Simulated Annealing For Dart

Introduction

Simulated annealing (SA) is an algorithm aimed at finding the global minimum of a function E(x₀, x₁, ..., x_n) for a given region ω(x₀, x₁, ..., x_n). The function to be minimized can be interpreted as the system energy. In that case, the global minimum represents the ground state of the system.

Simulated annealing works by randomly selecting a new point (y₀, y₁,  ..., y_n) in the neighbourhood of the current solution, evaluating the energy function E(y₀,  y₁, ..., y_n), and deciding if the new solution is accepted or rejected:

• If ΔE = E - E_min < 0 ,where E_min is a previously found energy minimum, the new solution is accepted with probability: 1.0.
• If ΔE > 0, the new solution is accepted with probability: P(ΔE > 0, T) = e^{-ΔE/(k_B·T)}. The Boltzmann constant k_B relates the system temperature with the kinetic energy of particles in a gas. In the context of SA, it is customary to set k_B ≡ 1.

Accepting up-hill moves provides a method of escaping from local energy minima. The acceptance probability for solution satisfying ΔE > 0 decreases with decreasing temperature. As such, the temperature is a parameter that controls the probability of up-hill moves.

The algorithm name was coined by ... click to show details.

Kirkpatrick et al. and was derived from the process of annealing a metal alloy or glass. The first step of the annealing process consists of heating a solid material above a critical temperature. This allows its atoms to gain sufficient kinetic energy to be able to rearrange themselves. Then the temperature is decreased sufficiently slowly in order to minimize atomic lattice defects as the material solidifies.

The expression above ensures that the acceptance probability decreases with decreasing temperature (for ΔE > 0). As such, the temperature is a parameter that controls the probability of up-hill moves.

The process is demonstrated in the animation above. The left figure shows a spherical 3D search space while the energy value is represented by colour. The figure on the right shows a projection of the energy function onto the x-y plane. Initially, random points are chosen from a large region encompasing the entire spherical search space. In the simulation shown above, intermediate solutions near the local minimum are followed by up-hill moves. As the temperature drops the search neighourhood is contracted and the solution converges to the global minimum.

Usage

To use this package include simulated_annealing as a dependency in your pubspec.yaml file.

The following steps are required to set up the SA algorithm.

1. Specify the search space ω. Common 2d and 3d search spaces (circle, sphere, rectangle, box, disk, cone, triangle) are predefined as static functions of the class SearchSpace.

   Click to show the source code.

   // Defining a spherical space in terms of Cartesian Coordinates
   // using parametric intervals.
   import 'dart:math';
   
   import 'package:list_operators/list_operators.dart';
   import 'package:simulated_annealing/simulated_annealing.dart';
   
   final radius = 2;
   final x = FixedInterval(-radius, radius, name: 'x');
   
   num yLimit() => sqrt(pow(radius, 2) - pow(x.next(), 2));
   final y = ParametricInterval(() => -yLimit(), yLimit, name: 'y');
   
   num zLimit() => sqrt(pow(radius, 2) - pow(y.next(), 2) - pow(x.next(), 2));
   final z = ParametricInterval(() => -zLimit(), zLimit, name: 'z');
   
   final deltaPositionMin = <num>[1e-6, 1e-6, 1e-6];
   final space = SearchSpace.parametric([x, y, z]);

2. Define the system EnergyField, an object encapsulating the energy function (cost function) and its domain: the search space.

   Click to show the source code.

   // Defining an energy function.
   final globalMin = [0.5, 0.7, 0.8];
   final localMin = [-1.0, -1.0, -0.5];
   num energy(List<num> position) {
     return 4.0 -
         4.0 * exp(-4 * globalMin.distance(position)) -
         0.3 * exp(-6 * localMin.distance(position));
   }
   
   final field = EnergyField(
     energy,
     space,
   );
   

3. Create an instance of LoggingSimulator or alternatively extend the abstract class Simulator.

   Click to show source code.

   import 'dart:io';
   import 'package:list_operators/list_operators.dart';
   import 'package:simulated_annealing/simulated_annealing.dart';
   import '../../test/src/energy_field_instance.dart';
   
     // Construct a simulator instance.
     final simulator = LoggingSimulator(
       field, // Defined in file `energy_field_example.dart'
       gammaStart: 0.8,
       gammaEnd: 0.05,
       outerIterations: 100,
       innerIterationsStart: 2,
       innerIterationsEnd: 10,
     );
   
     // The perturbation magnitude at the end of the annealing cycle.
     simulator.deltaPositionEnd = [1e-7, 1e-7, 1e-7];

4. Start the simulated annealing process.

   Click to show the source code.

