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

An artificial neural network approach for modelling the ward atmosphere in a medical unit

Artificial neural networks (ANNs) have been developed, implemented and tested on the basis of a four-year-long experimental data set, with the aim of analyzing the performance and clinical outcome of an existing medical ward, and predicting the effects that possible readjustments and/or interventions on the structure may produce on it. Advantages of the ANN technique over more traditional mathematical models are twofold: on one hand, this approach deals quite naturally with a large number of parameters/variables, and also allows to identify those variables which do not play a crucial role in the system dynamics; on the other hand, the implemented ANN can be more easily used by a staff of non-mathematicians in the unit, as an on-site predictive tool. As such, the ANN model is particularly suitable for the case study. The predictions from the ANN technique are then compared and contrasted with those obtained from a generalized kinetic approach previously proposed and tested by the authors. The comparison on the two case periods shows the ANN predictions to be somewhat closer to the experimental values. However, the mean deviations and the analysis of the statistical coefficients over a span of multiple years suggest the kinetic model to be more reliable in the long run, i.e., its predictions can be considered as acceptable even on periods that are quite far away from the two case periods over which the many parameters of the model had been optimized. The approach under study, referring to paradigms and methods of physical and mathematical models integrated with psychosocial sciences, has good chances of gaining the attention of the scientific community in both areas, and hence of eventually obtaining wider diffusion and generalization.

Unifying hybrid types and contracts

Contract systems and hybrid type systems provide two alternative approaches for enforcing precisely-defined interface specifications, with complementary advantages: contract systems excel at blame assignment, whereas hybrid type systems support type-based static analysis. We unify these two approaches by demonstrating that hybrid type checking is sufficiently expressive to encode higher-order contracts with proper blame assignment. In particular, a contract obligation that enforces both sides of a contract is decomposed into two type casts that each enforce one side of the contract. This expressiveness result provides several benefits, including allowing one of these casts to be lifted to earlier in the program’s execution, resulting in improved contract coverage.

An artificial neural network approach for modeling the ward atmosphere in a medical unit

Artificial neural networks (ANNs) have been developed, implemented and tested on the basis of a four-year-long experimental data set, with the aim of analyzing the performance and clinical outcome of an existing medical ward, and predicting the effects that possible readjustments and/or interventions on the structure may produce on it. Advantages of the ANN technique over more traditional mathematical models are twofold: on one hand, this approach deals quite naturally with a large number of parameters/variables, and also allows to identify those variables which do not play a crucial role in the system dynamics; on the other hand, the implemented ANN can be more easily used by a staff of non-mathematicians in the unit, as an on-site predictive tool. As such, the ANN model is particularly suitable for the case study. The predictions from the ANN technique are then compared and contrasted with those obtained from a generalized kinetic approach previously proposed and tested by the authors. The comparison on the two case periods shows the ANN predictions to be somewhat closer to the experimental values. However, the mean deviations and the analysis of the statistical coefficients over a span of multiple years suggest the kinetic model to be more reliable in the long run, i.e., its predictions can be considered as acceptable even on periods that are quite far away from the two case periods over which the many parameters of the model had been optimized. The approach under study, referring to paradigms and methods of physical and mathematical models integrated with psychosocial sciences, has good chances of gaining the attention of the scientific community in both areas, and hence of eventually obtaining wider diffusion and generalization