Structured barrier and penalty functions in infinite dimensional optimisation and analysis. Very large scale tightly-constrained optimisation problems are ubiquitous and include water management, traffic flow, and imaging at telescopes and hospitals. Massively parallel computers can solve such problems and provide physically realisable solution only if subtle design issues are mastered. Resolving such issues is the goal of this project.
Decomposition and Duality: New Approaches to Integer and Stochastic Integer Programming. Because of their rich modelling capabilities, integer programs are widely used in industry for decision making and planning. However their solution algorithms do not have the maturity of their cousins in convex optimisation, where the theory of strong duality is ubiquitous. Efficient methods for convex optimisation under uncertainty do not apply to the integer case, which is highly non-convex. Furthermore, i ....Decomposition and Duality: New Approaches to Integer and Stochastic Integer Programming. Because of their rich modelling capabilities, integer programs are widely used in industry for decision making and planning. However their solution algorithms do not have the maturity of their cousins in convex optimisation, where the theory of strong duality is ubiquitous. Efficient methods for convex optimisation under uncertainty do not apply to the integer case, which is highly non-convex. Furthermore, integer models usually assume the data is known with certainty, which is often not the case in the real world. This project will develop new theory and algorithms to enhance the analysis of integer models, including those that incorporating uncertainty, while also enabling the use of parallel computing paradigms. Read moreRead less
Fuzzy modelling and design of complex networked systems. This project aims to develop analysis and synthesis approaches for non-linear networked control systems, including modelling, stability analysis and design problems. The non-linear effects and analysis of networked control systems have received considerable attention because of the universal existence of nonlinearities in practice. Network-based non-linear systems are widely used but face problems from non-linearities and networks. This pr ....Fuzzy modelling and design of complex networked systems. This project aims to develop analysis and synthesis approaches for non-linear networked control systems, including modelling, stability analysis and design problems. The non-linear effects and analysis of networked control systems have received considerable attention because of the universal existence of nonlinearities in practice. Network-based non-linear systems are widely used but face problems from non-linearities and networks. This project will establish a software-based nonlinear networked control system platform to test the presented algorithms and strengthen the scenarios in applications. This project is expected to increase Australian excellence in cyber-security and advanced manufacturing.Read moreRead less
New Theory and Algorithms for Nonsmooth Optimisation with Application to Integer Programming. Mathematical optimisation plays a key role in a wide variety of applications in business, industry, engineering and science. For example, airlines cannot fly and radiation treatment for cancer cannot be delivered without solving (a series of) optimisation problems. Some classes of optimisation problem are very well solved, with clear mathematical foundations, efficient algorithms, and reliable software ....New Theory and Algorithms for Nonsmooth Optimisation with Application to Integer Programming. Mathematical optimisation plays a key role in a wide variety of applications in business, industry, engineering and science. For example, airlines cannot fly and radiation treatment for cancer cannot be delivered without solving (a series of) optimisation problems. Some classes of optimisation problem are very well solved, with clear mathematical foundations, efficient algorithms, and reliable software implementations. Both nonsmooth and integer optimisation problems have a good mathematical basis, but there are "gaps"; existing methods cannot always solve real industrial problems. This project will deliver better methods, built on better theory, and so will yield better solutions for important applications.Read moreRead less
Graph isomorphism and quantisation of longest cycles by means of determinants and spectra. A characterisation of the difficulty of the Hamiltonian cycle problem and the graphs isomorphism problem will be a significant conceptual advancement with repercussions in a number of fields including combinatorial optimisation and theoretical computer science, in particular, the Google PageRank. Applications of tensor networks technique will lead to a design of a quantum computer that enumerates all Hamil ....Graph isomorphism and quantisation of longest cycles by means of determinants and spectra. A characterisation of the difficulty of the Hamiltonian cycle problem and the graphs isomorphism problem will be a significant conceptual advancement with repercussions in a number of fields including combinatorial optimisation and theoretical computer science, in particular, the Google PageRank. Applications of tensor networks technique will lead to a design of a quantum computer that enumerates all Hamiltonian cycles in a graph. Analysis of the determinant objective function in terms of the eigenvalues may lead to new spectral properties of stochastic matrices. Algorithmic advances exploiting such a characterisation will significantly contribute to existing technologies for solving problems in a wide range of applications.Read moreRead less
