Many Physical Design Problems are Sparse QCQPs

Physical design refers to mathematical optimization of a desired objective (e.g. strong light--matter interactions, or complete quantum state transfer) subject to the governing dynamical equations, such as Maxwell's or Schrodinger's differential equations. Computing an optimal design is challenging: generically, these problems are highly nonconvex and finding global optima is NP hard. Here we show that for linear-differential-equation dynamics (as in linear electromagnetism, elasticity, quantum mechanics, etc.), the physical-design optimization problem can be transformed to a sparse-matrix, quadratically constrained quadratic program (QCQP). Sparse QCQPs can be tackled with convex optimization techniques (such as semidefinite programming) that have thrived for identifying global bounds and high-performance designs in other areas of science and engineering, but seemed inapplicable to the design problems of wave physics. We apply our formulation to prototypical photonic design problems, showing the possibility to compute fundamental limits for large-area metasurfaces, as well as the identification of designs approaching global optimality. Looking forward, our approach highlights the promise of developing bespoke algorithms tailored to specific physical design problems.Comment: 9 pages, 4 figures, plus references and Supplementary Material

In-situ Data Analytics In Cyber-Physical Systems

Cyber-Physical System (CPS) is an engineered system in which sensing, networking, and computing are tightly coupled with the control of the physical entities. To enable security, scalability and resiliency, new data analytics methodologies are required for computing, monitoring and optimization in CPS. This work investigates the data analytics related challenges in CPS through two study cases: Smart Grid and Seismic Imaging System. For smart grid, this work provides a complete solution for system management based on novel in-situ data analytics designs. We first propose methodologies for two important tasks of power system monitoring: grid topology change and power-line outage detection. To address the issue of low measurement redundancy in topology identification, particularly in the low-level distribution network, we develop a maximum a posterior based mechanism, which is capable of embedding prior information on the breakers status to enhance the identification accuracy. In power-line outage detection, existing approaches suer from high computational complexity and security issues raised from centralized implementation. Instead, this work presents a distributed data analytics framework, which carries out in-network processing and invokes low computational complexity, requiring only simple matrix-vector multiplications. To complete the system functionality, we also propose a new power grid restoration strategy involving data analytics for topology reconfiguration and resource planning after faults or changes. In seismic imaging system, we develop several innovative in-situ seismic imaging schemes in which each sensor node computes the tomography based on its partial information and through gossip with local neighbors. The seismic data are generated in a distributed fashion originally. Dierent from the conventional approach involving data collection and then processing in order, our proposed in-situ data computing methodology is much more ecient. The underlying mechanisms avoid the bottleneck problem on bandwidth since all the data are processed distributed in nature and only limited decisional information is communicated. Furthermore, the proposed algorithms can deliver quicker insights than the state-of-arts in seismic imaging. Hence they are more promising solutions for real-time in-situ data analytics, which is highly demanded in disaster monitoring related applications. Through extensive experiments, we demonstrate that the proposed data computing methods are able to achieve near-optimal high quality seismic tomography, retain low communication cost, and provide real-time seismic data analytics

Quantum computing for finance

Quantum computers are expected to surpass the computational capabilities of classical computers and have a transformative impact on numerous industry sectors. We present a comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on stochastic modeling, optimization, and machine learning. This Review is aimed at physicists, so it outlines the classical techniques used by the financial industry and discusses the potential advantages and limitations of quantum techniques. Finally, we look at the challenges that physicists could help tackle

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

A Perspective on Future Research Directions in Information Theory

Information theory is rapidly approaching its 70th birthday. What are promising future directions for research in information theory? Where will information theory be having the most impact in 10-20 years? What new and emerging areas are ripe for the most impact, of the sort that information theory has had on the telecommunications industry over the last 60 years? How should the IEEE Information Theory Society promote high-risk new research directions and broaden the reach of information theory, while continuing to be true to its ideals and insisting on the intellectual rigor that makes its breakthroughs so powerful? These are some of the questions that an ad hoc committee (composed of the present authors) explored over the past two years. We have discussed and debated these questions, and solicited detailed inputs from experts in fields including genomics, biology, economics, and neuroscience. This report is the result of these discussions

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

Optimization Methods in Electric Power Systems: Global Solutions for Optimal Power Flow and Algorithms for Resilient Design under Geomagnetic Disturbances

