Recursive McCormick Linearization of Multilinear Programs

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce the first systematic approach for identifying RMLs, in which we focus on the identification of linear relaxation with a small number of artificial variables and with strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation for the identification of minimum-size RMLs; we show that this problem is NP-hard in general, whereas a special case is fixed-parameter tractable. Moreover, we explore structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best possible relaxation bound is optimal. Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.Comment: 22 pages, 11 figures, Under Revie

Volumetric Guidance for Handling Triple Products in Spatial Branch-and-Bound

Spatial branch-and-bound (sBB) is the workhorse algorithmic framework used to globally solve mathematical mixed-integer non-linear optimization (MINLO) problems. Formulating a problem using this paradigm allows both the non-linearities of a system and any discrete design choices to be modeled effectively. Because of the generality of this approach, MINLO is used in a wide variety of applications, from chemical engineering problems and network design, to medical applications and problems in the airline industry. Due in part to their generality (and therefore wide applicability), MINLO problems are very difficult in general, and consequently, the best ways to implement many details of sBB are not wholly understood. In this work, we provide analytic results guiding the implementation of sBB for a simple but frequently occurring function ‘building block’. As opposed to computationally demonstrating that our techniques work only for a particular set of test problems, we analytically establish results that hold for all problems of the given form. In this way, we also demonstrate that analytic results are indeed obtainable for certain sBB implementation decisions. In particular, we use volume as a geometric measure to compare different convex relaxations for functions involving trilinear monomials (or any three quantities multiplied together). We consider different choices for convexifying the graph of a triple product, and obtain formulae for the volume (in terms of the variable upper and lower bounds) for each of these convexifications. We are then able to order the convexifications with regard to their volume. We also provide computational evidence to support our choice of volume as an effective comparison measure, and show that in the context of triple products, volume is an excellent predictor of the objective function gap. Finally, we use the volume measure to provide guidance regarding branching-point selection in the implementation of sBB.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/136973/1/eespeakm_1.pd

Convex envelopes of bounded monomials on two-variable cones

We consider an $n$-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Such functions are building blocks of several problems found in practical applications, and that fall under the class of Mixed Integer Nonlinear Optimization. We show that the upper envelope of the function in the given domain, for $n\ge 2$ is given by a conic inequality. We also present the lower envelope for $n=2$. To assess the applicability of branching rules based on homogeneous linear inequalities, we also derive the volume of the convex hull for $n=2$.Comment: 22 pages, 12 figure

Global dynamic optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2004.Includes bibliographical references (p. 247-256).(cont.) on a set composed of the Cartesian product between the parameter bounds and the state bounds. Furthermore, I show that the solution of the differential equations is affine in the parameters. Because the feasible set is convex pointwise in time, the standard result that a convex function composed with an affine function remains convex yields the desired result that the integrand is convex under composition. Additionally, methods are developed using interval arithmetic to derive the exact state bounds for the solution of a linear dynamic system. Given a nonzero tolerance, the method is rigorously shown to converge to the global solution in a finite time. An implementation is developed, and via a collection of case studies, the technique is shown to be very efficient in computing the global solutions. For problems with embedded nonlinear dynamic systems, the analysis requires a more sophisticated composition technique attributed to McCormick. McCormick's composition technique provides a method for computing a convex underestimator for for the integrand given an arbitrary nonlinear dynamic system provided that convex underestimators and concave overestimators can be given for the states. Because the states are known only implicitly via the solution of the nonlinear differential equations, deriving these convex underestimators and concave overestimators is a highly nontrivial task. Based on standard optimization results, outer approximation, the affine solution to linear dynamic systems, and differential inequalities, I present a novel method for constructing convex underestimators and concave overestimators for arbitrary nonlinear dynamic systems ...My thesis focuses on global optimization of nonconvex integral objective functions subject to parameter dependent ordinary differential equations. In particular, efficient, deterministic algorithms are developed for solving problems with both linear and nonlinear dynamics embedded. The techniques utilized for each problem classification are unified by an underlying composition principle transferring the nonconvexity of the embedded dynamics into the integral objective function. This composition, in conjunction with control parameterization, effectively transforms the problem into a finite dimensional optimization problem where the objective function is given implicitly via the solution of a dynamic system. A standard branch-and-bound algorithm is employed to converge to the global solution by systematically eliminating portions of the feasible space by solving an upper bounding problem and convex lower bounding problem at each node. The novel contributions of this work lie in the derivation and solution of these convex lower bounding relaxations. Separate algorithms exist for deriving convex relaxations for problems with linear dynamic systems embedded and problems with nonlinear dynamic systems embedded. However, the two techniques are unified by the method for relaxing the integral in the objective function. I show that integrating a pointwise in time convex relaxation of the original integrand yields a convex underestimator for the integral. Separate composition techniques, however, are required to derive relaxations for the integrand depending upon the nature of the embedded dynamics; each case is addressed separately. For problems with embedded linear dynamic systems, the nonconvex integrand is relaxed pointwise in timeby Adam Benjamin Singer.Ph.D

