Ant Colony Optimization Algorithm Model Based on the Continuous Space

—Ant colony algorithm is a heuristic algorithm which is fit for solving complicated combination optimization. It showed great advantage on solving combinatorial optimization problem since it was proposed. The algorithm uses distributed parallel computing and positive feedback mechanism, and is easy to combine with other algorithms. This ant colony algorithm has already been widespread used in the field of discrete space optimization, however, is has been rarely used for continuous space optimization question. On the basis of basic ant colony algorithm principles and mathematical model, this paper proposes an ant colony algorithm for solving continuous space optimization question. Comparing with the ant colony algorithm, the new algorithm improves the algorithm in aspects of ant colony initialization, information density function, distribution algorithms, direction of ant colony motion, and so on. The new algorithm uses multiple optimization strategy, such as polynomial time reduction and branching factor, and improves the ant colony algorithm effectively.


INTRODUCTION
Using for aging behaviour of ant colony as inspiration, Italian scholar Marco Dorigo and Vittorio Maniezzo designed the first ACO algorithm: Ant System (AS), which is a highly innovative meta-heuristic algorithm. The researchers have tried various methods to improve the algorithm since AS, the first algorithm that conforms to ACO, was proposed. Since the ant colony algorithm has developed for many years, the researches of ant colony algorithm have been extended from single TSP field to a lot of fields of application. The ant colony algorithm has developed form one-dimensional static optimization to multidimensional dynamic combinatorial optimization, from research in discrete domain to research in continuous domains. The algorithm makes break through on implementation of the hardware, also, there are many significant results on model improving and integration of the algorithm model with bionics evolution algorithm or local search algorithms. The great advance of research of ant colony algorithm makes a widespread application, this paper will optimize traditional ant colony algorithm for characteristic of continuous spaces in aspects of ant colony initialization, information density function, distribution algorithms, ant colony direction of motion, and so on.This paper will improve the ant colony algorithm by using optimization strategy of polynomial time reduction, branching factor, etc.

II. DESCRIPTION OF TRADITIONAL ANT ALGORITHM
Assume that there are m ants, the characteristics of each ant are given as follows: it chose next city according to the distance between two cities and probability functions for whose variables the amount of the pheromone on the is the amount of the ants for city i at moment t; is the amount of all ants, and for each ant, it has characteristics as follows: it chose next city according to the distance between two cities and probability functions for whose variable is the amount of the pheromone on the connected bound; it is provided to walk along the legitimate route and can't turn to the city which it has visited except it has travelled all over the cities, and this is controlled by tabu list; the ant leaves pheromone on each connected bound that it has visited.
According to the assumption, the amounts of information for each path are equal to each other at initial time. Assume that: For ant ( 1,2, , ) k k m = ! , it chose the turn direction according to amount of information when it is moving.
Assume that ( ) k ij p t is probability of that ant k turn from position i to position j at momentt, and it can obtain that: is the distance between city i and city j; ! is relative importance of the traces, ! is relative importance of visibility, ! is persistence of the traces and 1 ! " is information decay rate which means that the previous information will disappear gradually with time.
After time n, the ant finishes one cycle, and the amounts of information for each path are adjusted by using the following formula: Construction algorithm is a method that add solution element into solutions iteratively from initially solution space until the solution was completed Usually, one used greedy construction heuristic method to construct completed solution, i.e. in every construction procedure, one adds heuristic function to evaluate the solution, so that, one can obtain the solution which has maximum profit and uses this solution to construct the completed solution. The algorithm is given as follows: Procedure Usually, the optimal solution obtained by local search algorithmic local optimal solution and is not global optimal solution. Therefore, definition3 is given as: Definition 3: a local optimal solution (local minimum) of a minimization problem s should meet: Similarly, a local optimal solution (local maximum) of a maximization problem s should meet: Local search algorithm also needs to definite a neighbourhood cheek used to decide how to search neighbourhood and which adjacent solution can be accepted. Usually, the rule for accepting adjacent solution is best-improvement rule or first-improvement rule.
In the system, ant k in city i chooses city j as next city according pseudorandom proportional rule, i.e.:  For local pheromone, the update rule is given by: # # (13) The long-term effect of pheromone can reduce the scale of search space with good parameters. However, if the effect is too strong, the algorithm will be in the state of stagnates rapidly. So, one must take some methods to evaluate the algorithm for checking whether the algorithm is in the state of stagnates or not.
So, this paper used ! " branching factor (0< ! <1) which can calculate the distribution of pheromone more directly. For given city i, ! " can be defined as: Definition 4: ! " is the amount of boundaries which is related to node i and meets On the continuous space, the optimization process of ant colony algorithm includes: initialization of assign ant colony, determination of information distribution function, analysis of distribution algorithm and motion direction of ant colony. The algorithm is given by: Firstly, the amount of ant colony N should be determined according to the problem. The problem domain should be divided to N sub domain, so, ant , [1, ] i i N ! is placed in the N subdomain. For each ant, the motion distance is given by:

IV. SIMULATION STUDY
In the initialization, each pheromone is evaluated an initial value ! . In the iteration process of the algorithm, the ants are distributed to each nodes according to the algorithm rule, and for each node, the probability that it is chosen as next node is given by: Forderivation, the probability above can be written as: After a random iteration, the maximum increase value of pheromone of bound is . So that, the maximum increase value of pheromone in first iteration is given by: # + (18) In the second iteration, the maximum is: Similarly, due to the evaporation of the pheromone, the pheromone value in cycle ! should not larger than: Due to 0 1 ! < < , the summation of above equation advance gradually converge: x .
An ordinary floor probability of min p is given by: where c N is potential of set C.
So, the probability of obtaining a general solution s! , including any best solution * * s S ! , where n ! +" is the maximum length of sequence, is minˆ0 n p p ! > . * ( ) P ! has a floor as: It can obtain a probability larger than any 1 ! " when ! is large enough.So that:

=
Simulation test Comparing algorithm proposed this paper to ACO algorithm shows that the optimal algorithm is better than ACO algorithm on average and standard deviation of object function, as shown in Table 1 and Figure 1   An ant colony algorithm for solving continuous space optimization question is proposed in this paper. The value convergence and solution convergence of the algorithm is confirmed by theory method. Simulation test shows that the optimal algorithm is better than ACO algorithm on average and standard deviation of object function.