Interval simplex splines for scientific databases

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1995.Includes bibliographical references (p. 130-138).by Jingfang Zhou.Ph.D

Outer Approximation Algorithms for DC Programs and Beyond

We consider the well-known Canonical DC (CDC) optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems. We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively. As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Relaxation refinement for mixed-integer nonlinear programs with applications in engineering

Lösungsstrategien für Gemischt-Ganzzahlige Nichtlineare Programme (MINLPs) basieren häufig auf einer konvexen Relaxierung der zulässigen Menge. Diese Relaxierung wird benutzt um untere Schranken zu ermitteln und um die Qualität lokaler Lösungen zu beurteilen. In dieser Thesis diskutieren wir verschiedene Ansätze um geeignete Relaxierungen zu konstruieren und zu verbessern. Außerdem analysieren wir diese in Hinblick auf Strenge und Qualität der resultierenden unteren Schranken. Dabei betrachten wir sowohl allgemeine MINLPs als auch spezifische Probleme, die sich aus der Anwendung ergeben. Wir entwickeln ein Schnittebenenverfahren für die konvexe Hülle der zulässigen Menge von relativ allgemeinen MINLPs. Es basiert auf der simultanen Betrachtung von Nebenbedingungen und auf einem konvexen Optimierungsproblem. Dieses Separationsproblem ist nicht-differenzierbar und benötigt die konvexe Einhüllende von Linearkombinationen der Nebenbedingungen. Wir analysieren seine Struktur und Glätte ausführlich und diskutieren passende Lösungsansätze. Außerdem entwickeln wir Approximationen der konvexen Einhüllenden und ein ensprechendes approximatives Separationsproblem. Dieses führt zu schwächeren Resultaten aber zu einer höheren Anwendbarkeit. Das obige Schnittebenenverfahren wird außerdem auf eine Menge von Nebenbedingungen angewendet, die aus bivariaten quadratischen Absolutwertfunktionen besteht. Wir präsentieren allgemeine analytische Hilfsmittel und Konzepte und bestimmen die konvexe Einhüllende für diese Funktionen unter gewissen Voraussetzungen. Diese Klasse von Funktionen wird auch bei der Modellierung von Gasnetzwerken verwendet, was es uns erlaubt den Einfluss des Schnittebenenverfahrens auf Probleme aus der Anwendung zu untersuchen. Schließlich betrachten wir noch ein Beispiel eines optimalen Designproblems aus dem Bereich des Chemieingenieurwesens. Für das Modell einer Destillationskolonne bieten wir eine Reformulierung an und beweisen monotones Verhalten von bestimmten Folgen relevanter Variablen. Reformulierung und Monotonie werden benutzt um die Formulierung der zugehörigen zulässigen Menge zu verbessern. Insbesondere entwickeln wir eine problemspezifische Bound-Tightening-Strategie. Unsere Ergebnisse werden an einigen Testinstanzen computergestützt evaluiert.Solution strategies for Mixed-Integer Nonlinear Programs (MINLPs) often rely on a convex relaxation of the feasible set. This relaxation is used to derive lower bounds and to evaluate the quality of local solutions. In this thesis, we discuss different approaches of constructing and improving suitable relaxations. We further analyze these relaxations with respect to tightness and quality of the resulting lower bounds. This is done for general MINLPs as well as for specific problems arising from certain real world applications. We develop a cutting plane method for the convex hull of the feasible set of relatively general MINLPs. It is based on simultaneous considerations of the involved constraints and on solving a convex optimization problem. This underlying separation problem is non-differentiable and requires the convex envelope of linear combinations of the constraint functions. We analyze its structure and smoothness in detail, and discuss suitable solution approaches. Furthermore, we introduce approximation strategies for the convex envelope and discuss the resulting approximate version of the separation problem. This approximate version leads to weaker results but to a greater applicability. The proposed cutting plane approach is further applied to constraint sets consisting of bivariate quadratic absolute value functions. We present general analytic tools and concepts, and derive the convex envelope of the considered functions under certain assumptions. This type of functions also emerges from the modeling of gas networks, which allows us to computationally evaluate the impact of our cutting plane approach on a real world application. Finally, we consider an example of optimal design problems in chemical engineering. For a distillation column model, we introduce a suitable reformulation and prove monotonic behavior of several sequences of relevant variables. Reformulation and monotonicity are used to improve the formulation of the respective feasible set. In particular, we develop a problem specific bound tightening strategy. Our results are computationally evaluated on multiple test instances

