Academic literature on the topic 'Lagrangian relaxation bounds'

Introduction. Due to the fact that quadratic extremal problems are generally NP-hard, various convex relaxations to find bounds for their global extrema are used, namely, Lagrangian relaxation, SDP-relaxation, SOCP-relaxation, LP-relaxation, and others. This article investigates a dual bound that results from the Lagrangian relaxation of all constraints of quadratic extremal problem. The main issue when using this approach for solving quadratic extremal problems is the quality of the obtained bounds (the magnitude of the duality gap) and the possibility to improve them. While for quadratic convex optimization problems such bounds are exact, in other cases this issue is rather complicated. In non-convex cases, to improve the dual bounds (to reduce the duality gap) the techniques, based on ambiguity of the problem formulation, can be used. The most common of these techniques is an extension of the original quadratic formulation of the problem by introducing the so-called functionally superfluous constraints (additional constraints that result from available constraints). The ways to construct such constraints can be general in nature or they can use specific features of the concrete problems. The purpose of the article is to propose methods for improving the Lagrange dual bounds for quadratic extremal problems by using technique of functionally superfluous constraints; to present examples of constructing such constraints. Results. The general concept of using functionally superfluous constraints for improving the Lagrange dual bounds for quadratic extremal problems is considered. Methods of constructing such constraints are presented. In particular, the method proposed by N.Z. Shor for constructing functionally superfluous constraints for quadratic problems of general form is presented in generalized and schematized forms. Also it is pointed out that other special techniques, which employ the features of specific problems for constructing functionally superfluous constraints, can be used. Conclusions. In order to improve dual bounds for quadratic extremal problems, one can use various families of functionally superfluous constraints, both of general and specific type. In some cases, their application can improve bounds or even provide an opportunity to obtain exact values of global extrema.

We consider dynamic selection problems, where a decision maker repeatedly selects a set of items from a larger collection of available items. A classic example is the dynamic assortment problem with demand learning, where a retailer chooses items to offer for sale subject to a display space constraint. The retailer may adjust the assortment over time in response to the observed demand. These dynamic selection problems are naturally formulated as stochastic dynamic programs (DPs) but are difficult to solve because the optimal selection decisions depend on the states of all items. In this paper, we study heuristic policies for dynamic selection problems and provide upper bounds on the performance of an optimal policy that can be used to assess the performance of a heuristic policy. The policies and bounds that we consider are based on a Lagrangian relaxation of the DP that relaxes the constraint limiting the number of items that may be selected. We characterize the performance of the Lagrangian index policy and bound and show that, under mild conditions, these policies and bounds are asymptotically optimal for problems with many items; mixed policies and tiebreaking play an essential role in the analysis of these index policies and can have a surprising impact on performance. We demonstrate these policies and bounds in two large scale examples: a dynamic assortment problem with demand learning and an applicant screening problem. This paper was accepted by Yinyu Ye, optimization.

This paper explores the connections between the classical maximum clique problem and its edge-weighted generalization, the maximum edge weight clique (MEWC) problem. As a result, a new analytic upper bound on the clique number of a graph is obtained and an exact algorithm for solving the MEWC problem is developed. The bound on the clique number is derived using a Lagrangian relaxation of an integer (linear) programming formulation of the MEWC problem. Furthermore, coloring-based bounds on the clique number are used in a novel upper-bounding scheme for the MEWC problem. This scheme is employed within a combinatorial branch-and-bound framework, yielding an exact algorithm for the MEWC problem. Results of computational experiments demonstrate a superior performance of the proposed algorithm compared with existing approaches.

The fault rate in equipment increases significantly along with the service life of the equipment, especially for multiple fault. Typically, the Bayesian theory is used to construct the model of faults, and intelligent algorithm is used to solve the model. Lagrangian relaxation algorithm can be adopted to solve multiple fault diagnosis models. But the mathematical derivation process may be complex, while the updating method for Lagrangian multiplier is limited and it may fall into a local optimal solution. The particle swarm optimization (PSO) algorithm is a global search algorithm. In this paper, an improved Lagrange-particle swarm optimization algorithm is proposed. The updating of the Lagrangian multipliers is with the PSO algorithm for global searching. The difference between the upper and lower bounds is proposed to construct the fitness function of PSO. The multiple fault diagnosis model can be solved by the improved Lagrange-particle swarm optimization algorithm. Experiment on a case study of sensor data-based multiple fault diagnosis verifies the effectiveness and robustness of the proposed method.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O problema de dimensionamento de lotes é um problema de otimização da produção, em que o objetivo é planejar a quantidade de itens a ser produzida em várias, ou única, máquinas em cada período ao longo do horizonte de tempo, de modo a tender uma demanda e otimizar uma função objetivo. Este trabalho aborda o problema de dimensionamento de lotes em um único estágio em um ambiente com máquinas paralelas distintas. Cada item pode ser produzido em qualquer máquina, acarretando um tempo de preparação que é gasto antes de começar a produção. O objetivo do trabalho consiste em obter limitantes inferiores de boa qualidade para este problema. Para tanto, é desenvolvido um método de solução baseado numa reformulação do problema a e na relaxação lagrangiana de um conjunto de restrições. Alguns resultados computacionais são apresentados algumas propostas futuras para a continuidade do trabalho.
The lot-sizing problem is a production optimization problem, where the objective is to plan the quantity of items to be produced in multiple, or single, machines in each period over a time horizon, in order to satisfy a demand and optimize an objective function. This work addresses the single stage parallel machine lot-sizing problem. Each item can be produced on any machine, and incur a setup time before to start the production. The objective of this work is to lower bounds of good quality for this problem. A solution method is developed based on a reformulation of the problem and the Lagrangian relaxation of a set of constrainsts. Some computational results are presented comparing the proposed method with a method from the literature, and, some future researches are proposed.

