Lab 2: Local Search
Due Oct. 1 by midnight

Starting point code

Use Teammaker to form your team. You can log in to that site to indicate your partner preference. Once the lab is released and you and your partner have specified each other, a GitHub repository will be created for your team.

Introduction

The objectives of this lab are to:

• Use local search to solve traveling salesperson and vehicle routing problems.
• Explore the consequences of different ways to define local search problems.
• Compare the performance of different local search algorithms.

You will implement the following local search methods:

• hill climbing
• simulated annealing
• stochastic beam search

You will need to modify the following python files:

• TSP.py
• VRP.py
• HillClimbing.py
• SimulatedAnnealing.py
• BeamSearch.py

LocalSearch.py

./LocalSearch.py HC TSP coordinates/South_Africa_10.json
./LocalSearch.py SA VRP coordinates/India_15.json
./LocalSearch.py BS VRP coordinates/United_States_25.json -config my_config.json -plot USA25_map.pdf

NOTE: In order to execute python files directly (as shown above, without having to type python3), you'll neeed to make the file executable first. Type the following at the unix prompt to change the file's permissions:

    chmod u+x LocalSearch.py 
  

The first example runs hill climbing to solve a traveling salesperson problem on the 10-city South Africa map. The second example runs simulated annealing to solve a vehicle routing problem on the 15-city India map. The third example runs stochastic beam search, and demonstrates the non-required options, which allow you to load a non-default parameter file and/or save the resulting map to a non-default output file.

Local search is typically used on hard problems where informed search is not viable (for example both TSP and VRP are NP-complete problems). Thus the search often runs for an extended period of time. It is important to keep the user informed of the search's progress. You should print informative messages during the search execution. For instance, you should inform the user whenever the best value found so far is updated along with the current search step at that point.

Traveling Salesperson and Vehicle Routing Problems

A traveling salesperson tour visits each of a collection of cities once before returning to the starting point. The objective is to minimize the total distance traveled. We will represent a tour by listing cities in the order visited. Neighbors of a tour are produced by moving one city to a new position in the ordering.

A vehicle routing plan sends several vehicles on round trip tours such that at least one vehicle visits each city, and all vehicles eventually return to the starting point. The general goal is to minimize travel distance. We represent a plan as a list of tours that all have the same starting point. Neighboring plans are ones where a single city has been reassigned (to a different vehicle and/or a different position in the tour).

The class TSP in the file TSP.py and the class VRP in the file VRP.py represent instances of each problem. Both of these classes provide several useful methods:

• random_candidate() generates a tour/plan from which to begin a local search run.
• cost() evaluates the quality of a tour/plan. For TSP, this function is already implemented. For VRP, it points to one of two functions you will implement.
• get_neighbor() points to either random_neighbor() or best_neighbor() to generate adjacent candidates. For VRP, random_neighbor() is already implemented and best_neighbor() won't be used. For TSP, you need to implement both.

Instances of both TSP and VRP are constructed by passing in the name of a JSON file specifying the latitude and longitude of locations that must be visited. Example JSON files are located in the coordinates/ directory. The __init__() method also takes additional parameters read from a separate JSON file. The default file for these parameters is the file default_config.json.

The TSP and VRP classes leave the following methods for you to implement:

• TSP._all_neighbors() returns a list of all neighbors.
• TSP._best_neighbor() returns the lowest cost neighbor.
• TSP.random_neighbor() returns a random neighbor.
• VRP._max_cost() returns the maximum length of any vehicle's tour.
• VRP._total_cost() returns the total length of all vehicles' tours.

Note that most of these method names begin with a single underscore. This is a python convention for indicating a 'private' method. Python doesn't actually distinguish between private and public class members, but the underscore lets programmers know that the method isn't intended to be called directly. In this case, your local search functions should not call these methods. Instead, they should call TSP.get_neighbor(), or VRP.cost(), which will point to one of your functions based on the parameters from the config file. TSP.random_neighbor() does not begin with an underscore because the local search functions will sometimes need to call it directly.

You are strongly encouraged to incrementally test your code as you develop it. The best way to do this is to add tests inside the if at the bottom of the files TSP.py and VRP.py:

  if __name__ == '__main__':
     ...

