results in largest results than the current working solution.

Local search algorithms in unstipulated can get stuck in local optima. One tideway to write this problem is to restart the search from a new randomly selected starting point. The restart procedure can be performed many times and may be triggered without a stock-still number of function evaluations or if no remoter resurgence is seen for a given number of algorithm iterations. This algorithm is tabbed stochastic hill climbing with random restarts.

The simplest possibility to modernize upon a forfeit found by LocalSearch is to repeat the search from flipside starting point.

— Page 132, Handbook of Metaheuristics, 3rd edition 2019.

Iterated local search is similar to stochastic hill climbing with random restarts, except rather than selecting a random starting point for each restart, a point is selected based on a modified version of the weightier point found so far during the broader search.

The perturbation of the weightier solution so far is like a large jump in the search space to a new region, whereas the perturbations made by the stochastic hill climbing algorithm are much smaller, serving to a explicit region of the search space.

The heuristic here is that you can often find largest local optima near to the one you’re presently in, and walking from local optimum to local optimum in this way often outperforms just trying new locations entirely at random.

— Page 26, Essentials of Metaheuristics, 2011.

This allows the search to be performed at two levels. The hill climbing algorithm is the local search for getting the most out of a explicit candidate solution or region of the search space, and the restart tideway allows variegated regions of the search space to be explored.

In this way, the algorithm Iterated Local Search explores multiple local optima in the search space, increasing the likelihood of locating the global optima.

The Iterated Local Search was proposed for combinatorial optimization problems, such as the traveling salesman problem (TSP), although it can be unromantic to continuous function optimization by using variegated step sizes in the search space: smaller steps for the hill climbing and larger steps for the random restart.

Now that we are familiar with the Iterated Local Search algorithm, let’s explore how to implement the algorithm from scratch.

## Ackley Objective Function

First, let’s pinpoint a channeling optimization problem as the understructure for implementing the Iterated Local Search algorithm.

The Ackley function is an example of a multimodal objective function that has a each global optima and multiple local optima in which a local search might get stuck.

As such, a global optimization technique is required. It is a two-dimensional objective function that has a global optima at [0,0], which evaluates to 0.0.

The example unelevated implements the Ackley and creates a three-dimensional surface plot showing the global optima and multiple local optima.

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# ackley multimodal function from numpy import arange from numpy import exp from numpy import sqrt from numpy import cos from numpy import e from numpy import pi from numpy import meshgrid from matplotlib import pyplot from mpl_toolkits.mplot3d import Axes3D # objective function def objective(x, y): return –20.0 * exp(–0.2 * sqrt(0.5 * (x**2 y**2))) – exp(0.5 * (cos(2 * pi * x) cos(2 * pi * y))) e 20 # pinpoint range for input r_min, r_max = –5.0, 5.0 # sample input range uniformly at 0.1 increments xaxis = arange(r_min, r_max, 0.1) yaxis = arange(r_min, r_max, 0.1) # create a mesh from the axis x, y = meshgrid(xaxis, yaxis) # compute targets results = objective(x, y) # create a surface plot with the jet verisimilitude scheme figure = pyplot.figure() axis = figure.gca(projection=‘3d’) axis.plot_surface(x, y, results, cmap=‘jet’) # show the plot pyplot.show() |

Running the example creates the surface plot of the Ackley function showing the vast number of local optima.

We will use this as the understructure for implementing and comparing a simple stochastic hill climbing algorithm, stochastic hill climbing with random restarts, and finally iterated local search.

We would expect a stochastic hill climbing algorithm to get stuck hands in local minima. We would expect stochastic hill climbing with restarts to find many local minima, and we would expect iterated local search to perform largest than either method on this problem if configured appropriately.

## Stochastic Hill Climbing Algorithm

Core to the Iterated Local Search algorithm is a local search, and in this tutorial, we will use the Stochastic Hill Climbing algorithm for this purpose.

The Stochastic Hill Climbing algorithm involves first generating a random starting point and current working solution, then generating perturbed versions of the current working solution and unsuspicious them if they are largest than the current working solution.

Given that we are working on a continuous optimization problem, a solution is a vector of values to be evaluated by the objective function, in this case, a point in a two-dimensional space regional by -5 and 5.

