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

Topics in linear and nonlinear discrete optimization

This work contributes to modeling, theoretical, and practical aspects of structured Mathematical Programming problems. Many real-world applications have nonlinear characteristics and can be modeled as Mixed Integer Nonlinear Programming problems (MINLP). Modern global solvers have significant difficulty handling large-scale instances of them. Several convexification and underestimation techniques were proposed in the last decade as a part of the solution process, and we join this trend. The thesis has three major parts. The first part considers MINLP problems containing convex (in the sense of continuous relaxations) and posynomial terms (also called monomials), i.e. products of variables with some powers. Recently, a linear Mixed Integer Programming (MIP) approach was introduced for minimization the number of variables and transformations for convexification and underestimation of these structured problems. We provide polyhedral analysis together with separation for solving our variant of this minimization subproblem, containing binary and bounded continuous variables. Our novel mixed hyperedge method allows to outperform modern commercial MIP software, providing new families of facet-defining inequalities. As a byproduct, we introduce a new research area called mixed conflict hypergraphs. It merges mixed conflict graphs and 0-1 conflict hypergraphs. The second part applies our mixed hyperedge method to a linear subproblem of the same purpose for another class of structured MINLP problems. They contain signomial terms, i.e. posynomial terms of both positive and negative signs. We obtain new facet-defining inequalities in addition to those families from the first part. The final part is dedicated to managing guest flow in Georgia Aquarium after the Dolphin Tales opening with applying a large-scale MINLP. We consider arrival and departure processes related to scheduled shows and develop three stochastic models for them. If demand for the shows is high, all processes become interconnected and require a generalized model. We provide and solve a Signomial Programming problem with mixed variables for minimization resources to prevent and control congestions.Ph.D

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

Layout optimization and Sustainable development of waste water networks with the use of heuristic algorithms: The Luxemburgish case

Fresh water tends to increasingly comprise a scarcity today both in arid or demographically boosted regions of the world such as large and smaller cities. On this basis, research is directed towards minimization of fresh water supply into a Waste Water Network Topology (WWNT) and maximizing water re-use. This might be composed of a cluster of agents which have certain demands for fresh water as well as waste water dependent on their daily uses and living profiles. This work is divided into two parts. In the first part, different waste water flows within a reference building unit i.e. a typical household of four (4) occupants is simulated. This type of building represents a major part of the total building stock in Luxembourg. In its first part the present study attempts to examine the optimized fresh and waste water flow pathways between water using units of the building. Between water flows two domestic treatment units are adopted. The simulation of above mentioned system is attempted by adopting different algorithm methods such as the Sequential Quadratic Programming (SQP), the interior point and meta-heuristic optimization algorithms such as the Genetic Algorithms (GA’s).Suitable computational platform tools such as MATLAB and GAMS are incorporated. A comparison study on the most efficient approach is then realized on the single household unit by developing four (4) different mathematical model formulation versions. The second part of this study comprises simulation and development of the Waste Water Network Grid (WWNG) in the upscale level, such as the neighborhood level within or outside the urban context. This model encompasses all possible land uses and different kinds of buildings of different use envelopes thus demands. This range of units includes mainly building stock, agricultural and infrastructure of the tertiary sector. Integration of above mentioned model to the existing WWNG will enhance attempts to more closely reach the optimum points. The use of appropriate mathematical programming methods for the upscale level, will take place. Increased uncertainties within the built model will be attempted to be tackled by developing linear programming techniques and suitable assumptions without distorting initial condition largely. Assumptions are then drawn on the efficiency of the adopted method an additional essential task is the minimization of the overall infrastructure and network cost, which may in turn give rise to corresponding reduced waste effluents discharge off the proposed network. The case study comprises selected rural and semi-rural areas zone districts of similar living profiles outside the City of Luxembourg. Therefore a clustering of end users of similar demand will be attempted. Possible redesign of an optimized WWNG comprises a vital need within the context of large scale demographic growth of urban environments today.Open Acces

Modelo matemático y metodología para la optimización de redes de intercambio de calor

