A computational comparison of compact MILP formulations for the zero forcing number

Consider a graph where some of its vertices are colored. A colored vertex with a single uncolored neighbor forces that neighbor to become colored. A zero forcing set is a set of colored vertices that forces all vertices to become colored. The zero forcing number is the size of a minimum forcing set. Finding the minimum forcing set of a graph is NP-hard. We give a new compact mixed integer linear programming formulation (MILP) for this problem, and analyse this formulation and establish relation to an existing compact formulation and to two variants. In order to solve large size instances we propose a sequential search algorithm which can also be used as a heuristic to derive upper bounds for the zero forcing number. A computational study using Xpress (a MILP solver) is conducted to test the performances of the discussed compact formulations and the sequential search algorithm. We report results on cubic, Watts-Strogatz and randomly generated graphs with 10, 20 and 30 vertices.publishe

Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Compact Representation of Time-Index Job Shop Problems Using a Bit-Vector Formulation

The Job Shop Scheduling Problem (JSP) is a combinatorial optimization problem where jobs visit single-capacity machines while minimizing a cost function, typically the makespan. The problem can be extended to fit typical industrial scenarios such as flexible assembly shop floors or for coordinating fleets of automated vehicles. General purpose optimizers can handle extended versions of the problem that typically arise in industrial problems. Mixed Integer Linear Programming (MILP) solvers and recently optimizing Satisfiability Modulo Theory (SMT) solvers can be used as general solvers for JSP problems. There exist different formulations of JSP problems, among them the time-index (TI) model. The TI offers the advantage of providing strong lower bounds, though its drawback is the model size.In this paper we present a new formulation of the TI model suitable for optimizing SMT-solvers that support bit-vector theories. The new formulation is significantly more compact than the standard TI formulation and is thus reducing one of the major issues with the TI model.We benchmark two different optimizing SMT solvers supporting bit-vector theories, comparing the standard formulation of the TI to the new formulation on a set of benchmark instances. The computational analysis shows that the new formulation outperforms the standard one, being up to twice faster and regardless of the solver employed; moreover the model generated with the new formulation is considerably smaller than with the standard formulation

Stability Verification of Neural Network Controllers using Mixed-Integer Programming

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity

Models, methods and algorithms for supply chain planning

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.An outline of supply chains and differences in the problem types is given. The motivation for a generic framework is discussed and explored. A conceptual model is presented along with it application to real world situations; and from this a database model is developed. A MIP and CP implementations are presented; along with alternative formulation which can be use to solve the problems. A local search solution algorithm is presented and shown to have significant benefits. Problem instances are presented which are used to validate the generic models, including a large manufacture and distribution problem. This larger problem instance is not only used to explore the implementation of the models presented, but also to explore the practically of the use of alternative formulation and solving techniques within the generic framework and the effectiveness of such methods including the neighbourhood search solving method. A stochastic dimension to the generic framework is explored, and solution techniques for this extension are explored, demonstrating the use of solution analysis to allow problem simplification and better solutions to be found. Finally the local search algorithm is applied to the larger models that arise from inclusion of scenarios, and the methods is demonstrated to be powerful for finding solutions for these large model that were insoluble using the MIP on the same hardware

Approximated Perspective Relaxations: a Project&Lift Approach

The Perspective Reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (P^2R) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (AP^2R) whereby the projected formulation is "lifted" back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original P^2R development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the AP^2R bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications

A new mixed-integer modeling approach for capacity-constrained continuous-time scheduling problems

Nowadays, scheduling and resource management are increasingly important issues for organizations. Indeed, they do not only constitute an underlying necessity to make things work properly within the companies, but are and will always be more critical means to reduce costs and get competitive advantage in the market. Different approaches have been typically employed for these problems during the years. Among the others, linear programming techniques represent a valid tool that, despite applicable only to instances of limited dimension, offers an extremely flexible modeling opportunity, able to produce either optimal or approximate solutions of certified quality. In this spirit, the definition of suitable indicator variables and the use of particular constraints are proposed in the present work, with the aim of providing a useful basis for different mathematical models, taking into account scarce resources and other potential limitations. More in detail, a very well-known problem from the literature, the Resource Constrained Project Scheduling Problem, is investigated, and a new mixed-integer linear formulation is introduced, which treats time as a continuous variable. The considered model presents several advantages from the computational point of view, that are deeply studied and compared with those of one of the best methods recently developed in the same field. Extensive experiments reveal the good performances achieved by the proposed formulation over all the KPIs included in the analysis, thus motivating further applications to derived problems, such as the workforce planning and scheduling framework presented at the end of this dissertation

Scaling finite difference methods in large eddy simulation of jet engine noise to the petascale: numerical methods and their efficient and automated implementation

Reduction of jet engine noise has recently become a new arena of competition between aircraft manufacturers. As a relatively new field of research in computational fluid dynamics (CFD), computational aeroacoustics (CAA) prediction of jet engine noise based on large eddy simulation (LES) is a robust and accurate tool that complements the existing theoretical and experimental approaches. In order to satisfy the stringent requirements of CAA on numerical accuracy, finite difference methods in LES-based jet engine noise prediction rely on the implicitly formulated compact spatial partial differentiation and spatial filtering schemes, a crucial component of which is an embedded solver for tridiagonal linear systems spatially oriented along the three coordinate directions of the computational space. Traditionally, researchers and engineers in CAA have employed manually crafted implementations of solvers including the transposition method, the multiblock method and the Schur complement method. Algorithmically, these solvers force a trade-off between numerical accuracy and parallel scalability. Programmingwise, implementing them for each of the three coordinate directions is tediously repetitive and error-prone. ^ In this study, we attempt to tackle both of these two challenges faced by researchers and engineers. We first describe an accurate and scalable tridiagonal linear system solver as a specialization of the truncated SPIKE algorithm and strategies for efficient implementation of the compact spatial partial differentiation and spatial filtering schemes. We then elaborate on two programming models tailored for composing regular grid-based numerical applications including finite difference-based LES of jet engine noise, one based on generalized elemental subroutines and the other based on functional array programming, and the accompanying code optimization and generation methodologies. Through empirical experiments, we demonstrate that truncated SPIKE-based spatial partial differentiation and spatial filtering deliver the theoretically promised optimal scalability in weak scaling conditions and can be implemented using the two programming models with performance on par with handwritten code while significantly reducing the required programming effort

