용량 제약이 없는 부보상 문제의 혼합 이진 이차 문제로의 모형화를 통한 해법

학위논문(석사) -- 서울대학교대학원 : 공과대학 산업공학과, 2022. 8. 홍성필.부보상 문제는 비순환 유향 그래프 상에서 출발, 도착 마디를 잇는 경로와 그 경로 상의 흐름을 결정하는 문제이다. 부보상은 도시 1에서 n까지 이동하면서 각 도시를 경유하 거나 지나치며, 경유하는 도시에서만 상품을 매매할 수 있지만 이동 거리와 상품량에 따른 비용 또한 지불해야 한다. 이 때, 부보상은 자신의 수익, 즉 총 상품의 판매량에서 얻는 수익과 지불 비용의 차를 최대화하고자 한다. 본 연구에서는 용량 제약이 없는 경우만을 다루며 기존 부보상 문제를 혼합이진이차문제으로 재모형화하여 분지절단법 으로 문제를 푼다. 이 때 목적 함수를 볼록화하고 연속 완화시켜 얻을 수 있는 상한을 비교하기 위해 여러 볼록화 방법들을 비교하고 비교실험한 결과 또한 제시한다.Bubosang Problem is a problem set on a directed acyclic graph path concerning both the path and multi-commodity flow decisions. A merchant travels from city 1 through n, either transiting through a city and trading products or passing by the city to the next city on his route. He wants to choose the path and trading product quantity to maximize his net profit which is defined by the difference between the total sales revenue and the traveling cost. The scope of the study considers only the uncapacitated case. In this study, we reformulate BP into a mixed binary quadratic problem to employ the branch-and-cut algorithm to solve the problem. Specifically, we compare the upper bound obtained through the continuous relaxation and convexification of the objective by studying different convexification methods. Computational results of the comparison are also provided.Chapter 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Research Motivations 5 1.4 Organization of the Thesis 6 Chapter 2 Problem Definition and Mathematical Formulations 7 2.1 Problem Definition 7 2.2 Flow Arc Formulation 8 2.3 MBQP Formulation 11 2.3.1 MBQP 13 2.4 Branch-and-Cut Algorithm 14 2.4.1 Overall Setting 14 2.4.2 Cutset Inequality 14 2.4.3 Lower Bound 15 2.4.4 Upper Bound 18 Chapter 3 Convexification Methods 19 3.1 One Coefficient Case : Eigenvalue Method 21 3.2 Criteria for Convexification Evaluation 22 3.2.1 Criterion for Unweighted Methods 22 3.3 Two Coefficient Case : (α, β) - SDP method 23 3.4 Two Coefficient Case : (α, β) - Sum of Squares Method 24 3.5 Four Coefficient Case : (α, β, γ, δ) - method 26 3.5.1 (α, β, γ, δ) - SDP method 26 3.5.2 (α, β, γ, δ) - Sum of Squares method 28 3.6 Five Coefficient Case : (α, β, γ, δ, τ ) - Sum of Squares method 29 3.7 Weighted methods 30 3.7.1 Criterion for Weighted Methods 30 Chapter 4 Computational Experiments 32 Chapter 5 Conclusion 36 Bibliography 37 국문초록 41석

An Improved Linearization Strategy for Zero-One Quadratic Programming Problems

the date of receipt and acceptance should be inserted later Abstract We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed by Chaovalitwongse, Pardalos, and Prokopyev (Op-erations Research Letters, 2004), and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances

Uma nova relaxação quadrática para variáveis binárias com aplicações a confiabilidade de redes de energia elétrica, a segmentação de imagens médicas de nervos e a problemas de geometria de distâncias