We can generate a random point by sampling the search space with a uniform probability distribution. For example:

... # generate a random point in the search space solution = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) |

We can generate perturbed versions of a currently working solution using a Gaussian probability distribution with the midpoint of the current values in the solution and a standard deviation controlled by a hyperparameter that controls how far the search is unliable to explore from the current working solution.

We will refer to this hyperparameter as “*step_size*“, for example:

... # generate a perturbed version of a current working solution candidate = solution randn(len(bounds)) * step_size |

Importantly, we must trammels that generated solutions are within the search space.

This can be achieved with a custom function named *in_bounds()* that takes a candidate solution and the premises of the search space and returns True if the point is in the search space, *False* otherwise.

# trammels if a point is within the premises of the search def in_bounds(point, bounds): # enumerate all dimensions of the point for d in range(len(bounds)): # trammels if out of premises for this dimension if point[d] < bounds[d, 0] or point[d] > bounds[d, 1]: return False return True |

This function can then be tabbed during the hill climb to personize that new points are in the premises of the search space, and if not, new points can be generated.

Tying this together, the function *hillclimbing()* unelevated implements the stochastic hill climbing local search algorithm. It takes the name of the objective function, premises of the problem, number of iterations, and steps size as arguments and returns the weightier solution and its evaluation.

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# hill climbing local search algorithm def hillclimbing(objective, bounds, n_iterations, step_size): # generate an initial point solution = None while solution is None or not in_bounds(solution, bounds): solution = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # evaluate the initial point solution_eval = objective(solution) # run the hill climb for i in range(n_iterations): # take a step candidate = None while candidate is None or not in_bounds(candidate, bounds): candidate = solution randn(len(bounds)) * step_size # evaluate candidate point candidte_eval = objective(candidate) # trammels if we should alimony the new point if candidte_eval <= solution_eval: # store the new point solution, solution_eval = candidate, candidte_eval # report progress print(‘>%d f(%s) = %.5f’ % (i, solution, solution_eval)) return [solution, solution_eval] |

We can test this algorithm on the Ackley function.

We will fix the seed for the pseudorandom number generator to ensure we get the same results each time the lawmaking is run.

The algorithm will be run for 1,000 iterations and a step size of 0.05 units will be used; both hyperparameters were chosen without a little trial and error.

At the end of the run, we will report the weightier solution found.

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... # seed the pseudorandom number generator seed(1) # pinpoint range for input bounds = asarray([[–5.0, 5.0], [–5.0, 5.0]]) # pinpoint the total iterations n_iterations = 1000 # pinpoint the maximum step size step_size = 0.05 # perform the hill climbing search best, score = hillclimbing(objective, bounds, n_iterations, step_size) print(‘Done!’) print(‘f(%s) = %f’ % (best, score)) |

Tying this together, the well-constructed example of applying the stochastic hill climbing algorithm to the Ackley objective function is listed below.

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# hill climbing search of the ackley objective function from numpy import asarray from numpy import exp from numpy import sqrt from numpy import cos from numpy import e from numpy import pi from numpy.random import randn from numpy.random import rand from numpy.random import seed # objective function def objective(v): x, y = v return –20.0 * exp(–0.2 * sqrt(0.5 * (x**2 y**2))) – exp(0.5 * (cos(2 * pi * x) cos(2 * pi * y))) e 20 # trammels if a point is within the premises of the search def in_bounds(point, bounds): # enumerate all dimensions of the point for d in range(len(bounds)): # trammels if out of premises for this dimension if point[d] < bounds[d, 0] or point[d] > bounds[d, 1]: return False return True # hill climbing local search algorithm def hillclimbing(objective, bounds, n_iterations, step_size): # generate an initial point solution = None while solution is None or not in_bounds(solution, bounds): solution = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # evaluate the initial point solution_eval = objective(solution) # run the hill climb for i in range(n_iterations): # take a step candidate = None while candidate is None or not in_bounds(candidate, bounds): candidate = solution randn(len(bounds)) * step_size # evaluate candidate point candidte_eval = objective(candidate) # trammels if we should alimony the new point if candidte_eval <= solution_eval: # store the new point solution, solution_eval = candidate, candidte_eval # report progress print(‘>%d f(%s) = %.5f’ % (i, solution, solution_eval)) return [solution, solution_eval] # seed the pseudorandom number generator seed(1) # pinpoint range for input bounds = asarray([[–5.0, 5.0], [–5.0, 5.0]]) # pinpoint the total iterations n_iterations = 1000 # pinpoint the maximum step size step_size = 0.05 # perform the hill climbing search best, score = hillclimbing(objective, bounds, n_iterations, step_size) print(‘Done!’) print(‘f(%s) = %f’ % (best, score)) |

Running the example performs the stochastic hill climbing search on the objective function. Each resurgence found during the search is reported and the weightier solution is then reported at the end of the search.