   /// To run this program navigate to the root folder in your local
   /// copy of the package `simulated_annealing` and use the command:
   /// $ dart example/bin/simulated_annealing_example.dart
   void main() async {
     print(await simulator.info);
   
     print('Start annealing process ...');
     final xSol = await simulator.anneal(
       isRecursive: true,
       isVerbose: true,
     );
     print('Annealing ended.');
     print('Writing log to file: example/data/log.dat');
     await File('example/data/log.dat').writeAsString(simulator.export());
     print('Finished writing. ');
   
     print('Solution: $xSol');
     print('xSol - globalMin: ${xSol - globalMin}.');
   }

Algorithm Tuning

It can be shown that by selecting a sufficiently high initial temperature the algorithm converges to the global minimum if the temperature decreases on a logarithmic scale (slow cooling schedule) and the number of inner iterations (Markov chain length) is sufficiently high [1].

Practical implementations of the SA algorithm aim to generate an acceptable solution with minimal computational effort. For such fast cooling schedules, algorithm convergence to the global minimum is not strictly guaranteed. In that sense, SA is a heuristic approach and some degree of trial and error is required to determine which annealing schedule works best for a given problem.

The behaviour of the annealing simulator can be tuned using the following optional parameters of the Simulator constructor:

• gammaStart: Initial acceptance probability with default value 0.7. Useful values for γ_start are in the range of [0.7, 0.9]. If γ_start is too low, up-hill moves are unlikely (potentially) preventing the SA algorithm from escaping a local miniumum. If γ_start is set close to 1.0 the algorithm will accept too many up-hill moves at high temperatures wasting computational time and delaying convergence.
• gammaEnd: Final acceptance probability. Towards the end of the annealing process one assumes that the solution has converged towards the global minimum and up-hill moves should be restricted. For this reason γ_end has default value 0.05.
• outerIterations: Determines the number of temperature steps in the annealing schedule. It is recommended to start with a higher number of outer iterations (number of entries in the sequence of temperatures) and log quantities like the current system energy, temperature, and the intermediate solutions.
• innerIterationsStart: The number of inner iterations (at constant temperature) at the start of the annealing process.
• innerIterationsEnd: The number of inner iterations (at constant temperature) at the end of the annealing process.
• sampleSize: The size of the sample used to determine the initial and final annealing temperature.

Additionally, it is possible to set the class variable temperatureSequence to function of type TemperatureSequence that is used to determine the temperature at each outer iteration step.

The figure below shows a typical SA log where the x-coordinate of the solution (green dots) converges asymptotically to 0.5. The graph is discussed in more detail here.

The number of inner iterations (performed while the temperature is kept constant) is also referred to as Markov chain length and is determined by a function with typedef MarkovChainLength. It can be adjusted by setting the simulator arguments innerIterationsStart and innerIterationsEnd. In general, it is advisable to increase the number of inner Iterations towards the end of the annealing process in order to increase the algorithm precision.

Annealing Schedule

In general, the following information is required to define an annealing schedule:

• T_start, the initial temperature,
• T_end, the final temperature,
• the number of outer iterations (temperature steps),
• a function of type TemperatureSequence that is used to determine the temperature at each (outer) iteration step.

The class EnergyField provides the methods tStart and tEnd. These use an algorithm introduced by Ben-Ameur to calculate the initial and final annealing temperature [2].

Examples

Further information can be found in the folder example. The following topics are covered:

• search space,
• annealing schedule,
• system energy and logging simulator.

Features and bugs

Please file feature requests and bugs at the issue tracker.