Geometry in projection methods and fixed-point theory. This project aims to resolve mathematical challenges arising from problems with specific structure typical for key modern applications, such as big data optimisation, chemical engineering and medical imaging. We focus on developing new mathematical tools for the analysis of projection methods and accompanying fixed point theory, specifically targeting the refinement of the geometric intuition for algorithm design techniques to inform the imp ....Geometry in projection methods and fixed-point theory. This project aims to resolve mathematical challenges arising from problems with specific structure typical for key modern applications, such as big data optimisation, chemical engineering and medical imaging. We focus on developing new mathematical tools for the analysis of projection methods and accompanying fixed point theory, specifically targeting the refinement of the geometric intuition for algorithm design techniques to inform the implementation of optimal methods for huge-scale optimisation problems.Read moreRead less
An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, mos ....An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, most notably optimisation. The techniques and methods developed should also have significant benefits in the many disciplines where approximation problems appear, such as engineering, physics or data mining. The research outputs resulting from this project will be used in a wide range of fields to help implement programs, policies and improve decision making.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE240100006
Funder
Australian Research Council
Funding Amount
$444,847.00
Summary
Robust Derivative-Free Algorithms for Complex Optimisation Problems. Mathematical optimisation gives a systematic way for optimal decision-making. This project aims to develop new mathematical tools for complex optimisation problems where limited problem information is available. It will generate new foundational theories for alternative optimisation tools, introducing substantial new capability and rigour to the discipline. The project will create significant new mathematical optimisation techn ....Robust Derivative-Free Algorithms for Complex Optimisation Problems. Mathematical optimisation gives a systematic way for optimal decision-making. This project aims to develop new mathematical tools for complex optimisation problems where limited problem information is available. It will generate new foundational theories for alternative optimisation tools, introducing substantial new capability and rigour to the discipline. The project will create significant new mathematical optimisation techniques and create world-leading and publicly available software. These new techniques and software may ultimately be able to solve some of the most complex optimisation problems in research and industry, such as improving long-term climate predictions and designing 3D-printed medical implants.Read moreRead less
Innovations in sparse optimisation: big data nonsmooth optimisation. This project aims to produce innovative optimisation methods capable of solving a wide range of practical problems that are currently too complex to be solved. Optimisation involving huge data sets is ubiquitous. Sparse optimisation has emerged as a challenging frontier of modern optimisation because it effectively computes an optimal solution with desired low complexity structure so that a resulting solution can be efficiently ....Innovations in sparse optimisation: big data nonsmooth optimisation. This project aims to produce innovative optimisation methods capable of solving a wide range of practical problems that are currently too complex to be solved. Optimisation involving huge data sets is ubiquitous. Sparse optimisation has emerged as a challenging frontier of modern optimisation because it effectively computes an optimal solution with desired low complexity structure so that a resulting solution can be efficiently stored, implemented and utilised, and is robust to the data inexactness. This project aims at developing innovative mathematical techniques and efficient numerical schemes for solving sparse optimisation problems. The intended outcomes will have significant impact on many areas of science, medicine and engineering, where sparse optimisation is used, including cancer radiotherapy optimal planning.Read moreRead less
Relaxed reflection methods for feasibility and matrix completion problems. The project proposes to further develop the non-linear convergence theory, and to provide problem-specific implementations. Many applied and pure problems require solution of a large set of linear or nonlinear equations (or inequalities). Highly effective, parallelisable methods are based on iterated projection or reflection algorithms which aggregate information about individual equations. The theory is well developed in ....Relaxed reflection methods for feasibility and matrix completion problems. The project proposes to further develop the non-linear convergence theory, and to provide problem-specific implementations. Many applied and pure problems require solution of a large set of linear or nonlinear equations (or inequalities). Highly effective, parallelisable methods are based on iterated projection or reflection algorithms which aggregate information about individual equations. The theory is well developed in the linear case, but does not explain many important applications for which they are often highly successful (eg optical aberration correction, protein reconstruction, tomography, compressed sensing). The project also plans to provide heuristics to help explain why an algorithm performs well on one class of applications but fails on another.Read moreRead less
Data-Driven Multistage Robust Optimization—the New Frontier in Optimization. Robust optimisation is a powerful technology for decision-making in uncertain environments. Yet, developing numerically certifiable optimisation principles and data-driven methods that can be readily implemented by common computer algorithms remains an elusive goal for multistage robust optimisation. But it is crucial for the practical use of multistage optimisation. This project aims to develop this novel mathematical ....Data-Driven Multistage Robust Optimization—the New Frontier in Optimization. Robust optimisation is a powerful technology for decision-making in uncertain environments. Yet, developing numerically certifiable optimisation principles and data-driven methods that can be readily implemented by common computer algorithms remains an elusive goal for multistage robust optimisation. But it is crucial for the practical use of multistage optimisation. This project aims to develop this novel mathematical theory and methods by extending the investigators' recent award winning advances, including the von Neumann-prizewinning Lasserre-hierarchy approach. Results will provide a foundation and technologies for making superior decisions in the pervasive presence of big data uncertainty, enhancing data-driven innovation in AustraliaRead moreRead less
Discovery Early Career Researcher Award - Grant ID: DE240100674
Funder
Australian Research Council
Funding Amount
$370,237.00
Summary
New Frontiers in Large-Scale Polynomial Optimisation. Polynomial optimisation is ubiquitous in many areas of engineering and applied mathematics. The mathematical methods and algorithms used for polynomial problems of large size are not sufficiently developed, limiting their applicability for real-world problems. This project aims to develop a mathematical foundation and computational methods for large-scale polynomial optimisation. By using an innovative combination of a novel theory of algebra ....New Frontiers in Large-Scale Polynomial Optimisation. Polynomial optimisation is ubiquitous in many areas of engineering and applied mathematics. The mathematical methods and algorithms used for polynomial problems of large size are not sufficiently developed, limiting their applicability for real-world problems. This project aims to develop a mathematical foundation and computational methods for large-scale polynomial optimisation. By using an innovative combination of a novel theory of algebraic geometry and convex optimisation, this project expects to generate new knowledge and tools for solving these problems. Anticipated outcomes include a new generation of large-scale optimisation technologies, providing significant benefit to Australia's industries and international research standing.
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