An Improved Particle Swarm Optimization for Prediction Model of ...

An Improved Particle Swarm Optimization for Prediction Model of Macromolecular Structure Fuli RONG, Yang YI，Yang HU Information S cience S chool & Technology, Sun Yat-sen University GuangZhou, 510275, China E-mail:[email protected]

Abstract A novel particle swarm optimization combined with filter method (FM _PSO) is proposed in this paper. FM _PSO adopts the filter technique which can broaden the iteration acceptance criterion to improve the exploitation ability in the initial stage. Simultaneously, the divide and conquer method is considered to speed the convergence rate. FM _PSO can improve the ability of exploration and exploitation, despite of large scale optimization. Numerical tests show that FM_PSO is feasible and efficient both in the test function and prediction model of macromolecular structure, and outperforms other algorithms. Key words: PSO, Filter Method, Macromolecular Structure Prediction

1. Introduction Molecular modeling gives rise to a wide variety of global optimiza tion problems according to different models derived fro m chemistry, physics, biology, etc, wh ich plays an important part in discovering new material and producing new medicine. The main types of the macro molecular model include potential energy model, distance geometry model, emp irical force field model, etc, where potential energy minimizat ion received most attention. Owing to its properties of mult i-extreme and mu lti-dimensional, it turns out to be difficu lt for global optimization. The main methods applied in this model include random method [1], branch bound method [2], genetic algorith m [3], simulated annealing algorith m [4] etc. In order to solve the burden of cost time and shortages in large scale situation, this paper proposes an improved particle swarm optimization comb ining with filter technique and divide and conquer strategy, efficient for large scale optimization. The particle swarm optimization (PSO) originally proposed by Kennedy and Eberhart as a simulat ion of social behavior has developed greatly in recent years, widely used in many areas such as image process [5], job shop scheduling [6], dynamic resource allocation [7], etc. In view of the shortages of basic PSO, Sh i and Eberhart [8] introduced an inert ia weight to controls the impact of previous velocity of particle on its current one. Later, they adjusted it linear decrease with the generation [9], and

proposed a maximal speed vmax controlled the exp loring ability. Then Jiang [10] divided the entire space into mu ltip le sub-swarms, each of which is made to evolve based on PSO algorith m. Hu [11] proposed the improved PSO based on the simp le evolutionary equations and the extremu m disturbed arith metic operators to overcome the demerits of basic PSO. He [12] used a special mutation named escape operator to make particles explo re the search space more efficiently. Despite the above-mentioned methods have imp roved the performance of PSO, there are some shortages in solving the high-dimensional optimization problems and premature convergence. Inspired by the filter method we propose an improved particle swarm optimization (FM_PSO) in this paper. In FM_PSO, filter technique is adopted to broaden the iteration acceptance criterion, such that FM_PSO can improve the exp lo itation ability in the in itial stage. Simu ltaneously, the divide and conquer method is considered to partition the entire swarm into several sub-swarms, each of which is made to evolve based on FM_PSO, then merge together to speed the convergence rate. FM_PSO can avoid premature convergence effectively and run more efficiently especially for large scale optimization.

2. Overvie w of standard PSO The PSO is init ialized with a population of random particles, each particle ad justs itself by tracking two extreme called ind ividual best particle x_pbest and global best particle x_gbest for each generation. Suppose that the search space is n-dimensional, and then the particle i of the swarm can be represented by an n-dimensional vector Xi =(xi1 ,xi2 ,…,xin ), its velocity can be represented by Vi =(vi1 ,vi2 ,…,vin ), the other informat ion of particle can be denoted similarly. At each step, the velocity and position of particle will be updated according to the following two equations: vid (k+1)＝w*vid (k)+c1 *r1*[x_pbest id (k)－xid (k)]＋c2* r2 *[x_gbestgd (k)－xid (k)] (1) xid (k+1)= xid (k)+ vid (k+1) (2)

where, the inertia weight w updates as follo ws: w=(w1 -w0 )*(max_gen-gen)n /max_genn + w0 (3) w0 denotes initial inert ia weight, w1 denotes final inertia weight, max_gen denotes the maximu m generations, gen represents the current generation. In Eq. (1), cl and c2 are positive constant parameters called acceleration coefficients. r1 and r2 are random variables independently uniformly distributed with range (0,1).

3. The theory and description of FM_PSO 3.1. Filter method theory Filter method was first introduced by Fletcher and Leyffer [13] as a way to globalize SLP (sequential linear programming) and SQP (sequential quadratic programming). Filter methods are designed to solve nonlinear programming problem:  min x¡ n f ( x)    s.t.c( x)  0

(4)

where we assume f : R→ R and c : R→ R are t wice continuously differentiable. There are two co mpeting aims to measure whether to accept a new iteration. The first is the minimizat ion of the objective function and the second is the satisfaction of the constraints. Conceptually, these two conflicting aims can be written as follow: minimize f ( x)  minimize h(c( x))

(5)

m

m

j 1

j 1

where h(c(x))= h(c( x))   c j ( x)   max(0, c j ) is the l1 norm o f the constraint violation. In this paper, we modify the filter function inspired by [14] to control the searching space adaptively as follow. 0if u ( x)  0 Definiti on 1. Filter function. h( x)   u ( x)else

(6)

where u(x)=f(x)-min{f(x_lbest), f(xi ) | xi ∈ N(x)}, N (x) denotes the neighborhood of x. There are many topological structures in PSO, this paper considers the ring structure. Definiti on 2. Do mination. A pair (f(x(k)), h(x(k))) obtained on iteration k is said to dominate another pair (f(x(l ), h(x(l))) if and only if both f(x(k)) < f(x(l)) and h(x(k))< h(x(l)). Definiti on 3. Filter. A filter is a list of pairs(f(x),h(x)) such that no pair dominates each other. Definiti on 4. Acceptance criterion. Letting F denotes the filter, and (f(x(i)),h(x(i))), i=1,2,…,n are the elements in the filter. x can be accepted by filter F(k) if for all xi ∈F(k) it satisfies h(x)