Maximum Likelihood Estimation of Discrete Log-Concave Distributions with Applications

Shape-constrained methods specify a class of distributions instead of a single parametric family. The approach increases the robustness of the estimation without much loss of efficiency. Among these, log-concavity is an appealing shape constraint in distribution modeling, because it falls into the popular unimodal shape-constraint and many parametric models are log-concave. This is, therefore, the focus of our work. First, we propose a maximum likelihood estimator of discrete log-concave distributions in higher dimensions. We define a new class of log-concave distributions in multiple dimensional spaces and study its properties. We show how to compute the maximum likelihood estimator from an independent and identically distributed sample, and establish consistency of the estimator, even if the class has been incorrectly specified. For finite sample sizes, the proposed estimator outperforms a purely nonparametric approach (the empirical distribution), but is able to remain comparable to the correct parametric approach. Furthermore, the new class has a natural relationship with log-concave densities when data has been grouped or discretized. We show how this property can be used in a real data example. Secondly, we apply the discrete log-concave maximum likelihood estimator in one-dimensional space to a clustering problem. Our work mainly focuses on the categorical nominal data. We develop a log-concave mixture model using the discrete log-concave maximum likelihood estimator. We then apply the log-concave mixture model to our clustering algorithm. We compare our proposed clustering algorithm with the other two clustering methods. Comparing results show that our proposed algorithm has a good performance

Analysis and Parameterization of Triangulated Surfaces

This dissertation deals with the analysis and parameterization of surfaces represented by triangle meshes, that is, piecewise linear surfaces which enable a simple representation of 3D models commonly used in mathematics and computer science. Providing equivalent and high-level representations of a 3D triangle mesh M is of basic importance for approaching different computational problems and applications in the research fields of Computational Geometry, Computer Graphics, Geometry Processing, and Shape Modeling. The aim of the thesis is to show how high-level representations of a given surface M can be used to find other high-level or equivalent descriptions of M and vice versa. Furthermore, this analysis is related to the study of local and global properties of triangle meshes depending on the information that we want to capture and needed by the application context. The local analysis of an arbitrary triangle mesh M is based on a multi-scale segmentation of M together with the induced local parameterization, where we replace the common hypothesis of decomposing M into a family of disc-like patches (i.e., 0-genus and one boundary component) with a feature-based segmentation of M into regions of 0-genus without constraining the number of boundary components of each patch. This choice and extension is motivated by the necessity of identifying surface patches with features, of reducing the parameterization distortion, and of better supporting standard applications of the parameterization such as remeshing or more generally surface approximation, texture mapping, and compression. The global analysis, characterization, and abstraction of M take into account its topological and geometric aspects represented by the combinatorial structure of M (i.e., the mesh connectivity) with the associated embedding in R^3. Duality and dual Laplacian smoothing are the first characterizations of M presented with the final aim of a better understanding of the relations between mesh connectivity and geometry, as discussed by several works in this research area, and extended in the thesis to the case of 3D parameterization. The global analysis of M has been also approached by defining a real function on M which induces a Reeb graph invariant with respect to affine transformations and best suited for applications such as shape matching and comparison. Morse theory and the Reeb graph were also used for supporting a new and simple method for solving the global parameterization problem, that is, the search of a cut graph of an arbitrary triangle mesh M. The main characteristics of the proposed approach with respect to previous work are its capability of defining a family of cut graphs, instead of just one cut, of bordered and closed surfaces which are treated with a unique approach. Furthermore, each cut graph is smooth and the way it is built is based on the cutting procedure of 0-genus surfaces that was used for the local parameterization of M. As discussed in the thesis, defining a family of cut graphs provides a great flexibility and effective simplifications of the analysis, modeling, and visualization of (time-depending) scalar and vector fields; in fact, the global parameterization of M enables to reduce th