A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York

Bilevel Disjunctive Optimization on Affine Manifolds

Bilevel optimization is a special kind of optimization where one problem is embedded within another. The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lower-level optimization task. These problems involve two kinds of variables: upper-level variables and lower-level variables. Bilevel optimization was first realized in the field of game theory by a German economist von Stackelberg who published a book (1934) that described this hierarchical problem. Now the bilevel optimization problems are commonly found in a number of real-world problems: transportation, economics, decision science, business, engineering, and so on. In this chapter, we provide a general formulation for bilevel disjunctive optimization problem on affine manifolds. These problems contain two levels of optimization tasks where one optimization task is nested within the other. The outer optimization problem is commonly referred to as the leaders (upper level) optimization problem and the inner optimization problem is known as the followers (or lower level) optimization problem. The two levels have their own objectives and constraints. Topics affine convex functions, optimizations with auto-parallel restrictions, affine convexity of posynomial functions, bilevel disjunctive problem and algorithm, models of bilevel disjunctive programming problems, and properties of minimum functions

Optimization-based design of fault-tolerant avionics

This dissertation considers the problem of improving the self-consciousness for avionic systems using numerical optimization techniques, emphasizing UAV applications. This self-consciousness implies a sense of awareness for oneself to make a reliable decision on some crucial aspects. In the context of the avionics or aerospace industry, those aspects are SWaP-C as well as safety and reliability. The decision-making processes to optimize these aspects, which are the main contributions of this work, are presented. In addition, implementation on various types of applications related to avionics and UAV are also provided. The first half of this thesis lays out the background of avionics development ranging from a mechanical gyroscope to a current state-of-the-art electronics system. The relevant mathematics regarding convex optimization and its algorithms, which will be used for formulating this self-consciousness problem, are also provided. The latter half presents two problem formulations for redundancy design automation and reconfigurable middleware. The first formulation focuses on the minimization of SWaP-C while satisfying safety and reliability requirements. The other one aims to maximize the system safety and reliability by introducing a fault-tolerant capability via the task scheduler of middleware or RTOS. The usage of these two formulations is shown by four aerospace applications---reconfigurable multicore avionics, a SITL simulation of a UAV GNC system, a modular drone, and a HITL simulation of a fault-tolerant distributed engine control architecture.Ph.D

New Dependencies of Hierarchies in Polynomial Optimization

We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams (SA) hierarchies. We prove a collection of dependencies among these hierarchies both for general CPOPs and for optimization problems on the Boolean hypercube. Key results include for the general case that the SONC and SOS hierarchy are polynomially incomparable, while SDSOS is contained in SONC. A direct consequence is the non-existence of a Putinar-like Positivstellensatz for SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent. Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that provides a O(n) degree bound.Comment: 26 pages, 4 figure

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Power Allocation in Uplink NOMA-Aided Massive MIMO Systems

