Particle swarm optimization (PSO) Algorithm

History

Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling. The particle swarm concept originated as a simulation of simplified social system. The original intent was to graphically simulate the choreography of bird of a bird block or fish school. However, it was found that particle swarm model can be used as an optimizer.

Algorithm introduction

Each particle keeps track of its coordinates in the problem space which are associated with the best solution (fitness) it has achieved so far. (The fitness value is also stored.) This value is called pbest. Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the neighbors of the particle. This location is called lbest. when a particle takes all the population as its topological neighbors, the best value is a global best and is called gbest.

The particle swarm optimization concept consists of, at each time step, changing the velocity of (accelerating) each particle toward its pbest and lbest locations (local version of PSO). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and lbest locations.

Application

PSO has been successfully applied in many research and application areas. It is demonstrated that PSO gets better results in a faster, cheaper way compared with other methods.

Simulated Annealing(SA) Algorithm

Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more efficient than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.

Principle of the algorithm

In the simulated annealing (SA) method, each point s of the search space is analogous to a state of some physical system, and the function E(s) to be minimized is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy.

Steps

At each step, the SA heuristic considers some neighbouring state s' of the current state s, and probabilistically decides between moving the system to state s' or staying in state s. These probabilities ultimately lead the system to move to states of lower energy. Typically this step is repeated until the system reaches a state that is good enough for the application, or until a given computation budget has been exhausted.

State transition

The neighbours of a state are new states of the problem that are produced after altering the given state in some particular way. For example, in the traveling salesman problem, each state is typically defined as a particular permutation of the cities to be visited. The neighbours of some particular permutation are the permutations that are produced for example by interchanging a pair of adjacent cities. The action taken to alter the solution in order to find neighbouring solutions is called "move" and different "moves" give different neighbours. These moves usually result in minimal alterations of the solution, as the previous example depicts, in order to help an algorithm to optimize the solution to the maximum extent and also to retain the already optimum parts of the solution and affect only the suboptimum parts. In the previous example, the parts of the solution are the parts of the tour.

Probability function of the state transition

The probability of making the transition from the current state s to a candidate new state s' is specified by an acceptance probability function P(e,e',T), that depends on the energies e = E(s) and e' = E(s') of the two states, and on a global time-varying parameter T called the temperature. States with a smaller energy are better than those with a greater energy. The probability function P must be positive even when e' is greater than e. This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.

MoDeL

The Perl language is a powerful tool for dealing with regular expressions, and it manages to process complex problems in a timely way. For example, for a hash array with a few elements in the buckets almost get the same manipulating time with a big hash with millions of elements. This feature improves the speed of rule based modeling remarkably.

Dealing with regular expressions is also Perl's cup of tea. So the software can spend more running time saved by perl on providing a better user's interface, making it more convenient for users.

Our approach first realized by Liaochen and 2010igemers emphasize on the structure of the species.

Below is an illustration of how the algorithm processes the classical circuit example:

Here is a detailed explanation to the input file of the rule based modeling approach There are four blocks, respectively lists the definition of parameters, the definition of compartments, seed species and events.

The definition of parameters consists of two items. The left side term is the name of the variable, to the right is the expression of that parameter. Note that variables in that expression must be defined by the user. But it makes no difference whether they are defined before the expression. This is to avoid redundant definition and no definition.

The compartments definition is in this way:
[name ][outside][ruletable]{volume}{population}, where terms in the square brackets are mandatory, while terms in the curly braces are optional.
The default value of the volume is 1. The outside term means the compartment outside the compartment, which is usually the medium that held the compartment like the ecoli.
Ruletable is used to associate a rule in the data base with the compartment.

The third block, reads [compartment][name][structure][init_concentration]{const}, structure is the definition of the complex structure, const is optional since it's in the curly brace. If the substance has a const property, then after the substance, write a const. if not, leave it blank.

The last block is the events definition. The formats are [name][trigger_condition][event_assignments...] The name of the event is usr defined, trigger_condition is a bool expression, all the variables inside this bool expression must have been defined in one of the above three blocks. Event_assignments is a list of assignments, something like assignment1 assignment2 assignment3, with white space separated. There is no constraint on the number of assignments. Each assignment must be the format [variable]=[expression]

A general rule in the algorithm is that the name of the variable must be started by A-Z –z and other characters can be A-Z-a-z-0-9.