Orientador: Silvio Alexandrede Araujo
Banca: Bernardo Sobrinho Simões de Almada Lobo
Banca: Franklina Maria Bragion Toledo
Resumo: O problema de dimensionamento de lotes é um problema de otimização da produção, em que o objetivo é planejar a quantidade de itens a ser produzida em várias, ou única, máquinas em cada período ao longo do horizonte de tempo, de modo a tender uma demanda e otimizar uma função objetivo. Este trabalho aborda o problema de dimensionamento de lotes em um único estágio em um ambiente com máquinas paralelas distintas. Cada item pode ser produzido em qualquer máquina, acarretando um tempo de preparação que é gasto antes de começar a produção. O objetivo do trabalho consiste em obter limitantes inferiores de boa qualidade para este problema. Para tanto, é desenvolvido um método de solução baseado numa reformulação do problema a e na relaxação lagrangiana de um conjunto de restrições. Alguns resultados computacionais são apresentados algumas propostas futuras para a continuidade do trabalho.
Abstract: The lot-sizing problem is a production optimization problem, where the objective is to plan the quantity of items to be produced in multiple, or single, machines in each period over a time horizon, in order to satisfy a demand and optimize an objective function. This work addresses the single stage parallel machine lot-sizing problem. Each item can be produced on any machine, and incur a setup time before to start the production. The objective of this work is to lower bounds of good quality for this problem. A solution method is developed based on a reformulation of the problem and the Lagrangian relaxation of a set of constrainsts. Some computational results are presented comparing the proposed method with a method from the literature, and, some future researches are proposed.
Mestre

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Universidade Federal de Sao Carlos
The purpose of this work is to develop exact methods, based on Lagrangean and Surrogate relaxation, with good performance to solve the manufacturer s pallet loading problem. This problem consists of orthogonally arranging the maximum number of rectangles of sizes (l,w) and (w,l) into a larger rectangle (L,W) without overlapping. Such methods involve a tree search procedure of branch and bound type and they use, in each node of the branch and bound tree, bounds derived from Lagrangean and/or Surrogate relaxations of a 0-1 linear programming formulation. Subgradient optimization algorithms are used to optimize such bounds. Problem reduction tests and Lagrangean and Surrogate heuristics are also applied in the subgradient optimization to obtain good feasible solution. Computational experiments were performed with instances from the literature and also real instances obtained from a carrier. The results show that the methods are able to solve these instances, on average, more quickly than other exact methods, including the software GAMS/CPLEX.
O objetivo deste trabalho é desenvolver métodos exatos, baseados em relaxação Lagrangiana e Surrogate, com bom desempenho para resolver o problema de carregamento de paletes do produtor. Tal problema consiste em arranjar ortogonalmente e sem sobreposição o máximo número de retângulos de dimensões ( , ) l w ou ( , ) w l sobre um retângulo maior ( , ) L W . Tais métodos exatos são procedimentos de busca em árvore do tipo branch and bound que, em cada nó, utilizam limitantes derivados de relaxações Lagrangiana e/ou Surrogate de uma formulação de programação linear 0 1 − . Algoritmos de otimização do subgradiente são usados para otimizar estes limitantes. São aplicados ainda testes de redução do problema e heurísticas Lagrangiana e Surrogate na otimização do subgradiente para obter boas soluções factíveis. Testes computacionais foram realizados utilizando exemplos da literatura e exemplos reais, obtidos de uma transportadora. Os resultados mostram que os métodos são capazes de resolvê-los, em média, mais rapidamente do que outros métodos exatos, incluindo o software GAMS/CPLEX.