En este trabajo se aborda el desarrollo de elementos que apoyan la labor de síntesis de redes de intercambio de calor a través de una mejor valoración de la calidad estructural de un conjunto de diseños preliminares de red. La síntesis y la optimización de redes de intercambio de calor con un enfoque de programación matemática se sustentan en modelos de optimización que exhiben características no convexas dando lugar en muchos casos a la existencia de múltiples soluciones óptimas locales. El problema abordado fue la optimización de redes de intercambio de calor, asumiendo que la topología de red fue previamente establecida por alguna metodología de síntesis. Se buscaron los valores de las distribuciones de flujos, cargas térmicas, áreas de intercambio de calor y temperaturas intermedias de la red de intercambio de calor a través de la minimización de su costo total. El objetivo de este trabajo consiste en evaluar la calidad estructural de diferentes redes de intercambio de calor, para ello se desarrollaron una representación de las redes y con base en ella, un modelo matemático generalizado y una metodología de solución y búsqueda de diseños óptimos locales. En el Capítulo 1 se presenta una revisión del estado del arte en la optimización global de redes de intercambio de calor con métodos deterministas y estocásticos y en el Capítulo 2 se formula el problema de diseño óptimo de una red de intercambio de calor que se estudia. En el Capítulo 3 se propone una representación de las redes de intercambio de calor a través de un diagrama de malla con etapas a las que se asocia la existencia de un intercambiador de calor. Ligado a éste, en el Capítulo 4 se desarrolla un modelo de programación no lineal para la optimización de redes de intercambio de calor sin división de corrientes. En el modelado se definen una serie de conjuntos para las corrientes, los equipos y las etapas del diagrama; con base en ellos, se plantean las ecuaciones que describen los balances de materia y energía y las restricciones de diseño. Para la solución del modelo desarrollado, en el Capítulo 5 se presenta una metodología heurística de búsqueda de soluciones óptimas locales cuyo elemento principal es el multi-arranque, que consiste en la elección de manera aleatoria de diferentes puntos de inicio, a partir de los cuales se intenta resolver el modelo desde diferentes secciones de la región de búsqueda. El modelo propuesto se extiende en el Capítulo 6, para incluir el caso de la optimización de redes de intercambio de calor con división de corrientes. La aplicación de la representación, del modelo y de la metodología se ilustra con nueve ejemplos. Por último en el Capítulo 7, se resumen las aportaciones de este trabajo de investigación y se proponen posibles líneas para trabajo futuro

Optimal configuration, design and operation of batch distillation processes

The overall objective of this thesis is to study the optimal configuration. design and operating policy of batch distillation processes in different separation scenarios. In so doing, this work also aims to provide conceptual insights and compare the performance of the traditional regular column against unconventional columns. In the first part of the thesis, the optimal operation of extractive batch distillation is investigated. A rigorous dynamic optimisation approach based on a detailed model is employed. In addition to the regular column, the optimal operation of the process in the unconventional middle vessel column is examined. The liquid and vapour stream configurations at the middle section of the column is explored for the first time, resulting in improved process performance. The performance of both columns are compared and the results show how their relative performances are affected by different feed compositions. The second part of the thesis is concerned with the simultaneous design and operation of batch distillation processes. The thesis proposes a stochastic optimisation methodology based on genetic algorithm and penalty function. Using the proposed methodology, the simultaneous optimal designs and operations of the regular column for different design scenarios are investigated using rigorous models. Furthermore, the optimal design of the unconventional multivessel column for multicomponent separation is studied for the first time. The effect of different factors such as objective function, feed composition, relative volatility, product specification and number of components on the optimal design of the multivessel system is investigated. A comparison of the performance of the multivessel system with the regular column is also presented. In the final part of the thesis, the feasibility of the genetic algorithm-penalty function approach in tackling simultaneous configuration selection, column sizing and operation is explored. In the case of binary mixture separation, the regular column was found to be more profitable for feeds with a high fraction of the light component whilst the inverted column is optimal for heavier feeds. There exists a flip point, the location of which is case study specific. For the multicomponent separation case study, the multivessel system is found to be superior to both the regular and inverted configuration