On refinement strategies for solving MINLPs by piecewise linear relaxations: a generalized red refinement

We investigate the generalized red refinement for n-dimensional simplices that dates back to Freudenthal (Ann Math 43(3):580–582, 1942) in a mixed-integer nonlinear program (MINLP) context. We show that the red refinement meets sufficient convergence conditions for a known MINLP solution framework that is essentially based on solving piecewise linear relaxations. In addition, we prove that applying this refinement procedure results in piecewise linear relaxations that can be modeled by the well-known incremental method established by Markowitz and Manne (Econometrica 25(1):84–110, 1957). Finally, numerical results from the field of alternating current optimal power flow demonstrate the applicability of the red refinement in such MIP-based MINLP solution frameworks

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming

We study mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present two MIP relaxation methods for non-convex continuous variable products that enhance existing approaches. One is based on a separable reformulation, while the other extends the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT). In addition, we introduce a logarithmic MIP relaxation for univariate quadratic terms, called sawtooth relaxation, based on [4]. We combine the latter with the separable reformulation to derive MIP relaxations of MIQCQPs. We provide a comprehensive theoretical analysis of these techniques, and perform a broad computational study to demonstrate the effectiveness of the enhanced MIP relaxations in terms producing tight dual bounds for MIQCQP

An Approximation Algorithm for Optimal Piecewise Linear Interpolations of Bounded Variable Products

We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples.Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst http://dx.doi.org/10.13039/50110000456

On piecewise linear approximations of bilinear terms: structural comparison of univariate and bivariate mixed-integer programming formulations

Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an ε-approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error  ε, univariate ε-approximations require fewer simplices than bivariate ε-approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.Bayerisches Staatsministerium für Wirtschaft, Infrastruktur, Verkehr und Technologie http://dx.doi.org/10.13039/501100005017Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Bayerische Staatsregierun

CFD analysis of a dual heat recovery system

This paper presents a CFD Heat Transfer Analysis of an originally designed system for heat recovery in the building sector. The heat exchanger has a dual role, which means it will produce simultaneously hot water and warm air. The key to the efficiency of the heat exchanger is the heat pipe system which recovers thermal energy from residual hot water and transfers it to the secondary agents. The paper includes a case study structured by different mesh distributions and flow regimes. The purpose of the heat exchanger is to reduce the costs of producing thermal energy and to increase the overall energy efficiency of buildings

Simulation and modelling of microclimate in a building with high thermal mass during the winter season

This paper presents the simulation method for evaluating the heating system from a church. The inside climate has been evaluated by measures of temperature and humidity taken in the winter season. The aim of the paper is to model and validate the indoor climate measures thought numerical analysis and to evaluate the heating system performance. The paper include a case study over and representative category of buildings, used as worship place that can contain heritage values. Nowadays, to conserve the historical heritage is a fact studied in many countries of Europe

GasLib—A Library of Gas Network Instances

The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML (extensible markup language) format. In this paper, we explain this format and present the instances that are available in the library

GasLib — A Library of Gas Network Instances

The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML (extensible markup language) format. In this paper, we explain this format and present the instances that are available in the library

Adaptive gemischt-ganzzahlige Verfeinerungen zur Lösung nichtlinearer Probleme mit diskreten Entscheidungen

