Uma nova relaxação quadrática para variáveis binárias com aplicações a confiabilidade de redes de energia elétrica, a segmentação de imagens médicas de nervos e a problemas de geometria de distâncias

Orientador: Christiano Lyra FilhoTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Como o título sugere, o foco desta pesquisa é o desenvolvimento de uma nova relaxação quadrática para problemas binários, sua formalização em resultados teóricos, e a aplicação dos novos conceitos em aplicações à confiabilidade de redes de energia elétrica, à segmentação de imagens médicas de nervos e à problemas de geometria de distâncias. Modelos matemáticos contendo va-riáveis de decisões binárias podem ser usados para encontrar as melhores soluções em processos de tomada de decisões, normalmente caracterizando problemas de otimização combinatória difíceis. A solução desses problemas em aplicações de interesse prático requer um grande esforço computacional; por isso, ao longo dos últimos anos, têm sido objeto de pesquisas na área de metaheurísticas. As ideias aqui desenvolvidas abrem novas perspectivas para a abordagem desses problemas apoiando-se em métodos de otimização não-lineares, área que vem sendo povoada por "solvers" muito eficientes. Inicialmente, explorando aspectos formais, a relaxação desenvolvida é parti-cularizada para um problema de otimização quadrática binária irrestrita. O relaxamento permite o desenvolvimento de três estruturas para abordar esta classe de problemas, e explora a convexidade da função objetivo para obter melhorias computacionais. Estudos de casos compararam o relaxamento proposto com os relaxamentos similares apresentados na literatura. Foram desenvolvidas três aplicações para os desenvolvimentos teóricos da pesquisa. A primeira aplicação envolve a melhoria da confiabilidade de redes de energia elétrica. Especificamente, aborda o problema de definir a melhor alternativa para a alocação de sensores na rede, o que permite reduzir os efeitos de ocorrências indesejáveis e ampliar a resiliência das redes. A segunda aplicação envolve o problema de segmentação de imagens médicas associadas a estruturas de nervos. A abordagem proposta interpreta o problema de segmentação como um problema de otimização binária, onde medir cada axônio significa encontrar um ciclo Hamiltoniano, um caso do problema do caixeiro viajante; a solução desses problemas fornece a estatística descritiva para um conjunto de axônios, incluindo o número (de axônios), os diâmetros e as áreas ocupadas. A última aplicação elabora um modelo matemático para o problema de geometria de distâncias sem designação, área ainda pouco estudada e com muitos aspectos em aberto. A relaxação desenvolvida na pesquisa permitiu resolver instâncias com mais de vinte mil variáveis binárias. Esses resultados são bons indicadores dos benefícios alcançáveis com os aspectos teóricos da pesquisa, e abrem novas perspectivas para as aplicações, que incluem inovações em nanotecnologia e bioengenhariaAbstract: As the title suggests, the focus this research is the development of a new quadratic relaxation for binary problems, its formalization in theoretical results, and the application of the new concepts in applications to the reliability of electric power networks, segmentation of nerve root images, and distance geometry problems. Mathematical models with binary decision variables can be used to find the best solutions for decision-making process, usually leading to difficult combinatorial optimization problems. The solution to these problems in practical applications requires a high computational effort; therefore, over the past years it has been the subject of research in the area of metaheuristics. The ideas developed in this thesis open new perspectives for addressing these problems using nonlinear optimization approaches, an area that has been populated by very efficient solvers. The initial developments explore the formal aspects of the relaxation in the context of a quadratic unconstrained binary optimization problem. The use of the proposed relaxation allows to create three structures to deal with this class of problems, and explores the objective function convexity to improve the computational performance. Case studies compare the proposed relaxation with the previous relaxations proposed in the literature. Three new applications were developed to explore the theoretical developments of this research. The first application concerns the improvement of the reliability of electric power distribution networks. Specifically, it deals with the problem of defining the best allocation for remote fault sensor, allowing to reduce the consequence of the faults and to improve the resilience of the networks. The second application explores the segmentation of medical images related to nerve root structures. The proposed approach regards the segmentation problem as a binary optimization problem, where measuring each axon is equivalent to finding a Hamiltonian cycle for a variant of the traveling salesman problem; the solution to these problems provides the descriptive statistics of the axon set, including the number of axons, their diameters, and the area used by each axon. The last application designs a mathematical model for the unassigned distance geometry problem, an incipient research area with many open problems. The relaxation developed in this research allowed to solve instances with more than twenty thousand binary variables. These results can be seen as good indicators of the benefits attainable with the theoretical aspects of the research, and opens new perspectives for applications, which include innovations in nanotechnology and bio-engineeringDoutoradoAutomaçãoDoutora em Engenharia Elétrica148400/2016-7CNP

Sparse Recovery of Strong Reflectors With an Application to Non-Destructive Evaluation

In this paper we show that it is sufficient to recover the locations of K strong reflectors within an insonified medium from three receive elements and 2K+1 samples per element. The proposed approach leverages advances in sampling signals with a finite rate of innovation along each element and rank properties from the Euclidean distance matrix construction across elements. With the proposed approach, it is not necessary to construct an image in order to identify strong reflective sources, which is why much fewer receive elements are needed. However, the assumed transmit scheme still uses a standard linear array in order to excite the entire medium with sufficient energy. The approach is validated with simulated data and a measurement that emulates a scenario in non-destructive evaluation

3D unknown view tomography via rotation invariants

In this paper, we study the problem of reconstructing a 3D point source model from a set of 2D projections at unknown view angles. Our method obviates the need to recover the projection angles by extracting a set of rotation-invariant features from the noisy projection data. From the features, we reconstruct the density map through a constrained nonconvex optimization. We show that the features have geometric interpretations in the form of radial and pairwise distances of the model. We further perform an ablation study to examine the effect of various parameters on the quality of the estimated features from the projection data. Our results showcase the potential of the proposed method in reconstructing point source models in various noise regimes

On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

We consider the problem of recovering a complex vector $\mathbf{x}\in \mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes {identifiable}, and further prove isometry properties in the case when the matrices $\{A_i\}_{i=1}^m$ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex {optimization} formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.Comment: 21 page

Solving Complex Quadratic Systems with Full-Rank Random Matrices

We tackle the problem of recovering a complex signal $\boldsymbol x\in\mathbb{C}^n$ from quadratic measurements of the form $y_i=\boldsymbol x^*\boldsymbol A_i\boldsymbol x$, where $\boldsymbol A_i$ is a full-rank, complex random measurement matrix whose entries are generated from a rotation-invariant sub-Gaussian distribution. We formulate it as the minimization of a nonconvex loss. This problem is related to the well understood phase retrieval problem where the measurement matrix is a rank-1 positive semidefinite matrix. Here we study the general full-rank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior works either address the rank-1 case or focus on real measurements. The several papers that address the full-rank complex case adopt the computationally-demanding semidefinite relaxation approach. In this paper we prove that the general class of problems with rotation-invariant sub-Gaussian measurement models can be efficiently solved with high probability via the standard framework comprising a spectral initialization followed by iterative Wirtinger flow updates on a nonconvex loss. Numerical experiments on simulated data corroborate our theoretical analysis.Comment: This updated version of the manuscript addresses several important issues in the initial arXiv submissio