Modeling and Optimizing for NP-hard Problems in Graph Theory

This PhD thesis introduces optimization methods for graph problems classified as NP-hard. These are problems for which no deterministic algorithm is capable of solving them in polynomial time. More specifically, three graph problems were addressed, and for each, different optimization methods were used. These methods include standard methods, metaheuristics, and heuristics. In all cases, the performance of these methods was compared with those proposed in the literature, considering factors such as execution time and the quality of the solutions achieved. This comparative analysis aims to demonstrate the effectiveness of the proposed optimization methods

Intelligent optimisation of analogue circuits using particle swarm optimisation, genetic programming and genetic folding

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London.This research presents various intelligent optimisation methods which are: genetic algorithm (GA), particle swarm optimisation (PSO), artificial bee colony algorithm (ABCA), firefly algorithm (FA) and bacterial foraging optimisation (BFO). It attempts to minimise analogue electronic filter and amplifier circuits, taking a cascode amplifier design as a case study, and utilising the above-mentioned intelligent optimisation algorithms with the aim of determining the best among them to be used. Small signal analysis (SSA) conversion of the cascode circuit is performed while mesh analysis is applied to transform the circuit to matrices form. Computer programmes are developed in Matlab using the above mentioned intelligent optimisation algorithms to minimise the cascode amplifier circuit. The objective function is based on input resistance, output resistance, power consumption, gain, upperfrequency band and lower frequency band. The cascode circuit result presented, applied the above-mentioned existing intelligent optimisation algorithms to optimise the same circuit and compared the techniques with the one using Nelder-Mead and the original circuit simulated in PSpice. Four circuit element types (resistors, capacitors, transistors and operational amplifier (op-amp)) are targeted using the optimisation techniques and subsequently compared to the initial circuit. The PSO based optimised result has proven to be best followed by that of GA optimised technique regarding power consumption reduction and frequency response. This work modifies symbolic circuit analysis in Matlab (MSCAM) tool which utilises Netlist from PSpice or from simulation to generate matrices. These matrices are used for optimisation or to compute circuit parameters. The tool is modified to handle both active and passive elements such as inductors, resistors, capacitors, transistors and op-amps. The transistors are transformed into SSA and op-amp use the SSA that is easy to implement in programming. Results are presented to illustrate the potential of the algorithm. Results are compared to PSpice simulation and the approach handled larger matrices dimensions compared to that of existing symbolic circuit analysis in Matlab tool (SCAM). The SCAM formed matrices by adding additional rows and columns due to how the algorithm was developed which takes more computer resources and limit its performance. Next to this, this work attempts to reduce component count in high-pass, low-pass, and all- pass active filters. Also, it uses a lower order filter to realise same results as higher order filter regarding frequency response curve. The optimisers applied are GA, PSO (the best two methods among them) and Nelder-Mead (the worst method) are used subsequently for the filters optimisation. The filters are converted into their SSA while nodal analysis is applied to transform the circuit to matrices form. High-pass, low-pass, and all- pass active filters results are presented to demonstrate the effectiveness of the technique. Results presented have shown that with a computer code, a lower order op-amp filter can be applied to realise the same results as that of a higher order one. Furthermore, PSO can realise the best results regarding frequency response for the three results, followed by GA whereas Nelder- Mead has the worst results. Furthermore, this research introduced genetic folding (GF), MSCAM, and automatically simulated Netlist into existing genetic programming (GP), which is a new contribution in this work, which enhances the development of independent Matlab toolbox for the evolution of passive and active filter circuits. The active filter circuit evolution especially when operational amplifier is involved as a component is of it first kind in circuit evolution. In the work, only one software package is used instead of combining PSpice and Matlab in electronic circuit simulation. This saves the elapsed time for moving the simulation between the two platforms and reduces the cost of subscription. The evolving circuit from GP using Matlab simulation is automatically transformed into a symbolic Netlist also by Matlab simulation. The Netlist is fed into MSCAM; where MSCAM uses it to generate matrices for the simulation. The matrices enhance frequency response analysis of low-pass, high-pass, band-pass, band-stop of active and passive filter circuits. After the circuit evolution using the developed GP, PSO is then applied to optimise some of the circuits. The algorithm is tested with twelve different circuits (five examples of the active filter, four examples of passive filter circuits and three examples of transistor amplifier circuits) and the results presented have shown that the algorithm is efficient regarding design.Tertiary Education Trust Fund (TETFUND) through University of Calabar, Nigeria

Solving iterated functions using genetic programming

An iterated function f(x) is a function that when composed with itself, produces a given expression f(f(x))=g(x). Iterated functions are essential constructs in fractal theory and dynamical systems, but few analysis techniques exist for solving them analytically. Here we propose using genetic programming to find analytical solutions to iterated functions of arbitrary form. We demonstrate this technique on the notoriously hard iterated function problem of finding f(x) such that f(f(x))=x2–2. While some analytical techniques have been developed to find a specific solution to problems of this form, we show that it can be readily solved using genetic programming without recourse to deep mathematical insight. We find a previously unknown solution to this problem, suggesting that genetic programming may be an essential tool for finding solutions to arbitrary iterated functions