The rapid technological advancement in the past century resulted in a decreased life cycle of a large number of products and, consequently, increased the rate of technological obsolescence. The disposal of obsolete products has resulted in rapid landfilling and now poses a major environmental threat. The governments in many countries around the world have started imposing regulations to curb uncontrolled product disposal. The consumers, too, are now aware of adverse effects of product disposal on environment and increasingly favor environmentally benign products.

In the wake of imminent stringent government regulations and the consumer awareness about ecosystem-friendly products, the manufacturers need to think about the alternatives to product disposal. One way to deal with this problem is to disassemble an obsolete product and utilize some of its components/subassemblies in the manufacturing of new products. This seems to be a promising solution because products now-a-days are made in accordance with the highest quality standards and, although an obsolete product may not be in the required functional state as a whole, it is possible that several of its components or subassemblies are still in near perfect condition.

However, product disassembly is a complex task requiring human labor as well as automated processes and, consequently, a huge amount of monetary investment. This research addresses a disassembly optimization problem, which aims at minimizing the costs associated with the disassembly process (namely, the costs of breaking the joints and the sequence dependent set-up cost associated with disassembly operations), while maximizing the benefits resulting from recovery of components/subassemblies from a product. We provide a mathematical abstraction of the disassembly optimization problem in the form of integer-programming models. One of our formulations includes a new way of modeling the subtour elimination constraints (SECs), which are usually encountered in the well-known traveling salesman problems. Based on these SECs, a new valid formulation for asymmetric traveling salesman problem (ATSP) was developed. The ATSP formulation was further extended to obtain a valid formulation for the precedence constrained ATSP. A detailed experimentation was conducted to compare the performance of the proposed formulations with that of other well-known formulations discussed in the literature. Our results indicate that in comparison to other well-known formulations, the proposed formulations are quite promising in terms of the LP relaxation bounds obtained and the number of branch and bound nodes explored to reach an optimal integer solution. These new formulations along with the results of experimentation are presented in Appendix A.

To solve the disassembly optimization problem, a three-phase iterative solution procedure was developed that can determine optimal or near optimal disassembly plans for complex assemblies. The first phase helps in obtaining an upper bound on our maximization problem through an application of a Lagrangian relaxation scheme. The second phase helps to further improve this bound through addition of a few valid inequalities in our models. In the third phase, we fix some of our decision variables based on the solutions obtained in the iterations of phases 1 and 2 and then implement a branch and bound scheme to obtain the final solution. We test our procedure on several randomly generated data sets and identify the factors that render a problem to be computationally difficult. Also, we establish the practical usefulness of our approach through case studies on the disassembly of a computer processor and a laser printer.

