Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

AbstractExtending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Polyhedral techniques in combinatorial optimization

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

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

Strategic Surveillance System Design for Ports and Waterways

The purpose of this dissertation is to synthesize a methodology to prescribe a strategic design of a surveillance system to provide the required level of surveillance for ports and waterways. The method of approach to this problem is to formulate a linear integer programming model to prescribe a strategic surveillance system design (SSD) for ports or waterways, to devise branch-and-price decomposition (

Complete Randomized Cutting Plane Algorithms for Propositional Satisfiability

The propositional satisfiability problem (SAT) is a fundamental problem in computer science and combinatorial optimization. A considerable number of prior researchers have investigated SAT, and much is already known concerning limitations of known algorithms for SAT. In particular, some necessary conditions are known, such that any algorithm not meeting those conditions cannot be efficient. This paper reports a research to develop and test a new algorithm that meets the currently known necessary conditions. In chapter three, we give a new characterization of the convex integer hull of SAT, and two new algorithms for finding strong cutting planes. We also show the importance of choosing which vertex to cut, and present heuristics to find a vertex that allows a strong cutting plane. In chapter four, we describe an experiment to implement a SAT solving algorithm using the new algorithms and heuristics, and to examine their effectiveness on a set of problems. In chapter five, we describe the implementation of the algorithms, and present computational results. For an input SAT problem, the output of the implemented program provides either a witness to the satisfiability or a complete cutting plane proof of satisfiability. The description, implementation, and testing of these algorithms yields both empirical data to characterize the performance of the new algorithms, and additional insight to further advance the theory. We conclude from the computational study that cutting plane algorithms are efficient for the solution of a large class of SAT problems

Robust production optimization of gas-lifted oil fields

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2015.Com a crescente demanda por energia fóssil as operadoras petrolíferas têm buscado determinar planos operacionais que otimizam a produção dos campos em operação para satisfazer a demanda do mercado e reduzir os custos operacionais. Neste contexto, a pesquisa operacional tem se mostrado uma importante ferramenta para determinação dos planos de produção de curto prazo para campos de petróleo complexos. Alguns trabalhos já desenvolveram estratégias para a otimização integrada da produção que visam auxiliar engenheiros de produção e operadores a atingir condições de operação ótimas. Estes avanços científicos atestam o potencial da área de otimização integrada da produção de campos, justificando a busca por estratégias de otimização global e integradas de ativos. Contudo, a incerteza dos parâmetros que caracterizam o reservatório, os poços, fluidos e os diversos processos de produção não vem sendo considerada pelos modelos e algoritmos de otimização da produção diária. Considerando os modelos de produção de curto prazo, estas incertezas podem ser atribuídas a erros de medição , comportamento oscilatório dos sistemas, modelos imprecisos, entre outros. A influência da incerteza dos parâmetros em problemas de otimização tem, desde tempos, sido foco da comunidade de programação matemática. E já foi verificado que soluções de problemas de otimização podem apresentar significativa sensibilidade à pertubações nos parâmetros do dado problema, podendo levar a soluções não factíveis, subótimas ou ambas. Assim, buscando tornar as abordagens de otimização existentes mais confiáveis e robustas às incertezas intrínsecas dos sistemas de produção, esta dissertação investiga a modelagem e tratamento de incertezas na otimização diária da produção e propõe formulações em programação matemática para otimização robusta da produção de poços operados por gas-lift. As formulações representam curvas amostradas através de dados simulados ou medidos que refletem as incertezas dos sistemas de produção. Estas representações levam a formulações robustas em programação matemática inteira mista obtidas pela aproximação das curvas de produção através de linearização por partes. Além disso, este trabalho apresenta os resultados de uma analise computacional comparativa da aplicação da formulação robusta e da formulação nominal a um campo de petróleo em ambiente de simulação, porém considerando simuladores multifásicos amplamente empregados pela indústria do petróleo e gás, que representam a fenomenologia muito próximo da realidade. O primeiro capítulo apresenta a problemática em que estão envolvidos os desenvolvimentos realizados nesta dissertação e um resumo dos capítulos subsequentes. No segundo capítulo alguns conceitos fundamentais são apresentados para a compreensão do trabalho desenvolvido. Este capítulo é dividido em três partes. A primeira parte inicia apresentando brevemente a indústria de petróleo e gás com uma perspectiva histórica, econômica e dos processos envolvidos. Na sequência são expostos conceitos básicos de engenharia de petróleo necessários para o entendimento do sistema de produção utilizado ao longo a dissertação  i.e. gas-lift. Finalmente, o problema de otimização da produção é situado dentro do problema maior, que é o gerenciamento completo das operações de um campo de petróleo, seguido de uma revisão da literatura no que se refere a abordagens clássicas para otimização da produção de campos operados por gas-lift. A segunda parte é uma descrição compacta sobre modelagem de problemas de otimização utilizando programação matemática e na menção dos métodos de solução deste tipo de problema utilizados na parte experimental desta dissertação. A terceira parte começa com uma revisão sobre incerteza em problemas de otimização e sobre as decisões de modelagem enfrentadas quando na presença de problemas de otimização incertos. Na sequência o paradigma de otimização robusta é introduzido e é apresentada uma compilação de alguns dos principais resultados da área de otimização robusta linear. Além disso, ao fim, alguns pontos específicos da teoria de otimização robusta são apresentados pela suas relevâncias para o desenvolvimento da teoria dos capítulos seguintes. O terceiro capítulo inicia com uma discussão sobre as origens das incertezas nos modelos de produção para então prover uma revisão bibliográfica dos poucos trabalhos que mencionam ou lidam com incerteza em sistemas de produção. Na sequência, a incerteza é examinada na perspectiva do problema de otimização. Um sistema simples é usado para exemplificar a metodologia de otimização robusta desenvolvida nesta dissertação. O quarto capítulo apresenta dois problemas padrões de otimização da produção, um contendo poços satélites e outro com poços e completação submarina. Para ambos uma formulação em programação linear inteira mista é descrita considerando valores nominais para todos os parâmetros. Então, para cada problema uma reformulação robusta é implementada considerando incerteza nas curvas de produção do poço. A metodologia utilizada para o primeiro problema é a mesma detalhada no capítulo três, e para o segundo uma extensão da metodologia é proposta para poder lidar com restrições de igualdade incertas. No quinto capítulo são apresentados resultados experimentais de um problema de otimização da produção de um campo com poços satélites. Os resultados obtidos com otimização clássica (nominal) e com otimização robusta são então comparados em um campo de produção sintético instanciado em um simulador multifásico comercial. A solução robusta se mostrou indicada para cenários de operação mais críticos onde factibilidade e segurança são prioridade. No capítulo final uma análise dos resultados obtidos na dissertação é feita sob a perspectiva do possível emprego das técnicas desenvolvidas na indústria de óleo e gás. Apesar de à primeira vista os resultados serem conservadores e de sua utilização parecer limitada, existe potencial para a metodologia ser empregada no caso de situações que priorizam segurança. Além disso a metodologia aqui desenvolvida pode servir como ponto inicial para pesquisas e desenvolvimentos futuros. Uma breve descrição de possíveis trabalhos futuros é feita ao final deste capítulo. O apêndice traz a descrição de algoritmos de amostragem de curvas côncavas desenvolvidos para os experimentos numéricos realizados na dissertação.Abstract : Managing production of complex oil fields with multiple wells and coupled constraints remains a challenge for oil and gas operators. Some technical works developed strategies for integrated production optimization to assist production engineers in reaching best operating conditions. However, these works have neglected the uncertainties in the well-performance curves and production processes, which may have a significant impact on the operating practices. The uncertainties may be attributed to measurement errors, oscillating behavior, and model inaccuracy, among others. To this end, this dissertation investigates how uncertainty might be considered in daily production optimization and proposes formulations in mathematical programming for robust production optimization of gas-lifted oil fields. The formulations represent system-measured and simulated sample curves that reflect the underlying uncertainties of the production system. The representations lead to robust mixed-integer linear programming formulations obtained from piecewise-linear approximation of the production functions. Further, this work presents results from a computational analysis of the application of the robust and nominal formulations to a representative oil fields available in simulation software