Hill Climbing

candidate = problem.random_candidate()
cost = problem.cost(candidate)
return candidate, cost

./LocalSearch.py HC TSP coordinates/United_States_25.json

./LocalSearch.py HC TSP coordinates/India_15.json -plot random_India_tour.pdf

initialize best_state, best_cost = (None, float("inf"))
loop over runs
  current_state = random candidate
  current_cost = cost(state)
  loop over steps
    if random move
      current_state, current_cost = random neighbor of current_state
    else
      neighbor_state, neighbor_cost = get neighbor of current_state
      if neighbor_cost < current_cost:
         update current_state, current_cost
    if current_cost < best_cost
       update best_state, best_cost   	  
       print status information about new best state and cost
return best_state, best_cost

cp default_config.json my_config.json

./LocalSearch.py HC VRP coordinates/South_Africa_10.json -config my_config.json

Simulated Annealing

SimulatedAnnealing.py is set up much like HillClimbing.py. Your next task is to implement the algorithm that we discussed in class on Wednesday.

initialize best_state, best_cost = (None, float("inf"))
loop over runs
  initialize temp
  current_state = random candidate
  current_cost = cost(state)
  loop over steps
    candidate_state, candidate_cost = random neighbor of current state
    delta = current_cost - candidate_cost
    if delta > 0 or with probability e^(delta/temp)
      current_state = candidate_state
      current_cost = candidate_cost
    if current_cost < best_cost
      best_state = current_state
      best_cost = current_cost
    temp *= temp_decay
return best_state, best_cost

Edit the "SA" portion of the config file to set the defaults for simulated annealing. The init_temp specifies the starting temperature, and temp_decay gives the geometric rate at which the temperature decays. The runs and steps parameters have the same interpretation as in hill climbing: specifying how many random starting points simulated annealing should be run from, and how long each run should last.

Beam Search

Your final implementation task is to write stochastic beam search as discussed in class Friday.

initialize best_state, best_cost = (None, float("inf"))
initialize population with pop_size random candidates
initialize temp
loop over runs
generate max_neighbors candidates for each individual in pop
  find overall best_neighbor_state, best_neighbor_cost
  if best_neighbor_cost < best_cost
    update best_state and best_cost
  generate probabilities for selecting candidates based on costs:
        compute costs of neighbors 
        compute e^(-cost/temp) for neighbors
        normalize each by the sum
        use normalized values as sampling probabilities
        (NOTE: be sure to catch the ZeroDivisionError that may occur if
               probabilities get too small)
  pop = pop_size candidates selected by sampling 
  temp *= temp_decay
return best_state, best_cost

Stochastic Beam search uses the same temperature and steps parameters as simulated annealing. However, note that the algorithm uses the temperature somewhat differently. Additionally, instead of starting several separate runs, beam search maintains a population of candidates at each step. The pop_size parameter specifies the number of candidates.

Stochastic beam search considers neighbors of all individual candidates in the population and preserves each with probability proportional to its temperature score. It is recommended that you add a helper function for beam search that evaluates the cost of each neighbor and determines its temperature score, then randomly selects the neighbors that will appear in the next iteration. Remember to avoid excess calls to cost by saving the value whenever possible.

For the larger maps, the number of neighbors grows quite large, making it impractical to evaluate every neighbor of every candidate. Your helper function should therefore consider only a random subset of each individual's neighbors. The size of this subset is given by the max_neighbors parameter in the "BS" section of the config file.

A very useful python function for stochastic beam search is numpy.random.choice. This function takes a list of items to choose from, a number of samples to draw, and optionally another list representing a discrete probability distribution over the items. It returns a list of items that were selected. Here is an example:

import numpy.random
items = ["a", "b", "c", "d", "e"]
indices = list(range(len(items)))
probs = [0.1, 0.1, 0.2, 0.3, 0.3]
for i in range(5):
  choices = numpy.random.choice(indices, 10, p=probs)
  print(choices)

In this particular example, items "d" and "e" have the highest probability of being selected (30 percent chance each), and they are at indices 3 and 4. You can see in the sample results below that indices 3 and 4 are selected more often on average than indices 0, 1, and 2.

[3 2 2 2 4 3 4 4 4 4]
[3 4 2 4 4 4 4 4 4 3]
[4 4 4 3 4 2 3 0 3 2]
[4 2 3 0 1 0 3 4 4 2]
[4 0 1 4 4 2 4 0 4 3]

Unfortunately, numpy will treat a list of Tours as a multidimensional array, so it is better to select indices (as demonstrated above) rather than choosing from a list of neighbors directly.

Summarize your results

After all three of your local search algorithms are complete and working correctly, do some informal comparisons between the algorithms on one of the harder TSP problems, such as North_America_75.json. Be sure to spend a little time trying to tune the parameters of each algorithm to achieve its best results. If you find any particularly impressive tours, save them by renaming the file map.pdf to a more informative name.

In the file called summary.md, fill in the tables provided with your results and explain which local search algorithm was most successful at solving TSP problems.

Submitting your code