Extremal optimization

About: Extremal optimization is a research topic. Over the lifetime, 1168 publications have been published within this topic receiving 104943 citations.

The Tunnelling Salesman: Truncated Variational Approximations for Quantum Mechanical Global Optimization

Abstract: In this work we present a novel method for global optimization, exploiting the mathematics of quantum mechanics, and in particular the tunnelling phenomenon. We consider a quantum mechanical system with a single particle in a potential specified by the cost function of our optimization problem. We assume the groundstate of the system is localised to the global minimum of the potential function, a condition which can be assured by choosing the particle mass sufficiently large. We then approximate this groundstate using a variational approach to optimise an expansion in terms of a truncated basis set of harmonic oscillator wavefunctions. We show how to encode integer factoring and travelling salesman problems in terms of finding the global minimum of a quartic polynomial cost function. We demonstrate our quantum algorithm on oneand two-dimensional toy problems, and apply it to factoring biprimes with 10 bits. Introduction The state of the art in global optimization for general problems is simulated annealing and its derivatives. The algorithm can be viewed in the physical context of a classical thermal system. This inspires the novel approach presented here, in which we draw upon intuition from non-classical physical systems which obey quantum mechanics (QM), and in particular exhibit the phenomenon of quantum tunnelling. In QM, a potential function V (x) describes the energy terrain experienced by a particle, which may contain multiple minima separated by barriers. Combined with a term describing the kinetic energy of the particle, it defines the Hamiltonian operator: Ĥ = − 2 2m ∇2 + V (1) QM captures all available information on the state of the particle in a (complex) wave function ψ(x). For instance, |ψ(x)|2 is the probability distribution over the position x of the particle. To obtain ψ(x) one solves the eigenvalue problem Ĥψ = Eψ (2) known as the Schroedinger equation, where E is the energy. The solution is a set of wave functions termed eigenstates, and a matching set of energies. The eigenstate with the lowest energy is the ground state, ψ. By choosing the particle’s mass sufficiently large, the probability density associated with the ground state wave function becomes localized to the global minimum of the potential. Hence, a given optimization problem involving the minimization of a cost function can be restated as the problem of finding the ground state of the QM system with a potential specified by that cost function. In particular, computationally ‘hard’ integer problems, including factoring [1] and the travelling salesman problem, can be cast as QM problems involving a quartic polynomial potential, [3]. However, solving the Schroedinger equation exactly for these potentials is generally an analytically intractable problem. In our approach we approximate the groundstate of such a system using a method based on perturbation theory.

An improved extremal optimization based on the distribution knowledge of candidate solutions

Abstract: Extremal optimization (EO) is a phenomenon-mimicking algorithm inspired by the Bak-Sneppen model of self-organized criticality from the field of statistical physics. The canonical EO works on a single solution and only employs mutation operator, which is inclined to prematurely converge to local optima. In this paper, a population-based extremal optimization algorithm is developed to provide a parallel way for exploring the search space. In addition, a new mutation strategy named cloud mutation is proposed by analyzing the distribution knowledge of each component set in the solution set. The population-based extremal optimization with cloud mutation is characteristic of mining and recreating the uncertainty properties of candidate solutions in the search process. Finally, the proposed algorithm is applied to numerical optimization problems in comparison with other reported meta-heuristic algorithms. The statistical results show that the proposed algorithm can achieve a satisfactory optimization performance with regards to solution quality, successful rate, convergence speed, and computing robustness.

Integrating Local Search Methods in Metaheuristic Algorithms for Combinatorial Optimization: The Traveling Salesman Problem and its Variants

Abstract: Metaheuristic Algorithms (MAs) have demonstrated exceptional competence in solving Combinatorial Optimization Problems (COP) such as the Minimum Spanning Tree Problem (MSTP), the Features Selection Problem (FSP) with the most popular and oldest being the Traveling Salesman Problem (TSP). However, this class of algorithms many a time suffers from local optima stagnation leading to sub-optimal performance. Thus, there is a need for such algorithms to be supported with specific Local Search (LS) procedures either as an inner component or as a post-processing mechanism to enhance the search process for better performance. This paper presents a comprehensive review of the integration of LS methods in metaheuristic algorithms for solving single, multi, and many-objective COP with a focus on the TSP and its variants. The LS methods reviewed in this study were classified into one-way and two-way based on their mode of operation. In addition, practical suggestions were discussed and possible future directions were pointed out.

Combinatorial optimization with coupled chemical reaction cells

Abstract: Many combinatorial optimization problems in industry can be reduced to a so-called assignment problem. This assignment problem can be handled by an adapted form of the nonlinear differential equations which are used to model the macroscopic behavior of complex physical systems. To get the necessary adaption specific coupling terms are used to result in a suitable selection and feasible solutions as stable points of the dynamical system. In comparison to many other methods this approach has the advantage that additional constraints of the optimization problem can easily be considered. Furthermore, parallel hardware realizations of this approach are possible because of the similarity to models of complex physical and chemical systems. A realization with coupled chemical reaction cells is suggested in this paper.