Cooperative Multi-Simplex Algorithm: An Innovation from Localization to Globalization

This study suggests a novel cooperative multi-simplex algorithm that generalizes a local search optimizer to design a novel global search heuristic algorithm. The proposed algorithm exploits the vertex sharing strategy to enhance the search abilities of the working simplexes. The vertex sharing among the simplexes is carried out through cooperative step that is based on fitness of the underlying simplex. The proposed algorithm is applied to solve some systems of nonlinear equations by transforming them to optimization problems. Comparative analysis of results shows that the proposed method is practical and effective.


Consider
∈ ℝ ; 1 ≤ ≤ + 1 be the vertices of the Polytopes with corresponding function values arranged in ascending order ≤ ∀ ≤ . The NMS method calculates the centroid by relation (4) and then uses (5)- (8) to improve +1 by generating points , , or .

Figure 1. Operations on a simplex in ℝ 2
A general iteration of original NMS method in ℝ is restated as under [1,2,4].

Related works on the proposed Cooperative Multi-Simplex (CMS) algorithm
The proposed cooperative multi-simplex algorithm (CMS) algorithm starts by randomly generating , ∈ ℤ + , simplexes in the search space. The iterative process of the proposed CMS algorithm is comprised of a cooperative step and a rotational shrinkage based modified iteration of NMS method. The cooperative step establishes a probability based sharing among the vertices of various simplexes. Based upon a user-defined cooperative sharing probability ∈ [0, 1], the vertex sharing is divided in to mixed sharing and ascent sharing.

Initialization
Generate simplexes ( , ) : 1 ≤ ≤ , choose a suitable value of and set an integer as maximum number of function evaluations allowed.

Ordering
Sort all the vertices of the each simplex:

Cooperative step
The attribute of cooperative sharing and exploiting the information composed from the entire population are crucial tools of population based heuristic algorithms [5,6,7] which empower them to perform balanced exploration and exploitation in optimization process. In CMS algorithm, the cooperative step handles the sharing of vertices based on the fitness of the simplexes. It not only alters orientations of the corresponding simplexes but also enforces them to cluster around the promising locations in the search space. The fitness of a simplex is calculated by using Equations (10) and (11) in turn.
Two real numbers and ∈ [0, 1] are generated randomly. The sharing of a vertex of some ℎ simplex with another simplex takes place if > ( , ) , otherwise letting the simplexes proceed independently. If < , the mixed sharing exchanges the non-best vertex of a randomly simplex with some non-best vertex of the current simplex whereas the ascent sharing replaces the worst vertex of the current simplex by the worst vertex of some other simplex otherwise.

Applications of proposed CMS algorithm to physical systems and numerical results
In order to validate the effectiveness of our proposed CMS algorithm, two mathematical and one physical system of non-linear equations are utilized. Four state of the art algorithms, namely, Particle Swarm Optimization (PSO) [5], Differential Evolution (DE) [6], Artificial Bee Colony (ABC) [8] and Teaching Learning Based Optimization (TLBO) [9] are considered for the performance comparisons.

Mathematical test problem 1
The first test problem has been taken from [10][11][12][13]. This problem is described by the system (14) of non-linear equations.
The exact solution to the system reported is (4, 3, 1).

Thin wall rectangle girder section problem
Geometry size of thin wall rectangle girder section problem involves following system of equations [10,12,15,16].
Where 1 , 2 and 3 are height, width and thickness of the section respectively. The physical constraints on the system are:

Conclusion
This study presents a novel approach for solving a system of nonlinear equations as an optimization problem. The proposed method neither requires initial guess nor derivative information. The The proposed work can be extended to several disciplines of numerical optimization in collaboration with general purpose global search optimization algorithms.