In: Proc. IEEE Congr. Liu, H. Shi, Y.

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Evolutionary Computation, pp. Eberhart, R.

## Image Restoration Using Improved Particle Swarm Optimization - IEEE Conference Publication

Evolutionary Computation, vol. Clerc, M. Poli, R. In: Keijzer, M. EuroGP LNCS, vol. Springer, Heidelberg Google Scholar. Ting, T. Paquet, U. Parsopoulos, K. Grosan, C. Gehlhaar, Fogel: Tuning Evolutionary programming for conformationally flexible molecular docking. Pant, M.

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In: 3rd Indian Int. Krohling, R. Kimura, S. Nguyen, X. Brits, R. Chi, H. In: Sunderam, V. ICCS Engelbrecht, A. Riget, J. Technical report, EVAlife, Dept. In: Int. Millie Pant, T. If a system or design is defined as a process that consists of inputs that perform certain functions to produce outputs, then by producing a mathematical representation of the design and introducing an evaluation criterion, an optimization problem is created.

Therefore, an optimization problem consists of three components, a property to be minimized or maximized, restriction s or constraint s that needs to be satisfied and parameter s that can be adjusted.

## Particle Swarm Optimisation for Feature Selection in Classification

The parameter, either explicitly or implicitly, has an effect on both the property being optimized and the restriction that is being fulfilled. Many optimization methods have been developed and applied to produce better designs. Due to the advancement in computational power and the increasing need to solve optimization problems, researchers have developed various optimization methods to meet the demands to solve complex and multidimensional problems. This research focuses on the metaheuristic nature — inspired optimization method, the Particle Swarm Optimization PSO algorithm.

PSO is a stochastic optimization method that takes its inspiration 2 from the social behavior of animals looking for food such as a flock of birds or a school of fish [1].

Problems where some variables are required to be integers are termed as mixed integer programming problems. Whereas problems where any value can be selected within a predefined set are referred as continuous programming problems [2]. Nonlinear programming problems are problems where nonlinearity exists in any of the of the optimization functions [3]. In problems where the optimization function s are static throughout all iterations, this is known as a static problem [4]. Whereas a problem where the objective function s is solved without any restriction s is known as unconstrained problem.

While in cases where multiple objective functions are solved simultaneously, this type of problem is referred to as multiobjective function optimization problem. The following subsections describe the various classes of optimization methods. These methods perform an iterative calculation by initially starting with an estimate and attempts to improve the design until certain mathematical conditions are fulfilled. The earliest derivative based method developed is the steepest descent method [2].

The SQP method formulates a quadratic subproblem through the quadratic approximation of the Lagrange function. Only information from the objective function s and constraint s are used by these methods [7]. The key feature of this class of methods is that they do not utilize any qualitative information related to the function values and that no approximation of the optimization function s is made [8].

A common direct search method is the pattern search method, the Hooke-Jeeves method [9]. The Hooke-Jeeves method performs a series of exploratory and pattern moves to produce an improved design. Here also the iterative calculation is initiated with an initial design. In the exploratory phase, a single dimensional coordinate, i. If the function did not increase, the coordinate value is stored and if the function increased, the value is rejected and the step size is shortened by a factor and then depending on its performance, the value is either stored or rejected.

The difference between the present and previous base point constitutes the search direction taken in the pattern search phase. The pattern search equation is described by [9]:? The search continues until it 6 reaches a point where no further improvement is made [10].

Another popular direct search method is the Nelder — Mead simplex method [11]. However, the derivative — free methods approximates the optimization functions using a method called the response surface methods RSM. RSM is a technique, which is used to generate an approximate relationship between the design variables and the output parameter [12]. The original system is evaluated at several design points and after which a correlation is made using linear, quadratic or cubic functions. In order to produce a response surface, design points need to be selected.

### Associated Data

The selection of design points can be performed either randomly or by utilizing methods that attempt to cover the entire design space. Using the method of Design of Experiments DOE , sampling points can be determined for generation of the response surface model. Once the approximated model has been generated, any optimization method may be applied onto it. The response surface method, compared to the original system, reduces the computational demands; however, depending on the quality of the approximation, the result accuracy would be affected [14].

In this class, stochastic behavior is imbedded in the algorithm leading to a certain degree of randomness in the problem solving. This randomness is desirable since it allows the algorithm to escape a local result and increases the possibility of locating the global solution [15].

## A quantum behaved particle swarm approach to multi-objective optimization

Due to the advancement in computational power, this class of methods is becoming widely used to solve 7 optimization problems. These methods take their inspiration from naturally occurring phenomena such as natural selection and swarm intelligence [15]. Furthermore, these methods are general which means that they could be used to solve any type of optimization problem. The following subsections will cover some techniques within the nature — inspired class namely the Genetic Algorithm, Ant Colony Optimization and Particle Swarm Optimization.

GA solves for the global optimum by using the concept of natural selection [16]. In GA, design points are initialized randomly with respect to the allowable values for each design variable and the fitness of each randomly assigned design point is evaluated through the objective function. Two common terms used in GA are chromosome and gene [15]. Chromosome refers to the design point vector containing the value of all design variables and gene refers to the scalar quantity of an individual design variable. The GA algorithm usually represents the chromosome in the form of a binary string.

This algorithm contains three key steps: Selection, Crossover and Mutation. After the evaluation of each chromosome, the ones with the best potential are selected for reproduction which compromises the selection step [16].

In the crossover step, the algorithm merges two chromosomes parents to form a new design point child. Mutation is the following step in the GA algorithm where the objective is to reintroduce any lost genetic information in the preceding steps or to introduce any new information [17]. In this step, the binary value of the gene undergoes modifications from 0 to 1 or vice versa. Modification to the chromosome is necessary since it helps the chromosome in getting out of a local solution and continuing the search for the global solution [15].

An individual member will initially be randomly looking for food and upon detection of the pheromone trail, it will deduce whether to follow it or not. By following the same path, an ant secretes its own pheromone onto the trail thus increasing the likelihood of other ants following the same path [18]. Furthermore, it has been experimentally shown that this behavior pheromone trail gives rise to the shortest path between the colony and food [4].

All nodes are connected by a series of discrete points forming links. These links represent the trail that an ant would follow through when foraging. All links are initialized with a pheromone value and the virtual ants moves to the next node through a probabilistic method until they reach their final destination. Upon reaching the end point, evaporation of pheromone occurs leading to a reduction in the pheromone value across all links.

Following this, the ants make their way back to their initial position while reinforcing their chosen path by updating the pheromone value [18].