16 Application of Multi-agent Optimization Methods …
219
16.2 Description of Multi-agent Methods
Hereinafter, Sect. 16.2.1 discusses an optimization problem. Hybrid multi-agent optimization method of interpolation search is presented in Sect. 16.2.2. Multi-agent optimization algorithm using linear regulators for agents’ motion control is developed
in Sect. 16.2.3.
16.2.1 Optimization Problem
It is given the objective function f (x) = f (x 1 , x 2 , . . . , x n ) defined on the set of
admissible solutions D ⊆ R
n
. It is required to find the constrained global maximum
of a function f (x) on set D, i.e., such a point x
∗
∈ D, that
f (x
∗
) = max
x∈D
f (x),
(16.1)
where x = (x 1 , x 2 , . . . , x n )
T , D = {x|x i ∈ [a i , b i ], i = 1, 2, . . . , n}.
The task of finding the minimum of a function f (x) is replaced by the task of
finding the maximum by replacing the sign before the function with the opposite:
f (x
∗
) = min
x∈D
f (x) = − max
x∈D
[− f (x)]. Function f (x) can be multiextremal, so the
required solution in the general case is not unique.
16.2.2 Hybrid Multi-agent Optimization Method
of Interpolation Search
Solution search strategy. The search strategy includes interpolation search, which
uses several points of the current population and reduces the task of finding new solutions to the problems of one-dimensional parametric maximization, swarm intelligence method to maximize the objective function value along the interpolation curve,
and self-organizing migrating algorithm [7].
The considered objective function f (x) is called the fitness function, and the
vector of parameters x = (x 1 , x 2 , . . . , x n )
T of the objective function is an individual.
Each vector x = (x 1 , x 2 , . . . , x n )
T
∈ D is a possible solution of an optimization
problem. The smaller the value of the objective function f (x), the more the individual
x is adapted, i.e., suitable as a solution.
In solving problem (Eq. 16.1), finite sets I = {x
j
= (x
j
1 , x
j
2 , . . . , x
j
n )
T
, j =
1, 2, . . . , N P} ⊂ D of possible solutions are used. These solutions are called populations, where x
j is the individual with the number j and N P is the size of the
population.
219
16.2 Description of Multi-agent Methods
Hereinafter, Sect. 16.2.1 discusses an optimization problem. Hybrid multi-agent optimization method of interpolation search is presented in Sect. 16.2.2. Multi-agent optimization algorithm using linear regulators for agents’ motion control is developed
in Sect. 16.2.3.
16.2.1 Optimization Problem
It is given the objective function f (x) = f (x 1 , x 2 , . . . , x n ) defined on the set of
admissible solutions D ⊆ R
n
. It is required to find the constrained global maximum
of a function f (x) on set D, i.e., such a point x
∗
∈ D, that
f (x
∗
) = max
x∈D
f (x),
(16.1)
where x = (x 1 , x 2 , . . . , x n )
T , D = {x|x i ∈ [a i , b i ], i = 1, 2, . . . , n}.
The task of finding the minimum of a function f (x) is replaced by the task of
finding the maximum by replacing the sign before the function with the opposite:
f (x
∗
) = min
x∈D
f (x) = − max
x∈D
[− f (x)]. Function f (x) can be multiextremal, so the
required solution in the general case is not unique.
16.2.2 Hybrid Multi-agent Optimization Method
of Interpolation Search
Solution search strategy. The search strategy includes interpolation search, which
uses several points of the current population and reduces the task of finding new solutions to the problems of one-dimensional parametric maximization, swarm intelligence method to maximize the objective function value along the interpolation curve,
and self-organizing migrating algorithm [7].
The considered objective function f (x) is called the fitness function, and the
vector of parameters x = (x 1 , x 2 , . . . , x n )
T of the objective function is an individual.
Each vector x = (x 1 , x 2 , . . . , x n )
T
∈ D is a possible solution of an optimization
problem. The smaller the value of the objective function f (x), the more the individual
x is adapted, i.e., suitable as a solution.
In solving problem (Eq. 16.1), finite sets I = {x
j
= (x
j
1 , x
j
2 , . . . , x
j
n )
T
, j =
1, 2, . . . , N P} ⊂ D of possible solutions are used. These solutions are called populations, where x
j is the individual with the number j and N P is the size of the
population.