In the development of the fifth-generation (5G) as well as the vision for the future generations of wireless communications networks, massive multiple-input multiple-output (MIMO) technology has played an increasingly important role as a key enabler to meet the growing demand for very high data throughput. By equipping base stations (BSs) with hundreds to thousands antennas, the massive MIMO technology is capable of simultaneously serving multiple users in the same time-frequency resources with simple linear signal processing in both the downlink (DL) and uplink (UL) transmissions. Thanks to the asymptotically orthogonal property of users' wireless channels, the simple linear signal processing can effectively mitigate inter-user interference and noise while boosting the desired signal's gain, and hence achieves high data throughput. In order to realize this orthogonal property in a practical system, one critical requirement in the massive MIMO technology is to have the instantaneous channel state information (CSI), which is acquired via channel estimation with pilot signaling. Unfortunately, the connection capability of a conventional massive MIMO system is strictly limited by the time resource spent for channel estimation. Attempting to serve more users beyond the limit may result in a phenomenon known as pilot contamination, which causes correlated interference, lowers signal gain and hence, severely degrades the system's performance. A natural question is ``Is it at all possible to serve more users beyond the limit of a conventional massive MIMO system?''. The main contribution of this thesis is to provide a promising solution by integrating the concept of nonorthogonal multiple access (NOMA) into a massive MIMO system. The key concept of NOMA is based on assigning each unit of orthogonal radio resources, such as frequency carriers, time slots or spreading codes, to more than one user and utilize a non-linear signal processing technique like successive interference cancellation (SIC) or dirty paper coding (DPC) to mitigate inter-user interference. In a massive MIMO system, pilot sequences are also orthogonal resources, which can be allocated with the NOMA approach. By sharing a pilot sequence to more than one user and utilizing the SIC technique, a massive MIMO system can serve more users with a fixed amount of time spent for channel estimation. However, as a consequence of pilot reuse, correlated interference becomes the main challenge that limits the spectral efficiency (SE) of a massive MIMO-NOMA system. To address this issue, this thesis focuses on how to mitigate correlated interference when combining NOMA into a massive MIMO system in order to accommodate a higher number of wireless users. In the first part, we consider the problem of SIC in a single-cell massive MIMO system in order to serve twice the number of users with the aid of time-offset pilots. With the proposed time-offset pilots, users are divided into two groups and the uplink pilots from one group are transmitted simultaneously with the uplink data of the other group, which allows the system to accommodate more users for a given number of pilots. Successive interference cancellation is developed to ease the effect of pilot contamination and enhance data detection. In the second part, the work is extended to a cell-free network, where there is no cell boundary and a user can be served by multiple base stations. The chapter focuses on the NOMA approach for sharing pilot sequences among users. Unlike the conventional cell-free massive MIMO-NOMA systems in which the UL signals from different access points are equally combined over the backhaul network, we first develop an optimal backhaul combining (OBC) method to maximize the UL signal-to-interference-plus-noise ratio (SINR). It is shown that, by using OBC, the correlated interference can be effectively mitigated if the number of users assigned to each pilot sequence is less than or equal to the number of base stations. As a result, the cell-free massive MIMO-NOMA system with OBC can enjoy unlimited performance when the number of antennas at each BS tends to infinity. Finally, we investigate the impact of imperfect SIC to a NOMA cell-free massive MIMO system. Unlike the majority of existing research works on performance evaluation of NOMA, which assume perfect channel state information and perfect data detection for SIC, we take into account the effect of practical (hence imperfect) SIC. We show that the received signal at the backhaul network of a cell-free massive MIMO-NOMA system can be effectively treated as a signal received over an additive white Gaussian noised (AWGN) channel. As a result, a discrete joint distribution between the interfering signal and its detected version can be analytically found, from which an adaptive SIC scheme is proposed to improve performance of interference cancellation

Determination of Cost-Effective Range in Surface Finish for Single Pass Turning

Surface finish is considered a critical characteristic for manufacturing components when manufacturers strive to produce components with high-quality characteristics predefined by design engineers. The objective of this research is to provide a cost-effective range in surface finish for single pass turning that enables the design engineers to explore a wider spectrum of alternative solutions without significantly affecting the functionality of the part. Apart from the one optimal solution, the proposed methodology, which is based on Geometric Programming, would provide a range of cutting conditions solutions that satisfy the economic and functional needs for the designer. This can be achieved by switching cost reduction focus from tooling to labor cost, particularly by adjusting variables values such as spindle speed and feed. An algorithm has been developed to find the new variables values. In addition, a sensitivity analysis model, based on metaheuristic techniques, will also be developed to further give a set of possible solutions that are practically preferable to the practitioners. In addition, the developed methodology can be applied to other engineering applications. The proposed methodology will provide a tool that enhances the design for manufacturability for companies to become more competitive

Polynomial optimization: matrix factorization ranks, portfolio selection, and queueing theory

Inspired by Leonhard Euler’s belief that every event in the world can be understood in terms of maximizing or minimizing a specific quantity, this thesis delves into the realm of mathematical optimization. The thesis is divided into four parts, with optimization acting as the unifying thread. Part 1 introduces a particular class of optimization problems called generalized moment problems (GMPs) and explores the moment method, a powerful tool used to solve GMPs. We introduce the new concept of ideal sparsity, a technique that aids in solving GMPs by improving the bounds of their associated hierarchy of semidefinite programs. Part 2 focuses on matrix factorization ranks, in particular, the nonnegative rank, the completely positive rank, and the separable rank. These ranks are extensively studied using the moment method, and ideal sparsity is applied (whenever possible) to enhance the bounds on these ranks and speed-up their computation. Part 3 centers around portfolio optimization and the mean-variance-skewness kurtosis (MVSK) problem. Multi-objective optimization techniques are employed to uncover Pareto optimal solutions to the MVSK problem. We show that most linear scalarizations of the MVSK problem result in specific convex polynomial optimization problems which can be solved efficiently. Part 4 explores hypergraph-based polynomials emerging from queueing theory in the setting of parallel-server systems with job redundancy policies. By exploiting the symmetry inherent in the polynomials and some classical results on matrix algebras, the convexity of these polynomials is demonstrated, thereby allowing us to prove that the polynomials attain their optima at the barycenter of the simplex.<br/