Master of Science

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

Le problème d’ordre linéaire (LOP) a reçu beaucoup d’attention dans différents domaines d’application, allant de l’archéologie à l’ordonnancement en passant par l’économie et même de la psychologie mathématique. Ce problème est aussi connu pour être parmi les problèmes NP-difficiles. Nous considérons dans cette thèse une variante de (LOP) sous contrainte de cardinalité. Nous cherchons donc un ordre linéaire d’un sous-ensemble de sommets du graphe de préférences de cardinalité fixée et de poids maximum. Ce problème, appelé (FCLOP) pour ’fixed-cardinality linear ordering problem’, n’a pas été étudié en tant que tel dans la littérature scientifique même si plusieurs applications dans les domaines de macro-économie, de classification dominante ou de transport maritime existent concrètement. On retrouve en fait ses caractéristiques dans les modèles étendus de sous-graphes acycliques. Le problème d’ordre linéaire est déjà connu comme un problème NP-difficile et il a donné lieu à de nombreuses études, tant théoriques sur la structure polyédrale de l’ensemble des solutions réalisables en variables 0-1 que numériques grâce à des techniques de relaxation et de séparation progressive. Cependant on voit qu’il existe de nombreux cas dans la littérature, dans lesquelles des solveurs de Programmation Linéaire en nombres entiers comme CPLEX peuvent en résoudre certaines instances en moins de 10 secondes, mais une fois que la cardinalité est limitée, ces mêmes instances deviennent très difficiles à résoudre. Sur les aspects polyédraux, nous avons étudié le polytope de FCLOP, défini plusieurs classes d’inégalités valides et identifié la dimension ainsi que certaines inégalités qui définissent des facettes pour le polytope de FCLOP. Nous avons introduit un algorithme Relax-and-Cut basé sur ces résultats pour résoudre les instances du problème. Dans cette étude, nous nous sommes également concentrés sur la relaxation Lagrangienne pour résoudre ces cas difficiles. Nous avons étudié différentes stratégies de relaxation et nous avons comparé les bornes duales par rapport à la consolidation obtenue à partir de chaque stratégie de relâcher les contraintes afin de détecter le sous-ensemble des contraintes le plus approprié. Les résultats numériques montrent que nous pouvons trouver des bornes duales de très haute qualité. Nous avons également mis en place une méthode de décomposition Lagrangienne. Dans ce but, nous avons décomposé le modèle de FCLOP en trois sous-problèmes (au lieu de seulement deux) associés aux contraintes de ’tournoi’, de ’graphes sans circuits’ et de ’cardinalité’. Les résultats numériques montrent une amélioration significative de la qualité des bornes duales pour plusieurs cas. Nous avons aussi mis en oeuvre une méthode de plans sécants (cutting plane algorithm) basée sur la relaxation pure des contraintes de circuits. Dans cette méthode, on a relâché une partie des contraintes et on les a ajoutées au modèle au cas où il y a des de/des violations. Les résultats numériques montrent des performances prometteuses quant à la réduction du temps de calcul et à la résolution d’instances difficiles hors d’atteinte des solveurs classiques en PLNE
Linear Ordering Problem (LOP) has receive significant attention in different areas of application, ranging from transportation and scheduling to economics and even archeology and mathematical psychology. It is classified as a NP-hard problem. Assume a complete weighted directed graph on V n , |V n |= n. A permutation of the elements of this finite set of vertices is a linear order. Now let p be a given fixed integer number, 0 ≤ p ≤ n. The p-Fixed Cardinality Linear Ordering Problem (FCLOP) is looking for a subset of vertices containing p nodes and a linear order on the nodes in S. Graphically, there exists exactly one directed arc between every pair of vertices in an LOP feasible solution, which is also a complete cycle-free digraph and the objective is to maximize the sum of the weights of all the arcs in a feasible solution. In the FCLOP, we are looking for a subset S ⊆ V n such that |S|= p and an LOP on these S nodes. Hence the objective is to find the best subset of the nodes and an LOP over these p nodes that maximize the sum of the weights of all the arcs in the solution. Graphically, a feasible solution of the FCLOP is a complete cycle-free digraph on S plus a set of n − p vertices that are not connected to any of the other vertices. There are several studies available in the literature focused on polyhedral aspects of the linear ordering problem as well as various exact and heuristic solution methods. The fixed cardinality linear ordering problem is presented for the first time in this PhD study, so as far as we know, there is no other study in the literature that has studied this problem. The linear ordering problem is already known as a NP-hard problem. However one sees that there exist many instances in the literature that can be solved by CPLEX in less than 10 seconds (when p = n), but once the cardinality number is limited to p (p < n), the instance is not anymore solvable due to the memory issue. We have studied the polytope corresponding to the FCLOP for different cardinality values. We have identified dimension of the polytope, proposed several classes of valid inequalities and showed that among these sets of valid inequalities, some of them are defining facets for the FCLOP polytope for different cardinality values. We have then introduced a Relax-and-Cut algorithm based on these results to solve instances of the FCLOP. To solve the instances of the problem, in the beginning, we have applied the Lagrangian relaxation algorithm. We have studied different relaxation strategies and compared the dual bound obtained from each case to detect the most suitable subproblem. Numerical results show that some of the relaxation strategies result better dual bound and some other contribute more in reducing the computational time and provide a relatively good dual bound in a shorter time. We have also implemented a Lagrangian decomposition algorithm, decom-6 posing the FCLOP model to three subproblems (instead of only two subproblems). The interest of decomposing the FCLOP model to three subproblems comes mostly from the nature of the three subproblems, which are relatively quite easier to solve compared to the initial FCLOP model. Numerical results show a significant improvement in the quality of dual bounds for several instances. We could also obtain relatively quite better dual bounds in a shorter time comparing to the other relaxation strategies. We have proposed a cutting plane algorithm based on the pure relaxation strategy. In this algorithm, we firstly relax a subset of constraints that due to the problem structure, a very few number of them are active. Then in the course of the branch-and-bound tree we verify if there exist any violated constraint among the relaxed constraints or. Then the characterized violated constraints will be globally added to the model. (...)

Building a linear fitting model for a given interval-valued data set is challenging since the minimization of the residue function leads to a huge combinatorial problem. To overcome such a difficulty, this article proposes a new semidefinite programming-based method for implementing linear fitting to interval-valued data. First, the fitting model is cast to a problem of quadratically constrained quadratic programming (QCQP), and then two formulae are derived to develop the lower bound on the optimal value of the nonconvex QCQP by semidefinite relaxation and Lagrangian relaxation. In many cases, this method can solve the fitting problem by giving the exact optimal solution. Even though the lower bound is not the optimal value, it is still a good approximation of the global optimal solution. Experimental studies on different fitting problems of different scales demonstrate the good performance and stability of our method. Furthermore, the proposed method performs very well in solving relatively large-scale interval-fitting problems.

This paper addressed the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (e.g., robustness to bounded norm adversarial perturbations). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime, i.e., it can be stopped at any time and a valid bound on the maximum violation can be obtained. Finally, we highlight how this approach can be used to train models that are amenable to verification.