An electric power system is a network of various components that generates and delivers power to end users. Since 1881, U.S. electric utilities have supplied power to billions of industrial, commercial, public, and residential customers continuously. Given the rapid growth of power utilities, power system optimization has evolved with developments in computing and optimization theory. In this dissertation, we focus on two optimization problems associated with power system planning: the AC optimal power flow (ACOPF) problem and the optimal transmission line switching (OTS) problem under geomagnetic disturbances (GMDs). The former problem is formulated as a nonlinear, non-convex network optimization problem, while the latter is the network design version of the ACOPF problem that allows topology reconfiguration and considers space weather-induced effects on power systems. Overall, the goal of this research includes: (1) developing computationally efficient approaches for the ACOPF problem in order to improve power dispatch efficiency and (2) identifying an optimal topology configuration to help ISO operate power systems reliably and efficiently under geomagnetic disturbances. Chapter 1 introduces the problems we are studying and motivates the proposed research. We present the ACOPF problem and the state-of-the-art solution methods developed in recent years. Next, we introduce geomagnetic disturbances and describe how they can impact electrical power systems. In Chapter 2, we revisit the polar power-voltage formulation of the ACOPF problem and focus on convex relaxation methods to develop lower bounds on the problem objective. Based on these approaches, we propose an adaptive, multivariate partitioning algorithm with bound tightening and heuristic branching strategies that progressively improves these relaxations and, given sufficient time, converges to the globally optimal solution. Computational results show that our methodology provides a computationally tractable approach to obtain tight relaxation bounds for hard ACOPF cases from the literature. In Chapter 3, we focus on the impact that extreme GMD events could potentially have on the ability of a power system to deliver power reliably. We develop a mixed-integer, nonlinear model which captures and mitigates GMD effects through line switching, generator dispatch, and load shedding. In addition, we present a heuristic algorithm that provides high-quality solutions quickly. Our work demonstrates that line switching is an effective way to mitigate GIC impacts. In Chapter 4, we extend the preliminary study presented in Chapter 3 and further consider the uncertain nature of GMD events. We propose a two-stage distributionally robust (DR) optimization model that captures geo-electric fields induced by uncertain GMDs. Additionally, we present a reformulation of a two-stage DRO that creates a decomposition framework for solving our problem. Computational results show that our DRO approach provides solutions that are robust to errors in GMD event predictions. Finally, in Chapter 5, we summarize the research contributions of our work and provide directions for future research

Applied Harmonic Analysis and Data Processing

Massive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations

DESIGN AND OPTIMIZATION OF SIMULTANEOUS WIRELESS INFORMATION AND POWER TRANSFER SYSTEMS

The recent trends in the domain of wireless communications indicate severe upcoming challenges, both in terms of infrastructure as well as design of novel techniques. On the other hand, the world population keeps witnessing or hearing about new generations of mobile/wireless technologies within every half to one decade. It is certain the wireless communication systems have enabled the exchange of information without any physical cable(s), however, the dependence of the mobile devices on the power cables still persist. Each passing year unveils several critical challenges related to the increasing capacity and performance needs, power optimization at complex hardware circuitries, mobility of the users, and demand for even better energy efficiency algorithms at the wireless devices. Moreover, an additional issue is raised in the form of continuous battery drainage at these limited-power devices for sufficing their assertive demands. In this regard, optimal performance at any device is heavily constrained by either wired, or an inductive based wireless recharging of the equipment on a continuous basis. This process is very inconvenient and such a problem is foreseen to persist in future, irrespective of the wireless communication method used. Recently, a promising idea for simultaneous wireless radio-frequency (RF) transmission of information and energy came into spotlight during the last decade. This technique does not only guarantee a more flexible recharging alternative, but also ensures its co-existence with any of the existing (RF-based) or alternatively proposed methods of wireless communications, such as visible light communications (VLC) (e.g., Light Fidelity (Li-Fi)), optical communications (e.g., LASER-equipped communication systems), and far-envisioned quantum-based communication systems. In addition, this scheme is expected to cater to the needs of many current and future technologies like wearable devices, sensors used in hazardous areas, 5G and beyond, etc. This Thesis presents a detailed investigation of several interesting scenarios in this direction, specifically concerning design and optimization of such RF-based power transfer systems. The first chapter of this Thesis provides a detailed overview of the considered topic, which serves as the foundation step. The details include the highlights about its main contributions, discussion about the adopted mathematical (optimization) tools, and further refined minutiae about its organization. Following this, a detailed survey on the wireless power transmission (WPT) techniques is provided, which includes the discussion about historical developments of WPT comprising its present forms, consideration of WPT with wireless communications, and its compatibility with the existing techniques. Moreover, a review on various types of RF energy harvesting (EH) modules is incorporated, along with a brief and general overview on the system modeling, the modeling assumptions, and recent industrial considerations. Furthermore, this Thesis work has been divided into three main research topics, as follows. Firstly, the notion of simultaneous wireless information and power transmission (SWIPT) is investigated in conjunction with the cooperative systems framework consisting of single source, multiple relays and multiple users. In this context, several interesting aspects like relay selection, multi-carrier, and resource allocation are considered, along with problem formulations dealing with either maximization of throughput, maximization of harvested energy, or both. Secondly, this Thesis builds up on the idea of transmit precoder design for wireless multigroup multicasting systems in conjunction with SWIPT. Herein, the advantages of adopting separate multicasting and energy precoder designs are illustrated, where we investigate the benefits of multiple antenna transmitters by exploiting the similarities between broadcasting information and wirelessly transferring power. The proposed design does not only facilitates the SWIPT mechanism, but may also serve as a potential candidate to complement the separate waveform designing mechanism with exclusive RF signals meant for information and power transmissions, respectively. Lastly, a novel mechanism is developed to establish a relationship between the SWIPT and cache-enabled cooperative systems. In this direction, benefits of adopting the SWIPT-caching framework are illustrated, with special emphasis on an enhanced rate-energy (R-E) trade-off in contrast to the traditional SWIPT systems. The common notion in the context of SWIPT revolves around the transmission of information, and storage of power. In this vein, the proposed work investigates the system wherein both information and power can be transmitted and stored. The Thesis finally concludes with insights on the future directions and open research challenges associated with the considered framework