**Note**: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the stereotype outcome.

In this case, we can see well-nigh 13 improvements during the search and a final solution of well-nigh f(-0.981, 1.965), resulting in an evaluation of well-nigh 5.381, which is far from f(0.0, 0.0) = 0.

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>0 f([-0.85618854 2.1495965 ]) = 6.46986 >1 f([-0.81291816 2.03451957]) = 6.07149 >5 f([-0.82903902 2.01531685]) = 5.93526 >7 f([-0.83766043 1.97142393]) = 5.82047 >9 f([-0.89269139 2.02866012]) = 5.68283 >12 f([-0.8988359 1.98187164]) = 5.55899 >13 f([-0.9122303 2.00838942]) = 5.55566 >14 f([-0.94681334 1.98855174]) = 5.43024 >15 f([-0.98117198 1.94629146]) = 5.39010 >23 f([-0.97516403 1.97715161]) = 5.38735 >39 f([-0.98628044 1.96711371]) = 5.38241 >362 f([-0.9808789 1.96858459]) = 5.38233 >629 f([-0.98102417 1.96555308]) = 5.38194 Done! f([-0.98102417 1.96555308]) = 5.381939 |

Next, we will modify the algorithm to perform random restarts and see if we can unzip largest results.

## Stochastic Hill Climbing With Random Restarts

The Stochastic Hill Climbing With Random Restarts algorithm involves the repeated running of the Stochastic Hill Climbing algorithm and keeping track of the weightier solution found.

First, let’s modify the *hillclimbing()* function to take the starting point of the search rather than generating it randomly. This will help later when we implement the Iterated Local Search algorithm later.

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# hill climbing local search algorithm def hillclimbing(objective, bounds, n_iterations, step_size, start_pt): # store the initial point solution = start_pt # evaluate the initial point solution_eval = objective(solution) # run the hill climb for i in range(n_iterations): # take a step candidate = None while candidate is None or not in_bounds(candidate, bounds): candidate = solution randn(len(bounds)) * step_size # evaluate candidate point candidte_eval = objective(candidate) # trammels if we should alimony the new point if candidte_eval <= solution_eval: # store the new point solution, solution_eval = candidate, candidte_eval return [solution, solution_eval] |

Next, we can implement the random restart algorithm by repeatedly calling the *hillclimbing()* function a stock-still number of times.

Each call, we will generate a new randomly selected starting point for the hill climbing search.

... # generate a random initial point for the search start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # perform a stochastic hill climbing search solution, solution_eval = hillclimbing(objective, bounds, n_iter, step_size, start_pt) |

We can then inspect the result and alimony it if it is largest than any result of the search we have seen so far.

... # trammels for new best if solution_eval < best_eval: best, best_eval = solution, solution_eval print(‘Restart %d, best: f(%s) = %.5f’ % (n, best, best_eval)) |

Tying this together, the *random_restarts()* function implemented the stochastic hill climbing algorithm with random restarts.

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# hill climbing with random restarts algorithm def random_restarts(objective, bounds, n_iter, step_size, n_restarts): best, best_eval = None, 1e 10 # enumerate restarts for n in range(n_restarts): # generate a random initial point for the search start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # perform a stochastic hill climbing search solution, solution_eval = hillclimbing(objective, bounds, n_iter, step_size, start_pt) # trammels for new best if solution_eval < best_eval: best, best_eval = solution, solution_eval print(‘Restart %d, best: f(%s) = %.5f’ % (n, best, best_eval)) return [best, best_eval] |

We can then wield this algorithm to the Ackley objective function. In this case, we will limit the number of random restarts to 30, chosen arbitrarily.

The well-constructed example is listed below.