Effect of Chain Rigidity on Network Architecture and Deformation Behavior of Glassy Polymer Networks

Processing carbon fiber composite laminates creates molecular-level strains in the thermoset matrix upon curing and cooling which can lead to failures such as geometry deformations, micro-cracking, and other issues. It is known strain creation is attributed to the significant volume and physical state changes undergone by the polymer matrix throughout the curing process, though storage and relaxation of cure-induced strains remain poorly understood. This dissertation establishes two approaches to address the issue. The first establishes testing methods to simultaneously measure key volumetric properties of a carbon fiber composite laminate and its polymer matrix. The second approach considers the rigidity of the polymer matrix in regards to strain storage and relaxation mechanisms which ultimately control composite performance throughout manufacturing and use. Through the use of a non-contact, full-field strain measurement technique known as digital image correlation (DIC), we describe and implement useful experiments which quantify matrix and composite parameters necessary for simulation efforts and failure models. The methods are compared to more traditional techniques and show excellent correlation. Further, we established relationships which represent matrix-fiber compatibility in regards to critical processing constraints. The second approach involves a systematic study of epoxy-amine networks which are chemically-similar but differ in chain segment rigidity. Prior research has investigated the isomer effect of glassy polymers, showing sizeable differences in thermal, volumetric, physical, and mechanical properties. This work builds on these themes, and shows the apparent isomer effect is rather an effect of chain rigidity. Indeed, it was found that structurally-dissimilar polymer networks exhibit very similar properties as a consequence of their shared average network rigidity. Differences in chain packing, as a consequence of chain rigidity, were shown to alter the physical, volumetric, and mechanical properties of the glassy networks. Chain rigidity was found to directly control deformation mechanisms, which were related to the yielding behavior of the epoxy network series. The unique benefit to our approach is the ability to separate the role of rigidity – an intramolecular parameter – from intermolecular phenomena which otherwise influence network properties

