Global Optimization, Optimization under Uncertainty, Mathematical Programming, Convex Analysis and Optimization, Quadratic Optimization, Polynomial Optimization.
Field of Research (FoR)
I am an applied mathematician. My research interests are in mathematical optimization. It is an area of mathematics that directly deals with the problem of making the best possible choice from a set of feasible choices. It seeks to understand how we achieve the best possible choice and how we can use this knowledge to improve management and technical decisions in science, engineering and commerce. Thinking in terms of...view more
I am an applied mathematician. My research interests are in mathematical optimization. It is an area of mathematics that directly deals with the problem of making the best possible choice from a set of feasible choices. It seeks to understand how we achieve the best possible choice and how we can use this knowledge to improve management and technical decisions in science, engineering and commerce. Thinking in terms of choices is common in our cognitive culture and searching for the best possible choice is a basic human desire.Thus models of optimization arise everyday as management and technical decisions in many areas of human activity. I develop mathematical principles and methods for identifying and locating solutions of a wide range of optimization models.
PhD in Optimization, University of Melbourne
AWARDS & ACHIEVEMENTS
- 2017: Winner of the 2015 Optimization Letters Best Paper Award, http://link.springer.com/article/10.1007/s11590-016-1099-0
- 2017 UNSW Science Gold Star Award
- 2011: UNSW Gold Star Award
- 2010: International Collaboration Award, Australian Research Council
- 2007: Linkage International Award, Australian Research Council and Korean Science and Engineering Foundation
RECENT COMPETITIVE GRANTS
- 2004-2006, ARC Discovery Project Grant ($283,752) (with A.M. Rubinov): Necessary and Sufficient Conditions in Global Continuous Optimization
- 2005-2007, ARC Discovery Project Grant ($403,000) (with T. Hoang and B. Vo) : Convex optimization for control, signal processing and communications
- 2007-2009, ARC Discovery Project Grant ($246,000): Quadratic support function technique to solving hard global nonconvex optimization problems
- 2010-2013, ARC Discovery Project Grant($285,543): A new improved solution to global optimization over multi-variate polynomials
- 2012-2014, ARC Discovery Project Grant($330,000) (with G. Li): New theory and methods of robust global optimization: Modern decision making under uncertain conditions
- 2018-2020, ARC Discovery Project Grant ($401,706) (with G. Li) : New mathematics for multi-extremal optimization and diffusion tensor imaging
My Research Goals
- To understand and improve Mathematics of Decision Making
- To develop and characterize mathematical principles of optimization
- To apply the principles of optimization to bring rigorous approaches to improve management and technical decisions
- To provide advanced mathematical framework to solve practical optimization problems in science, engineering, commerce and medicine
My Research in Detail
I am interested in a range of topics in Optimization. One strand of my research examines optimization models in the face of data uncertainty. We examine mathematical approaches to finding robust best solutions of uncertain optimization models that are immunized against data uncertainty. We develop mathematical principles and methods for identifying and locating such solutions of a wide range of optimization models.
One paper, grown out our recent work in this area, won the Optimization Letters journal Best Paper Award for 2015.
A current project examines multi-objective optimization models in the presence of conflicting objectives and data uncertainty. It is a joint work with Dr Guoyin Li (Associate Professor), who is a member of the Optimization group at UNSW.
A second area of my research focuses on analysis, methods and applications of duality theory and associated techniques of convex optimization. We have examined a variety of topics such as the convex optimization models arising from the machine learning problems of large-scale data classification.
Some of my recent research projects in this area have involved technologically significant problems such as the development of screening algorithms for HIV-associated neurological disorders, produced by a new optimization procedure. This was a result of successful research collaboration between the research group in Optimization within Mathematics and the HIV Epidemiology and Clinical Research Group at St Vincent's hospital. The research outcome on screening algorithms was published in the British Journal, HIV Medicine.
Much of my studies in this area is of fundamental in nature, but these mathematical studies have a flow-through effect for addressing major challenges of modern problems such as the development of clinically based decision support tools in medical science. A current project is examining two-stage convex optimization with applications to medical decision-making optimization models of radiation therapy planning.
A third strand of my research looks at semi-algebraic global optimization problems, where the set of feasible choices is defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles which can identify and locate the globally best solutions and can also be readily validated by computer methods using commonly available algorithms and software. A current project is examining bi-level global optimization models involving polynomials.
Nowadays, optimization together with sophisticated computer models is a widely used technology to improve performance in many industrial problems and emerging scientific applications. It is an interesting and challenging area of mathematical research. My current research project contains exciting components that are suitable for honours projects and PhD programs.
One study examines emerging applications of algebraic geometry to optimization over polynomials. A good understanding of convex sets in algebraic geometry will lead to insights into solving complex optimization problems involving polynomials.
Another project investigates bi-level optimization models. These models arise when two independent decision makers, ordered within a hierarchical structure, have conflicting objectives. Optimization models of this kind appear in resource allocation and planning problems.
A new project starting this year is examining optimization approaches to multi-stage optimization problems in the face of evolving uncertainty. The dynamic decision-making problems under uncertainty are multi-stage robust optimization problems, where uncertainties evolve over time (in stages), rendering traditional optimization solutions sub-optimal. This issue is particularly prevalent in medical decision-making, where a patient's condition can change during the course of the treatment. The aim of the project is to develop optimization principles to identify true optimal solutions of these multi-stage problems and develop associated methods to find those solutions.
Recently completed thesis topics include:
(i) Bilevel optimization and Principal-Agent Problems
(ii) Portfolio Optimization under Data Uncertainty
(iii) Optimization approaches to simultaneous classification and feature selection
(iv) Convex optimization approaches to data-mining
I am interested in talented prospective postgraduate students/postdocs/honours students with interests in these areas.
TEACHING & OUTREACH
Courses I teach
Professional affiliations and service positions
- Associate Editor, The Journal of Optimization Theory and Applications
- Associate Editor, Journal of Global Optimization
- Associate Editor, Journal of the Australian Mathematical Society
- Associate Editor, Minimax Theory and Its Applications