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# hill climbing search with random restarts of the ackley objective function from numpy import asarray from numpy import exp from numpy import sqrt from numpy import cos from numpy import e from numpy import pi from numpy.random import randn from numpy.random import rand from numpy.random import seed # objective function def objective(v): x, y = v return –20.0 * exp(–0.2 * sqrt(0.5 * (x**2 y**2))) – exp(0.5 * (cos(2 * pi * x) cos(2 * pi * y))) e 20 # trammels if a point is within the premises of the search def in_bounds(point, bounds): # enumerate all dimensions of the point for d in range(len(bounds)): # trammels if out of premises for this dimension if point[d] < bounds[d, 0] or point[d] > bounds[d, 1]: return False return True # hill climbing local search algorithm def hillclimbing(objective, bounds, n_iterations, step_size, start_pt): # store the initial point solution = start_pt # evaluate the initial point solution_eval = objective(solution) # run the hill climb for i in range(n_iterations): # take a step candidate = None while candidate is None or not in_bounds(candidate, bounds): candidate = solution randn(len(bounds)) * step_size # evaluate candidate point candidte_eval = objective(candidate) # trammels if we should alimony the new point if candidte_eval <= solution_eval: # store the new point solution, solution_eval = candidate, candidte_eval return [solution, solution_eval] # hill climbing with random restarts algorithm def random_restarts(objective, bounds, n_iter, step_size, n_restarts): best, best_eval = None, 1e 10 # enumerate restarts for n in range(n_restarts): # generate a random initial point for the search start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # perform a stochastic hill climbing search solution, solution_eval = hillclimbing(objective, bounds, n_iter, step_size, start_pt) # trammels for new best if solution_eval < best_eval: best, best_eval = solution, solution_eval print(‘Restart %d, best: f(%s) = %.5f’ % (n, best, best_eval)) return [best, best_eval] # seed the pseudorandom number generator seed(1) # pinpoint range for input bounds = asarray([[–5.0, 5.0], [–5.0, 5.0]]) # pinpoint the total iterations n_iter = 1000 # pinpoint the maximum step size step_size = 0.05 # total number of random restarts n_restarts = 30 # perform the hill climbing search best, score = random_restarts(objective, bounds, n_iter, step_size, n_restarts) print(‘Done!’) print(‘f(%s) = %f’ % (best, score)) |

Running the example will perform a stochastic hill climbing with random restarts search for the Ackley objective function. Each time an improved overall solution is discovered, it is reported and the final weightier solution found by the search is summarized.

**Note**: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the stereotype outcome.

In this case, we can see three improvements during the search and that the weightier solution found was approximately f(0.002, 0.002), which evaluated to well-nigh 0.009, which is much largest than a each run of the hill climbing algorithm.

Restart 0, best: f([-0.98102417 1.96555308]) = 5.38194 Restart 2, best: f([1.96522236 0.98120013]) = 5.38191 Restart 4, best: f([0.00223194 0.00258853]) = 0.00998 Done! f([0.00223194 0.00258853]) = 0.009978 |

Next, let’s squint at how we can implement the iterated local search algorithm.

## Iterated Local Search Algorithm

The Iterated Local Search algorithm is a modified version of the stochastic hill climbing with random restarts algorithm.

The important difference is that the starting point for each using of the stochastic hill climbing algorithm is a perturbed version of the weightier point found so far.

We can implement this algorithm by using the *random_restarts()* function as a starting point. Each restart iteration, we can generate a modified version of the weightier solution found so far instead of a random starting point.

This can be achieved by using a step size hyperparameter, much like is used in the stochastic hill climber. In this case, a larger step size value will be used given the need for larger perturbations in the search space.

... # generate an initial point as a perturbed version of the last best start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = best randn(len(bounds)) * p_size |

Tying this together, the *iterated_local_search()* function is specified below.

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# iterated local search algorithm def iterated_local_search(objective, bounds, n_iter, step_size, n_restarts, p_size): # pinpoint starting point best = None while best is None or not in_bounds(best, bounds): best = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # evaluate current weightier point best_eval = objective(best) # enumerate restarts for n in range(n_restarts): # generate an initial point as a perturbed version of the last best start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = best randn(len(bounds)) * p_size # perform a stochastic hill climbing search solution, solution_eval = hillclimbing(objective, bounds, n_iter, step_size, start_pt) # trammels for new best if solution_eval < best_eval: best, best_eval = solution, solution_eval print(‘Restart %d, best: f(%s) = %.5f’ % (n, best, best_eval)) return [best, best_eval] |

We can then wield the algorithm to the Ackley objective function. In this case, we will use a larger step size value of 1.0 for the random restarts, chosen without a little trial and error.