Development of 1H-NMR Serum Profiling Methods for High-Throughput Metabolomics

El perfilat de sèrum per ressonància magnètica nuclear de protó (1H-RMN) està especialment indicat per a anàlisi a gran escala en estudis epidemiològics, nutricionals o farmacològics. L’espectroscòpia 1H-RMN requereix mínima manipulació de mostra i gràcies a la seva resposta quantitativa permet la comparació directa entre laboratoris. Un perfilat complet de sèrum per 1H-RMN requereix de tres mesures que es corresponen amb tres espècies moleculars diferents: lipoproteïnes, metabòlits de baix pes molecular i lípids. El perfilat de sèrum per 1H-RMN permet obtenir informació de grandària, nombre de partícules i contingut lipídic de les subfraccions lipoproteiques, així com l'abundància d'aminoàcids, productes de la glicòlisi, cossos cetònics, àcids grassos i fosfolípids, entre d'altres. No obstant això, la complexitat espectral afavoreix la inclusió d'errors en l'anàlisi manual de les dades, mentre que les múltiples interaccions moleculars en el sèrum comprometen la seva precisió quantitativa. És per tant necessari desenvolupar mètodes robustos de perfilat metabòlic per consolidar la 1H-RMN en la pràctica clínica. Per a això, aquesta tesi presenta diverses estratègies metodològiques i computacionals. En el primer treball, es van desenvolupar mètodes de regressió dels lípids del perfil lipídic clàssic, generalitzables a mostres de població sana i amb valors de lípids i lipoproteïnes anormals. Aquests lípids representen els principals indicadors de risc cardiovascular i els objectius terapèutics primaris. En el segon estudi caracteritzem els errors de quantificació en el perfilat 1H-RMN de metabòlits clínicament rellevants, que són deguts a la seva agregació a la proteïna sanguínia. També proposem un mètode que fomenta la competició per l'agregació i que ens permet obtenir quantificacions dels nostres metabòlits properes a les absolutes. Finalment, el tercer treball presenta LipSpin: una eina bioinformàtica de codi obert específicament dissenyada per al perfilat de lípids per 1H-RMN. A més, aquest estudi exposa alguns aspectes metodològics per millorar l'anàlisi de lípids per RMN.El perfilado de suero por resonancia magnética nuclear de protón (1H-RMN) está especialmente indicado para el análisis a gran escala en estudios epidemiológicos, nutricionales o farmacológicos. La espectroscopía 1H-RMN requiere mínima manipulación de muestra y gracias a su respuesta cuantitativa permite la comparación directa entre laboratorios. Un perfilado completo de suero por 1H-RMN requiere de tres mediciones que se corresponden con tres especies moleculares distintas: lipoproteínas, metabolitos de bajo peso molecular y lípidos. El perfilado de suero por 1H-RMN permite obtener información de tamaño, número de partículas y contenido lipídico de las subfracciones lipoproteicas, así como la abundancia de aminoácidos, productos de la glicólisis, cuerpos cetónicos, ácidos grasos y fosfolípidos, entre otros. Sin embargo, la complejidad espectral favorece la inclusión de errores en el análisis manual de los datos, mientras que las múltiples interacciones moleculares en el suero comprometen su precisión cuantitativa. Es por tanto necesario desarrollar métodos robustos de perfilado metabólico para consolidar la 1H-RMN en la práctica clínica. Para ello, esta tesis presenta varias estrategias metodológicas y computacionales. En el primer trabajo, se desarrollaron métodos de regresión de los lípidos del perfil lipídico clásico, generalizables a muestras de población sana y con valores de lípidos y lipoproteínas anormales. Estos lípidos representan los principales indicadores de riesgo cardiovascular y los objetivos terapéuticos primarios. En el segundo estudio caracterizamos los errores de cuantificación en el perfilado 1H-RMN de metabolitos clínicamente relevantes, que son debidos a su agregación a la proteína sanguínea. También proponemos un método que fomenta la competición por la agregación y que nos permite obtener cuantificaciones de nuestros metabolitos cercanas a las absolutas. Por último, el tercer trabajo presenta LipSpin: una herramienta bioinformática de código abierto específicamente diseñada para el perfilado de lípidos por 1H-RMN. Además, este estudio expone algunos aspectos metodológicos para mejorar el análisis de lípidos por RMN.1H-NMR serum profiling is especially suitable for high-throughput epidemiological studies and large-scale nutritional studies and drug monitoring. It requires minimal sample manipulation and its quantitative response allows inter-laboratory comparison. A comprehensive 1H-NMR serum profiling consists of three measurements encoding different molecular species: lipoproteins, low-molecular-weight metabolites and lipids. 1H-NMR serum profiling provides information of size, particle number and lipid content of lipoprotein subclasses, as well as abundance of amino acids, glycolysis-related metabolites, ketone bodies, fatty acids and phospholipids, among others. However, the spectral complexity promotes errors in manual data analysis and the multiple molecular interactions within the sample compromise reliable quantifications. Developing robust methods of metabolite serum profiling is therefore desirable to consolidate high-throughput 1H-NMR in the clinical practice. This thesis presents several methodological and computational strategies to that end. In the first study, we developed generalizable regression methods for lipids in routine clinical practice (known as “lipid panel”), to be applied in healthy population and in a wide spectrum of lipid and lipoprotein abnormalities. These standard lipids are still the main measurements of cardiovascular risk and therapy targets. In the second study, we characterised the quantitative errors introduced by protein binding in 1H-NMR profiling of clinically-relevant LMWM in native serum. Then, we proposed a competitive binding strategy to achieve quantifications closer to absolute concentrations, being fully compatible with high-throughput NMR. Finally, the third study presents LipSpin: an open source bioinformatics tool specifically designed for 1H-NMR profiling of serum lipids. Moreover, some methodological aspects to improve NMR-based serum lipid analysis are discussed