Perturbation and approximation methods for linear operators with applications to train control, water resource management and evolution of physical systems. Linear equations are used to solve practical problems. In realistic problems the equations and their solutions depend on parameters obtained by measurement of physical quantities and on data derived from observations and experiments. Changes to the values of the key parameters will lead to changes in the solutions. This project will devel ....Perturbation and approximation methods for linear operators with applications to train control, water resource management and evolution of physical systems. Linear equations are used to solve practical problems. In realistic problems the equations and their solutions depend on parameters obtained by measurement of physical quantities and on data derived from observations and experiments. Changes to the values of the key parameters will lead to changes in the solutions. This project will develop methods to better understand the relationships between the key parameters and the solutions and will apply the new insights to practical problems such as the minimization of fuel consumption in trains, optimal resource management in water supply systems and the evolution of physical systems.Read moreRead less
Strategic integration of renewable energy systems into the electricity grid. The Intergovernmental Panel on Climate Change states that 'warming of the climate system is unequivocal' and there is high probability it is anthropogenic. In response to the growing awareness of climate change, there is an expansion in Australia in the use of renewable energy resources in electricity generation, albeit from a low base. The various renewable energy systems have differing patterns of availability and vol ....Strategic integration of renewable energy systems into the electricity grid. The Intergovernmental Panel on Climate Change states that 'warming of the climate system is unequivocal' and there is high probability it is anthropogenic. In response to the growing awareness of climate change, there is an expansion in Australia in the use of renewable energy resources in electricity generation, albeit from a low base. The various renewable energy systems have differing patterns of availability and volatility, and it is difficult to determine the right mixture to best match the demand. It is imperative that future growth be structured so that both maximum grid penetration, and required greenhouse gas reductions be attained. Read moreRead less
Doubly Stochastic Matrices & The Hamiltonian Cycle Problem. The classical hard problem of determining whether a given graph possesses a Hamiltonian cycle contains the essential difficulty of the famous 'Travelling Salesman Problem'. A characterisation of this difficulty in terms of variability of returns (to the initial state) in a controlled stochastic process will be a significant conceptual advance with repercussions in a number of fields including optimisation and theoretical computer scien ....Doubly Stochastic Matrices & The Hamiltonian Cycle Problem. The classical hard problem of determining whether a given graph possesses a Hamiltonian cycle contains the essential difficulty of the famous 'Travelling Salesman Problem'. A characterisation of this difficulty in terms of variability of returns (to the initial state) in a controlled stochastic process will be a significant conceptual advance with repercussions in a number of fields including optimisation and theoretical computer science. Algorithmic advances exploiting such a characterisation will significantly contribute to existing technologies for solving problems in applications ranging from logistics to cryptography. Since TSP describes certain efficient ways of routing its applicability to information networks is clear.Read moreRead less
New Analytical Perspectives on the Algorithmic Complexity of the Hamiltonian Cycle Problem. Hamiltonian Cycle Problem (HCP), known - in the complexity theory of
algorithms -to be NP-hard is proposed for study, from three innovative,
separate (yet related) analytical perspectives: singularly perturbed
(controlled) Markov chains, that links the HCP with systems and control
theories; parametric nonconvex optimization, that links HCP with fast
interior point methods of modern optimization an ....New Analytical Perspectives on the Algorithmic Complexity of the Hamiltonian Cycle Problem. Hamiltonian Cycle Problem (HCP), known - in the complexity theory of
algorithms -to be NP-hard is proposed for study, from three innovative,
separate (yet related) analytical perspectives: singularly perturbed
(controlled) Markov chains, that links the HCP with systems and control
theories; parametric nonconvex optimization, that links HCP with fast
interior point methods of modern optimization and the spectral approach
based on a novel adaptation of Ihara-Selberg trace formula for regular
graphs. Our mathematical approach to this archetypal complex problem of graph
theory and discrete optimization promises to enhance the fundamental
understanding - and ultimate "managibility" - of the underlying
difficulty of HCP.
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Optimal Transforms of Random Vectors. This proposal focusses on development of optimal transforms to describe and model nonlinear phenomena when only statistical information is known. An optimal transform is a mathematical procedure that enables us to process information in a way that is most suited to the task in hand. These transforms have been successfully used in approximation, information theory, communications, control theory and signal and image processing. Applications include modelli ....Optimal Transforms of Random Vectors. This proposal focusses on development of optimal transforms to describe and model nonlinear phenomena when only statistical information is known. An optimal transform is a mathematical procedure that enables us to process information in a way that is most suited to the task in hand. These transforms have been successfully used in approximation, information theory, communications, control theory and signal and image processing. Applications include modelling of physical, chemical and biological systems, filtering and compression of signals and data classification and clustering. We propose two new hybrid models for realistic transforms in a general structural framework.
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