A new graph perspective on max-min fairness in Gaussian parallel channels

In this work we are concerned with the problem of achieving max-min fairness in Gaussian parallel channels with respect to a general performance function, including channel capacity or decoding reliability as special cases. As our central results, we characterize the laws which determine the value of the achievable max-min fair performance as a function of channel sharing policy and power allocation (to channels and users). In particular, we show that the max-min fair performance behaves as a specialized version of the Lovasz function, or Delsarte bound, of a certain graph induced by channel sharing combinatorics. We also prove that, in addition to such graph, merely a certain 2-norm distance dependent on the allowable power allocations and used performance functions, is sufficient for the characterization of max-min fair performance up to some candidate interval. Our results show also a specific role played by odd cycles in the graph induced by the channel sharing policy and we present an interesting relation between max-min fairness in parallel channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information Theory on August the 6th, 200

Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. In a recent publication, we studied the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device

Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. We study the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device.</p

Managing uncertainty in modelling of wicked problems: theory and application to Sustainable Aquifer Yield

This thesis presents two approaches to help manage uncertainty in modelling for the resolution of wicked problems , which have no clear problem definition, solution or measure of success. It focuses on Sustainable Aquifer Yield (SAY) as an example. SAY is defined as the pumping volume obtained by a management plan that is expected to satisfy objectives under future conditions within a groundwater system. Integrated modelling can help express, systematise and use knowledge of relevant behaviour of the system, while engaging diverse stakeholders and addressing their interests. Uncertainty is however a key and multifaceted issue when dealing with wicked problems. While many modelling methods exist to help address this uncertainty, there is a need for modellers to be able to integrate these methods purposefully for an applied problem. The research presented involved iteratively proposing two approaches to manage uncertainties in integrated modelling that supports decision making, and exploring the value of each approach by applying it to case studies. For each approach, the applications specifically a) address a technical problem, b) push boundaries on how the problem is viewed, specifically identifying hitherto neglected aspects, and c) address a context where accounting for contested views and surprise is imperative. This research process is described in terms of Critical Systems Practice and resulted in a compilation of linked publications. The first approach proposed is an Uncertainty Management Framework that can be used to help audit the treatment of uncertainty in a step-wise description of an analysis (e.g. evaluating a management plan). The framework provides a formal structure for managing uncertainty by incorporating an uncertainty typology and a set of fundamental uncertainty management actions, but may be too restrictive and demanding for some contexts. To address these limitations, a complementary second approach, designated Iterative Closed Question Modelling, addresses uncertainty by constructing models to test whether each possible answer to a closed question is plausible. The question, assumptions about plausibility and the process of constructing models are all considered uncertain and therefore themselves iteratively critiqued. This approach is formalised in terms of Boundary Critique such that it provides a philosophical foundation justifying the use of a broad range of methods to manage uncertainty in predictive modelling. The thesis concludes that uncertainty needs to be embraced as a natural part of researchers, policy makers and community coming to grips with an evolving situation, rather than being an obstacle to be eliminated. Training of modellers to manage uncertainty needs to specifically address: identification of model scenarios that contradict dominant conclusions; critique of model assumptions and questions from multiple stakeholders’ points of view; and negotiation of the modeller’s role in anticipating surprise (e.g. through understanding consequences of error, design of monitoring, contingency planning and adaptive management). The resulting emphasis on critical thinking about alternative models helps to remind the user that modelling is not a magic trick for seeing the future, but a structured way to reason about both what we do and do not know