Constrained Learning And Inference

Data and learning have become core components of the information processing and autonomous systems upon which we increasingly rely on to select job applicants, analyze medical data, and drive cars. As these systems become ubiquitous, so does the need to curtail their behavior. Left untethered, they are susceptible to tampering (adversarial examples) and prone to prejudiced and unsafe actions. Currently, the response of these systems is tailored by leveraging domain expert knowledge to either construct models that embed the desired properties or tune the training objective so as to promote them. While effective, these solutions are often targeted to specific behaviors, contexts, and sometimes even problem instances and are typically not transferable across models and applications. What is more, the growing scale and complexity of modern information processing and autonomous systems renders this manual behavior tuning infeasible. Already today, explainability, interpretability, and transparency combined with human judgment are no longer enough to design systems that perform according to specifications. The present thesis addresses these issues by leveraging constrained statistical optimization. More specifically, it develops the theoretical underpinnings of constrained learning and constrained inference to provide tools that enable solving statistical problems under requirements. Starting with the task of learning under requirements, it develops a generalization theory of constrained learning akin to the existing unconstrained one. By formalizing the concept of probability approximately correct constrained (PACC) learning, it shows that constrained learning is as hard as its unconstrained learning and establishes the constrained counterpart of empirical risk minimization (ERM) as a PACC learner. To overcome challenges involved in solving such non-convex constrained optimization problems, it derives a dual learning rule that enables constrained learning tasks to be tackled by through unconstrained learning problems only. It therefore concludes that if we can deal with classical, unconstrained learning tasks, then we can deal with learning tasks with requirements. The second part of this thesis addresses the issue of constrained inference. In particular, the issue of performing inference using sparse nonlinear function models, combinatorial constrained with quadratic objectives, and risk constraints. Such models arise in nonlinear line spectrum estimation, functional data analysis, sensor selection, actuator scheduling, experimental design, and risk-aware estimation. Although inference problems assume that models and distributions are known, each of these constraints pose serious challenges that hinder their use in practice. Sparse nonlinear functional models lead to infinite dimensional, non-convex optimization programs that cannot be discretized without leading to combinatorial, often NP-hard, problems. Rather than using surrogates and relaxations, this work relies on duality to show that despite their apparent complexity, these models can be fit efficiently, i.e., in polynomial time. While quadratic objectives are typically tractable (often even in closed form), they lead to non-submodular optimization problems when subject to cardinality or matroid constraints. While submodular functions are sometimes used as surrogates, this work instead shows that quadratic functions are close to submodular and can also be optimized near-optimally. The last chapter of this thesis is dedicated to problems involving risk constraints, in particular, bounded predictive mean square error variance estimation. Despite being non-convex, such problems are equivalent to a quadratically constrained quadratic program from which a closed-form estimator can be extracted. These results are used throughout this thesis to tackle problems in signal processing, machine learning, and control, such as fair learning, robust learning, nonlinear line spectrum estimation, actuator scheduling, experimental design, and risk-aware estimation. Yet, they are applicable much beyond these illustrations to perform safe reinforcement learning, sensor selection, multiresolution kernel estimation, and wireless resource allocation, to name a few