The well-constructed example is listed below.

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# iterated local search of the ackley objective function from numpy import asarray from numpy import exp from numpy import sqrt from numpy import cos from numpy import e from numpy import pi from numpy.random import randn from numpy.random import rand from numpy.random import seed # objective function def objective(v): x, y = v # trammels if a point is within the premises of the search def in_bounds(point, bounds): # enumerate all dimensions of the point for d in range(len(bounds)): # trammels if out of premises for this dimension if point[d] < bounds[d, 0] or point[d] > bounds[d, 1]: return False return True # hill climbing local search algorithm def hillclimbing(objective, bounds, n_iterations, step_size, start_pt): # store the initial point solution = start_pt # evaluate the initial point solution_eval = objective(solution) # run the hill climb for i in range(n_iterations): # take a step candidate = None while candidate is None or not in_bounds(candidate, bounds): candidate = solution randn(len(bounds)) * step_size # evaluate candidate point candidte_eval = objective(candidate) # trammels if we should alimony the new point if candidte_eval <= solution_eval: # store the new point solution, solution_eval = candidate, candidte_eval return [solution, solution_eval] # iterated local search algorithm def iterated_local_search(objective, bounds, n_iter, step_size, n_restarts, p_size): # pinpoint starting point best = None while best is None or not in_bounds(best, bounds): best = bounds[:, 0] rand(len(bounds)) * (bounds[:, 1] – bounds[:, 0]) # evaluate current weightier point best_eval = objective(best) # enumerate restarts for n in range(n_restarts): # generate an initial point as a perturbed version of the last best start_pt = None while start_pt is None or not in_bounds(start_pt, bounds): start_pt = best randn(len(bounds)) * p_size # perform a stochastic hill climbing search solution, solution_eval = hillclimbing(objective, bounds, n_iter, step_size, start_pt) # trammels for new best if solution_eval < best_eval: best, best_eval = solution, solution_eval print(‘Restart %d, best: f(%s) = %.5f’ % (n, best, best_eval)) return [best, best_eval] # seed the pseudorandom number generator seed(1) # pinpoint range for input bounds = asarray([[–5.0, 5.0], [–5.0, 5.0]]) # pinpoint the total iterations n_iter = 1000 # pinpoint the maximum step size s_size = 0.05 # total number of random restarts n_restarts = 30 # perturbation step size p_size = 1.0 # perform the hill climbing search best, score = iterated_local_search(objective, bounds, n_iter, s_size, n_restarts, p_size) print(‘Done!’) print(‘f(%s) = %f’ % (best, score)) |

Running the example will perform an Iterated Local Search of the Ackley objective function.

Each time an improved overall solution is discovered, it is reported and the final weightier solution found by the search is summarized at the end of the run.

**Note**: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the stereotype outcome.

In this case, we can see four improvements during the search and that the weightier solution found was two very small inputs that are tropical to zero, which evaluated to well-nigh 0.0003, which is largest than either a each run of the hill climber or the hill climber with restarts.

Restart 0, best: f([-0.96775653 0.96853129]) = 3.57447 Restart 3, best: f([-4.50618519e-04 9.51020713e-01]) = 2.57996 Restart 5, best: f([ 0.00137423 -0.00047059]) = 0.00416 Restart 22, best: f([ 1.16431936e-04 -3.31358206e-06]) = 0.00033 Done! f([ 1.16431936e-04 -3.31358206e-06]) = 0.000330 |

## Further Reading

This section provides increasingly resources on the topic if you are looking to go deeper.

### Books

### Articles

## Summary

In this tutorial, you discovered how to implement the iterated local search algorithm from scratch.

Specifically, you learned:

- Iterated local search is a stochastic global search optimization algorithm that is a smarter version of stochastic hill climbing with random restarts.
- How to implement stochastic hill climbing with random restarts from scratch.
- How to implement and wield the iterated local search algorithm to a nonlinear objective function.

**Do you have any questions?**

Ask your questions in the comments unelevated and I will do my weightier to answer.

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