Orientador: Christiano Lyra FilhoTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Como o título sugere, o foco desta pesquisa é o desenvolvimento de uma nova relaxação quadrática para problemas binários, sua formalização em resultados teóricos, e a aplicação dos novos conceitos em aplicações à confiabilidade de redes de energia elétrica, à segmentação de imagens médicas de nervos e à problemas de geometria de distâncias. Modelos matemáticos contendo va-riáveis de decisões binárias podem ser usados para encontrar as melhores soluções em processos de tomada de decisões, normalmente caracterizando problemas de otimização combinatória difíceis. A solução desses problemas em aplicações de interesse prático requer um grande esforço computacional; por isso, ao longo dos últimos anos, têm sido objeto de pesquisas na área de metaheurísticas. As ideias aqui desenvolvidas abrem novas perspectivas para a abordagem desses problemas apoiando-se em métodos de otimização não-lineares, área que vem sendo povoada por "solvers" muito eficientes. Inicialmente, explorando aspectos formais, a relaxação desenvolvida é parti-cularizada para um problema de otimização quadrática binária irrestrita. O relaxamento permite o desenvolvimento de três estruturas para abordar esta classe de problemas, e explora a convexidade da função objetivo para obter melhorias computacionais. Estudos de casos compararam o relaxamento proposto com os relaxamentos similares apresentados na literatura. Foram desenvolvidas três aplicações para os desenvolvimentos teóricos da pesquisa. A primeira aplicação envolve a melhoria da confiabilidade de redes de energia elétrica. Especificamente, aborda o problema de definir a melhor alternativa para a alocação de sensores na rede, o que permite reduzir os efeitos de ocorrências indesejáveis e ampliar a resiliência das redes. A segunda aplicação envolve o problema de segmentação de imagens médicas associadas a estruturas de nervos. A abordagem proposta interpreta o problema de segmentação como um problema de otimização binária, onde medir cada axônio significa encontrar um ciclo Hamiltoniano, um caso do problema do caixeiro viajante; a solução desses problemas fornece a estatística descritiva para um conjunto de axônios, incluindo o número (de axônios), os diâmetros e as áreas ocupadas. A última aplicação elabora um modelo matemático para o problema de geometria de distâncias sem designação, área ainda pouco estudada e com muitos aspectos em aberto. A relaxação desenvolvida na pesquisa permitiu resolver instâncias com mais de vinte mil variáveis binárias. Esses resultados são bons indicadores dos benefícios alcançáveis com os aspectos teóricos da pesquisa, e abrem novas perspectivas para as aplicações, que incluem inovações em nanotecnologia e bioengenhariaAbstract: As the title suggests, the focus this research is the development of a new quadratic relaxation for binary problems, its formalization in theoretical results, and the application of the new concepts in applications to the reliability of electric power networks, segmentation of nerve root images, and distance geometry problems. Mathematical models with binary decision variables can be used to find the best solutions for decision-making process, usually leading to difficult combinatorial optimization problems. The solution to these problems in practical applications requires a high computational effort; therefore, over the past years it has been the subject of research in the area of metaheuristics. The ideas developed in this thesis open new perspectives for addressing these problems using nonlinear optimization approaches, an area that has been populated by very efficient solvers. The initial developments explore the formal aspects of the relaxation in the context of a quadratic unconstrained binary optimization problem. The use of the proposed relaxation allows to create three structures to deal with this class of problems, and explores the objective function convexity to improve the computational performance. Case studies compare the proposed relaxation with the previous relaxations proposed in the literature. Three new applications were developed to explore the theoretical developments of this research. The first application concerns the improvement of the reliability of electric power distribution networks. Specifically, it deals with the problem of defining the best allocation for remote fault sensor, allowing to reduce the consequence of the faults and to improve the resilience of the networks. The second application explores the segmentation of medical images related to nerve root structures. The proposed approach regards the segmentation problem as a binary optimization problem, where measuring each axon is equivalent to finding a Hamiltonian cycle for a variant of the traveling salesman problem; the solution to these problems provides the descriptive statistics of the axon set, including the number of axons, their diameters, and the area used by each axon. The last application designs a mathematical model for the unassigned distance geometry problem, an incipient research area with many open problems. The relaxation developed in this research allowed to solve instances with more than twenty thousand binary variables. These results can be seen as good indicators of the benefits attainable with the theoretical aspects of the research, and opens new perspectives for applications, which include innovations in nanotechnology and bio-engineeringDoutoradoAutomaçãoDoutora em Engenharia Elétrica148400/2016-7CNP

Mathematical Programming Formulations for the Optimal Placement of Imperfect Detectors with Applications to Flammable Gas Detection and Mitigation Systems

The placement of detectors in mitigation systems is a difficult problem usually addressed in the industry via qualitative and semiquantative approaches. Simplifications are used to circumvent difficulties regarding problem size, parameter uncertainty, and lack of information concerning leak development. Given recent improvement of consequence modeling tools, the use of a stochastic Mixed-Integer Linear Programming (MILP) formulation (SP) was previously proposed to quantitatively approach this problem. This formulation minimizes the expected damage over a large set of gas leak scenarios while assuming perfect detectors. In reality gas detectors are prone to false positives and false negatives. Two solutions are usually implemented in the process industries. First, additional confirmation from several detectors (i.e., voting) is required before emergency actions are triggered in order to avoid false positives. Second, in order to avoid false negatives, the unavailability of the detectors is considered in the placement strategy. Unavailability corresponds to the probability that the detector will not be able to perform its intended function when required. In the first part of this dissertation, two problem formulations were developed and validated to address the issue of imperfect detectors: minimization of expected damage considering unavailability (SP-U) and minimization of the expected damage considering unavailability and voting (SP-UV). SP-U and SP-UV placement results were compared with those obtained assuming perfect detectors. Results demonstrate that explicit consideration of unavailability and voting effects alters the final detector placement. Quantitative risk can be significantly higher if we neglect these issues when solving for the optimal placement. Furthermore, SP-UV placement results were compared with those of four existing approaches for gas detector placement using three different performance metrics in accordance to the objectives of gas detection systems. Results provide further evidence on the effectiveness of the use of dispersion simulations, and mathematical programming, to supplement the gas detector placement problem. Formulation SP-U assumes a uniform unavailability across all detector types and locations. In the second part of this work, this assumption is relaxed via formulation SPqt, which considers non-uniform dynamic detector unavailabilities. Relaxing this assumption results in a Mixed-Integer NonLinear Programming (MINLP) formulation. SPqt, being an extension of SP-U, explicitly considers di↵erent backup detection levels, allowing an approximation where the maximum degree of the nonlinear products considered can be determined by the modeler. The effect of reducing the number of detection levels was analyzed. For the problem, results shown that two detection levels are sufficient to find objective values within 1% of the optimal solution. Considering two detection levels reduces the MINLP formulation to a zero-one quadratic formulation (SPqt-Q). A solution quality comparison between SPqt-Q and approximate solution strategies previously proposed in the literature demonstrates its suitability to obtain approximate answers for the general nonlinear problem. Two exact linear reformulation strategies (SPqt-L1 and SPqt-L2) were proposed for SPqt-Q and validated from the computationally efficiency perspective. All the results presented were obtained by using four real data sets provided by Gex-Con. The data corresponds to FLACS CFD dispersion simulations including the full geometric features of an offshore facility and capturing the uncertainty in the leak characteristics. Additionally, real unavailability values were obtained from industry gas detector reliability databases. The work presented here constitutes a step forward toward the achievement of a realistic detector placement formulation that includes current industrial practice for these important safety systems

