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

Über die Maximal Mediated Set Struktur und die Anwendungen Nichtnegativer Circuit Polynome

Certifying the nonnegativity of a polynomial is a significant task both for mathematical and for scientific applications. In general, showing the nonnegativity of a random polynomial is hard. However, for certain classes of polynomials one can find easier conditions that imply their nonnegativity. In this work we investigate both the theoretic and the applied aspects of a special class of polynomials called circuit polynomials. On the theoretical side, we study the relationship of this class of polynomials with another very well studied class called sums of squares using the notion of the maximal mediated set (MMS). We show that MMS is a property of an equivalence class, rather than a property of a single circuit polynomial. With this in mind, we generate a large database of MMS using the software Polymake, and present some statistical and computational observations. On the applied side, we address to the problem of multistationarity in the chemical reaction networks theory by employing a symbolic nonnegativity certification technique via circuit polynomials. The existence of multiple stationary states for a given reaction network with a given starting point is important, as this is closely related to cellular communication in the context of biochemical reaction networks. The existence of multistationarity can be decided by studying the signs of a relevant polynomial whose coefficients are parameterized by the reaction rates. As a case study, we consider the (de)phosphorylation cycle, and use the theory of nonnegative circuit polynomials in order to find a symbolic nonnegativity certificates for the aforementioned polynomial. We provide a method that describes a non-empty open region in the parameter space that enables multistationarity for the (de)phosphorylation cycle. Moreover, we provide an explicit description of such an open region for 2 and 3-site cases.Der Nachweis der Nichtnegativität eines Polynoms ist eine wichtige Aufgabe sowohl für mathematische als auch für wissenschaftliche Anwendungen. Im Allgemeinen ist es schwierig, die Nichtnegativität eines Zufallspolynoms zu zeigen. Für bestimmte Klassen von Polynomen kann man jedoch einfachere Bedingungen finden, die ihre Nichtnegativität implizieren. In dieser Arbeit untersuchen wir sowohl die theoretischen als auch die angewandten Aspekte einer speziellen Klasse von Polynomen, die als circuit Polynome bezeichnet werden. Auf der theoretischen Seite untersuchen wir die Beziehung dieser Klasse von Polynomen mit einer anderen sehr gut untersuchten Klasse namens sums of squares unter Verwendung des Begriffs der maximal mediated set (MMS). Wir zeigen, dass MMS eher eine Eigenschaft einer Äquivalenzklasse als eine Eigenschaft eines circuit polynom ist. Vor diesem Hintergrund erstellen wir mit der Polymake-Software eine große MMS-Datenbank und präsentieren einige statistische und rechnerische Beobachtungen. Auf der angewandten Seite adressieren wir das Problem der Multistationarität in der Theorie chemischer Reaktionsnetzwerke durch die Anwendung einer symbolischen Nichtnegativitäts-Zertifizierungstechnik über circuit Polynome. Die Existenz mehrerer stationärer Zustände für ein gegebenes Reaktionsnetzwerk mit einem gegebenen Startpunkt ist wichtig, da dies eng mit der zellulären Kommunikation im Kontext biochemischer Reaktionsnetzwerke zusammenhängt. Die Existenz von Multistationarität kann durch Studium der Vorzeichen eines relevanten Polynoms entschieden werden, dessen Koeffizienten durch die Reaktionsgeschwindigkeiten parametrisiert werden. Betrachten Sie als Fallbeispiel den (De)Phosphorylierungszyklus und verwenden Sie die Theorie der circuit Polynome, um ein symbolisches Nichtnegativitätszertifikat für das obige Polynom zu finden. Darüber hinaus bieten wir eine explizite Beschreibung einer solchen offenen Region für 2- und 3-Site-Fälle

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

Bridging the gap : an optimization-based framework for fast, simultaneous circuit & system design space exploration

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 107-110).Design of modern mixed signal integrated circuits is becoming increasingly difficult. Continued MOSFET scaling is approaching the global power dissipation limits while increasing transistor variability, thus requiring careful allocation of power and area resources to achieve increasingly more aggressive performance specifications. In this tightly constrained environment traditional iterative system-to-circuit redesign loop, is becoming inefficient. With complex system architectures and circuit specifications approaching technological limits of the process employed, the designers have less room to margin for the overhead of strict system and circuit design interdependencies. Severely constrained modern mixed IC design can take many iterations to converge in such a design flow. This is an expensive and time consuming process. The situation is particularly acute in high-speed links. As an important building block of many systems (high speed I/O, on-chip communication, ...) power efficiency and area footprint are of utmost importance. Design of these systems is challenging in both system and circuit domain. On one hand system architectures are becoming increasingly complex to provide necessary performance increase. On the other, circuit implementation of these increasingly complicated systems is difficult to achieve under tight power and area budget. To bridge this gap between system and circuit design, we formulate a circuit-to-system optimization-driven framework. It is an equation-based description, powered by a human designer. Provided with equation-based model we use fast optimization tools to quickly scout the available design space. Presence of a designer in the flow is invaluable resource enabling significant saving by simplifying the models to capture only the relevant information and constraining the search space to areas where meaningful solutions might be expected to be found.(cont) Thus, the computational effort overhead that plagues the simulation-based design space exploration and design optimization is greatly reduced. The flow is powered by a signomial optimization engine. The key challenge is to bring, from the modeling point of view, very different problems such as circuit design and system design into the realm of an optimization engine that can solve them jointly, thus breaking the re-design loop or at least cutting it shorter. Relying on signomial programming is necessary in order to accurately model all the necessary phenomenons that arise in electrical circuits and at system level. For example, defining regions of operation of transistors under polarization conditions can not be modeled accurately with simpler type of equations. Similarly, calculating the effect of filtering to a signal also requires possibility to handle signomial equations. Thus, signomial programming is necessary yet not fully explored and finding suitable formulation might take some experimenting as we will see in this thesis. Signomial programming, as a general non-convex optimization problem, is still an active research area. Most of the solutions proposed so far involve local convexification of the problem in addition to branch & bound type of search. Furthermore, most of the non-convex problems are solved for one particular system of equations, and general methodology that is reliable and efficient is not known. Thus, a big part the work to be presented in this thesis is detailing how to construct a system formulation that the optimization engine can solve efficiently and reliably. We tested different formulations and their performance measured in terms of parsing and solving speed and accuracy. From these tests we motivate and explain how a series of transformations we introduce improve our formulation and arrive to a well-behaved and reliable form. We show how to apply our design flow in high-speed link design.(cont) By restructuring the traditional design flow we derive system and circuit abstractions. These sub-problems are interfaced through a set of well defined interface variables, which enables code level separation of problem descriptions, thus building a modular and easy to read and maintain system and circuit model. Finally we develop a set of scripts to automate formulating parametrized system level description. We explain how our transformations influence the speed of this process as well as the size of the model produced.by Ranko Sredojević.S.M

Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability

Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies

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