Many optimization problems in science and technology are subject to a system of nonlinear constraints combined with discrete decisions. Mathematically, these problems can be modeled as mixed-integer nonlinear programs (MINLPs), since they can represent both nonlinear correlations and discrete decisions. However, the interaction of integrality and nonlinearity poses a major challenge in solving these problems. In this thesis, we propose a method for solving MINLPs to global optimality by discretization of the occurring nonlinearities. Our approach requires only continuous nonlinearities with bounded domains and is thus suitable for a wide range of MINLP problems. The emphasis is on using sophisticated and reliable mixed-integer linear programming technology to solve MINLPs. Similarly to the solution of mixed-integer linear programs (MIPs), which relies on solving linear programming relaxations, we develop a framework for solving MINLPs by MIP relaxations. To this end, we use piecewise linear functions to construct MIP relaxations of the underlying MINLP. An iterative algorithm constructs MIP relaxations that are subsequently refined and solved until a globally optimal solution is found. The refinement of the nonlinearities is crucial for the outcome of the algorithm. For that reason, we study different refinement strategies with a focus on embedding them in our adaptive MIP-based framework. We prove convergence results for the presented refinement methods. In addition, we present first results on the size of an MIP relaxation that is required to achieve an a priori given accuracy for the nonlinearities. Finally, we illustrate the practicalness of our approach by numerical results for MINLPs that are difficult to solve by state-of-the-art global MINLP solvers. These problems arise in the context of gas transport network optimization and optimal power flow. They combine non-convex nonlinearities that describe certain physical phenomena with integer restrictions that model discrete decisions for switchable elements. Such elements are, for instance, compressors in case of gas networks or generator units in case of power networks. On the basis of theses MINLP instances, we demonstrate the advantage of our approach over other global MINLP solvers.Viele Optimierungsprobleme der Wissenschaft und Technik werden anhand eines Systems nichtlinearer Restriktionen in Kombination mit diskreten Entscheidungen gelöst. Mathematisch lassen sich diese Probleme als gemischt-ganzzahlige nichtlineare Programme (MINLPs) modellieren, da diese sowohl nichtlineare Zusammenhänge als auch diskrete Entscheidungen darstellen können. Das Zusammenspiel von Ganzzahligkeit und Nichtlinearität stellt dabei eine große Herausforderung bei der Lösung dieser Probleme dar. In dieser Arbeit behandeln wir eine Methode zur global optimalen Lösung von MINLPs durch Diskretisierung der auftretenden Nichtlinearitäten. Unser Ansatz erfordert nur kontinuierliche Nichtlinearitäten mit beschränktem Definitionsbereich und ist daher für eine Vielzahl von MINLP-Problemen geeignet. Die grundlegende Idee hierbei ist, ausgereifte und zuverlässige Technologie der gemischt-ganzzahligen Programmierung zur Lösung von MINLPs einzusetzen. Ähnlich wie das Lösen von gemischt-ganzzahligen linearen Programmen (MIPs) auf der Lösung linearer Programme, d.h. von Relaxierungen der MIPs, beruht, entwickeln wir einen Ansatz zur Lösung von MINLPs, der auf der Lösung von MIP-Relaxierungen beruht. Hierzu verwenden wir stückweise lineare Funktionen, um MIP-Relaxierungen des zugrunde liegenden MINLPs zu konstruieren. Der iterative Algorithmus konstruiert MIP-Relaxierungen, die abwechselnd verfeinert und gelöst werden, bis eine global optimale Lösung gefunden wird. Die Verfeinerung der Nichtlinearitäten ist entscheidend für die Konvergenz und Korrektheit des Algorithmus. Aus diesem Grund untersuchen wir verschiedene Verfeinerungsstrategien mit dem Schwerpunkt auf deren Einbettung in unseren adaptiven MIP-basierten Ansatz. Wir liefern zunächst Konvergenzresultate für die vorgestellten Verfeinerungsmethoden. Darüber hinaus leiten wir erste Ergebnisse zur Größe einer MIP-Relaxierung her, die benötigt wird, um eine a priori gegebene Genauigkeit für die Nichtlinearitäten zu erreichen. Abschließend veranschaulichen wir die Anwendbarkeit unseres Ansatzes in der Praxis durch numerische Ergebnisse für MINLPs, die mit den modernsten globalen MINLP-Lösern nur schwer zu lösen sind. Diese Probleme entstehen im Kontext der Gastransportoptimierung und der optimalen Lastflussberechnung. Sie kombinieren nichtkonvexe Nichtlinearitäten, die spezifische physikalische Phänomene beschreiben, mit ganzzahligen Restriktionen, die diskrete Entscheidungen für schaltbare Elemente modellieren. Solche Elemente sind beispielsweise Kompressoren bei Gasnetzwerken oder Generatoreinheiten bei Stromnetzwerken. Anhand dieser MINLP-Instanzen zeigen wir den Mehrwert unseres Ansatzes gegenüber anderen globalen MINLP-Lösern auf

On refinement strategies for solving MINLPs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\textsc {MINLP}\mathrm{s}}\end{document} by piecewise linear relaxations: a generalized red refinement

We investigate the generalized red refinement for n-dimensional simplices that dates back to Freudenthal (Ann Math 43(3):580–582, 1942) in a mixed-integer nonlinear program (MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\textsc {MINLP}}\end{document}) context. We show that the red refinement meets sufficient convergence conditions for a known MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\textsc {MINLP}}\end{document} solution framework that is essentially based on solving piecewise linear relaxations. In addition, we prove that applying this refinement procedure results in piecewise linear relaxations that can be modeled by the well-known incremental method established by Markowitz and Manne (Econometrica 25(1):84–110, 1957). Finally, numerical results from the field of alternating current optimal power flow demonstrate the applicability of the red refinement in such MIP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\textsc {MIP}}\end{document}-based MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\textsc {MINLP}}\end{document} solution frameworks