Bilevel linear programs: generalized models for the lower-level reaction set and related problems

Bilevel programming forms a class of optimization problems that model hierarchical relation between two independent decision-makers, namely, the leader and the follower, in a collaborative or conflicting setting. Decisions in this hierarchical structure are made sequentially where the leader decides first and then the follower responds by solving an optimization problem, which is parameterized by the leader's decisions. The follower's reaction, in return, affects the leader's decision, usually through shaping the leader's objective function. Thus, the leader should take into account the follower's response in the decision-making process. A key assumption in bilevel optimization is that both participants, the leader and the follower, solve their problems optimally. However, this assumption does not hold in many important application areas because: (i) there is no known efficient method to solve the lower-level formulation to optimality; (ii) the follower either is not sufficiently sophisticated or does not have the required computational resources to find an optimal solution to the lower-level problem in a timely manner; or (iii) the follower might be willing to give up a portion of his/her optimal objective function value in order to inflict more damage to the leader. This dissertation mainly focuses on developing approaches to model such situations in which the follower does not necessarily return an optimal solution of the lower-level problem as a response to the leader's action. That is, we assume that the follower's reaction set may include both exact and inexact solutions of the lower-level problem. Therefore, we study a generalized class of the follower's reaction sets. This is arguably the case in many application areas in practice, thus our approach contributes to closing the gap between the theory and practice in the bilevel optimization area. In addition, we develop a method to solve bilevel problems through single-level reformulations under the assumption that the lower-level problem is a linear program. The most common technique for such transformations is to replace the lower-level linear optimization problem by its KKT optimality conditions. We propose an alternative technique for a broad class of bilevel linear integer problems, based on the strong duality property of linear programs and compare its performance against the current methods. Finally, we explore bilevel models in an application setting of the pediatric vaccine pricing problem

New Techniques for Polynomial System Solving

Since any encryption map may be viewed as a polynomial map between finite dimensional vector spaces over finite fields, the security of a cryptosystem can be examined by studying the difficulty of solving large systems of multivariate polynomial equations. Therefore, algebraic attacks lead to the task of solving polynomial systems over finite fields. In this thesis, we study several new algebraic techniques for polynomial system solving over finite fields, especially over the finite field with two elements. Instead of using traditional Gröbner basis techniques we focus on highly developed methods from several other areas like linear algebra, discrete optimization, numerical analysis and number theory. We study some techniques from combinatorial optimization to transform a polynomial system solving problem into a (sparse) linear algebra problem. We highlight two new kinds of hybrid techniques. The first kind combines the concept of transforming combinatorial infeasibility proofs to large systems of linear equations and the concept of mutants (finding special lower degree polynomials). The second kind uses the concept of mutants to optimize the Border Basis Algorithm. We study recent suggestions of transferring a system of polynomial equations over the finite field with two elements into a system of polynomial equalities and inequalities over the set of integers (respectively over the set of reals). In particular, we develop several techniques and strategies for converting the polynomial system of equations over the field with two elements to a polynomial system of equalities and inequalities over the reals (respectively over the set of integers). This enables us to make use of several algorithms in the field of discrete optimization and number theory. Furthermore, this also enables us to investigate the use of numerical analysis techniques such as the homotopy continuation methods and Newton's method. In each case several conversion techniques have been developed, optimized and implemented. Finally, the efficiency of the developed techniques and strategies is examined using standard cryptographic examples such as CTC and HFE. Our experimental results show that most of the techniques developed are highly competitive to state-of-the-art algebraic techniques