Applications of Convex Analysis to Signomial and Polynomial Nonnegativity Problems

Here is a question that is easy to state, but often hard to answer: Is this function nonnegative on this set? When faced with such a question, one often makes appeals to known inequalities. One crafts arguments that are sufficient to establish the nonnegativity of the function, rather than determining the function's precise range of values. This thesis studies sufficient conditions for nonnegativity of signomials and polynomials. Conceptually, signomials may be viewed as generalized polynomials that feature arbitrary real exponents, but with variables restricted to the positive orthant. Our methods leverage efficient algorithms for a type of convex optimization known as relative entropy programming (REP). By virtue of this integration with REP, our methods can help answer questions like the following: Is there some function, in this particular space of functions, that is nonnegative on this set? The ability to answer such questions is extremely useful in applied mathematics. Alternative approaches in this same vein (e.g., methods for polynomials based on semidefinite programming) have been used successfully as convex relaxation frameworks for nonconvex optimization, as mechanisms for analyzing dynamical systems, and even as tools for solving nonlinear partial differential equations. This thesis builds from the sums of arithmetic-geometric exponentials or SAGE approach to signomial nonnegativity. The term "exponential" appears in the SAGE acronym because SAGE parameterizes signomials in terms of exponential functions. Our first round of contributions concern the original SAGE approach. We employ basic techniques in convex analysis and convex geometry to derive structural results for spaces of SAGE signomials and exactness results for SAGE-based REP relaxations of nonconvex signomial optimization problems. We frame our analysis primarily in terms of the coefficients of a signomial's basis expansion rather than in terms of signomials themselves. The effect of this framing is that our results for signomials readily transfer to polynomials. In particular, we are led to define a new concept of SAGE polynomials. For sparse polynomials, this method offers an exponential efficiency improvement relative to certificates of nonnegativity obtained through semidefinite programming. We go on to create the conditional SAGE methodology for exploiting convex substructure in constrained signomial nonnegativity problems. The basic insight here is that since the standard relative entropy representation of SAGE signomials is obtained by a suitable application of convex duality, we are free to add additional convex constraints into the duality argument. In the course of explaining this idea we provide some illustrative examples in signomial optimization and analysis of chemical dynamics. The majority of this thesis is dedicated to exploring fundamental questions surrounding conditional SAGE signomials. We approach these questions through analysis frameworks of sublinear circuits and signomial rings. These sublinear circuits generalize simplicial circuits of affine-linear matroids, and lead to rich modes of analysis for sets that are simultaneously convex in the usual sense and convex under a logarithmic transformation. The concept of signomial rings lets us develop a powerful signomial Positivstellensatz and an elementary signomial moment theory. The Positivstellensatz provides for an effective hierarchy of REP relaxations for approaching the value of a nonconvex signomial minimization problem from below, as well as a first-of-its-kind hierarchy for approaching the same value from above. In parallel with our mathematical work, we have developed the sageopt python package. Sageopt drives all the examples and experiments used throughout this thesis, and has been used by engineers to solve high-degree polynomial optimization problems at scales unattainable by alternative methods. We conclude this thesis with an explanation of how our theoretical results affected sageopt's design.</p