Replica Exchange using q-Gaussian Swarm Quantum Particle Intelligence Method

We present a newly developed Replica Exchange algorithm using q -Gaussian Swarm Quantum Particle Optimization (REX@q-GSQPO) method for solving the problem of finding the global optimum. The basis of the algorithm is to run multiple copies of independent swarms at different values of q parameter. Based on an energy criterion, chosen to satisfy the detailed balance, we are swapping the particle coordinates of neighboring swarms at regular iteration intervals. The swarm replicas with high q values are characterized by high diversity of particles allowing escaping local minima faster, while the low q replicas, characterized by low diversity of particles, are used to sample more efficiently the local basins. We compare the new algorithm with the standard Gaussian Swarm Quantum Particle Optimization (GSQPO) and q-Gaussian Swarm Quantum Particle Optimization (q-GSQPO) algorithms, and we found that the new algorithm is more robust in terms of the number of fitness function calls, and more efficient in terms ability convergence to the global minimum. In additional, we also provide a method of optimally allocating the swarm replicas among different q values. Our algorithm is tested for three benchmark functions, which are known to be multimodal problems, at different dimensionalities. In addition, we considered a polyalanine peptide of 12 residues modeled using a Gō coarse-graining potential energy function. Introduction The problem of finding the global optimum in a multimodal and multidimensional space can be extremely difficult since the number of stable optima increases as the search space increases, for instance, the search for the global minimum energy in a surface energy landscape of the atomic structures. [1, 2] Swarm Particle Optimization (SPO) is population-based optimization technique, similar to evolutionary algorithms. [3] Kennedy & Eberhart introduced the method to solve the problem of finding the global optimum of a d dimensional function. [4] The basis of SPO method are the swarm intelligence algorithms, which concern with the design of intelligent multi-agent systems based on the collective behavior of insects (ants, termites, bees, and wasps) or other animal societies (flocks of birds and schools of fish). [4] In SPO method, the swarm particles, representing possible solutions, search the phase space, defined by their velocities and coordinates, which are updated based on the particle’s own experience and experience of the particle’s neighbors or the experience of the whole swarm. The method has already been used to solve many optimization problems, [5] with some interests also in other fields, such as statistical mechanics. [6] Since the standard SPO algorithm has a low convergence rate, [7, 8] several improvements and variants of the SPO algorithm have been proposed. [9, 10, 11, 12, 13] The new variant of the SPO method, the so-called Swarm Quantum Particle Optimization (SQPO), has been considered as an improvement against the classical SPO method, since there is a nonzero probability to escape the local minima even for very high barriers. [14] Efforts have been made to improve the SQPO method. [15, 16, 17, 18, 19, 20, 21, 22] These improvements focus primarily on parameter selection criteria, [18, 22] and maintaining diversity of the swarm. [19, 20, 21] A detailed review of all these methods is described in Ref. [23] Use of different forms of attractive potential-energy surfaces for SQPO algorithm is also considered for improvement of the algorithm. [14] Different potentials yield different probability distributions, which describe the probability of finding the swarm quantum-like particle at a certain position in the phase space. [14] 1 Corresponding author: [email protected]; Tel. +389(0)75462189; Fax. +389(0)23214832; Address: International Balkan University, Tashko Karadza 11A, 1000 Skopje, R. of Macedonia