Contributions to robust and bilevel optimization models for decision-making

Los problemas de optimización combinatorios han sido ampliamente estudiados en la literatura especializada desde mediados del siglo pasado. No obstante, en las últimas décadas ha habido un cambio de paradigma en el tratamiento de problemas cada vez más realistas, en los que se incluyen fuentes de aleatoriedad e incertidumbre en los datos, múltiples criterios de optimización y múltiples niveles de decisión. Esta tesis se desarrolla en este contexto. El objetivo principal de la misma es el de construir modelos de optimización que incorporen aspectos inciertos en los parámetros que de nen el problema así como el desarrollo de modelos que incluyan múltiples niveles de decisión. Para dar respuesta a problemas con incertidumbre usaremos los modelos Minmax Regret de Optimización Robusta, mientras que las situaciones con múltiples decisiones secuenciales serán analizadas usando Optimización Binivel. En los Capítulos 2, 3 y 4 se estudian diferentes problemas de decisión bajo incertidumbre a los que se dará una solución robusta que proteja al decisor minimizando el máximo regret en el que puede incurrir. El criterio minmax regret analiza el comportamiento del modelo bajo distintos escenarios posibles, comparando su e ciencia con la e ciencia óptima bajo cada escenario factible. El resultado es una solución con una eviciencia lo más próxima posible a la óptima en el conjunto de las posibles realizaciones de los parámetros desconocidos. En el Capítulo 2 se estudia un problema de diseño de redes en el que los costes, los pares proveedor-cliente y las demandas pueden ser inciertos, y además se utilizan poliedros para modelar la incertidumbre, permitiendo de este modo relaciones de dependencia entre los parámetros. En el Capítulo 3 se proponen, en el contexto de la secuenciación de tareas o la computación grid, versiones del problema del camino más corto y del problema del viajante de comercio en el que el coste de recorrer un arco depende de la posición que este ocupa en el camino, y además algunos de los parámetros que de nen esta función de costes son inciertos. La combinación de la dependencia en los costes y la incertidumbre en los parámetros da lugar a dependencias entre los parámetros desconocidos, que obliga a modelar los posibles escenarios usando conjuntos más generales que los hipercubos, habitualmente utilizados en este contexto. En este capítulo, usaremos poliedros generales para este cometido. Para analizar este primer bloque de aplicaciones, en el Capítulo 4, se analiza un modelo de optimización en el que el conjunto de posibles escenarios puede ser alterado mediante la realización de inversiones en el sistema. En los problemas estudiados en este primer bloque, cada decisión factible es evaluada en base a la reacción más desfavorable que pueda darse en el sistema. En los Capítulos 5 y 6 seguiremos usando esta idea pero ahora se supondrá que esa reacción a la decisión factible inicial está en manos de un adversario o follower. Estos dos capítulos se centran en el estudio de diferentes modelos binivel. La Optimización Binivel aborda problemas en los que existen dos niveles de decisión, con diferentes decisores en cada uno ellos y la decisión se toma de manera jerárquica. En concreto, en el Capítulo 5 se estudian distintos modelos de jación de precios en el contexto de selección de carteras de valores, en los que el intermediario nanciero, que se convierte en decisor, debe jar los costes de invertir en determinados activos y el inversor debe seleccionar su cartera de acuerdo a distintos criterios. Finalmente, en el Capítulo 6 se estudia un problema de localización en el que hay distintos decisores, con intereses contrapuestos, que deben determinar secuencialmente la ubicación de distintas localizaciones. Este modelo de localización binivel se puede aplicar en contextos como la localización de servicios no deseados o peligrosos (plantas de reciclaje, centrales térmicas, etcétera) o en problemas de ataque-defensa. Todos estos modelos se abordan mediante el uso de técnicas de Programación Matemática. De cada uno de ellos se analizan algunas de sus propiedades y se desarrollan formulaciones y algoritmos, que son examinados también desde el punto de vista computacional. Además, se justica la validez de los modelos desde un enfoque de las aplicaciones prácticas. Los modelos presentados en esta tesis comparten la peculiaridad de requerir resolver distintos problemas de optimización encajados.Combinatorial optimization problems have been extensively studied in the specialized literature since the mid-twentieth century. However, in recent decades, there has been a paradigm shift to the treatment of ever more realistic problems, which include sources of randomness and uncertainty in the data, multiple optimization criteria and multiple levels of decision. This thesis concerns the development of such concepts. Our objective is to study optimization models that incorporate uncertainty elements in the parameters de ning the model, as well as the development of optimization models integrating multiple decision levels. In order to consider problems under uncertainty, we use Minmax Regret models from Robust Optimization; whereas the multiplicity and hierarchy in the decision levels is addressed using Bilevel Optimization. In Chapters 2, 3 and 4, we study di erent decision problems under uncertainty to which we give a robust solution that protects the decision-maker minimizing the maximum regret that may occur. This robust criterion analyzes the performance of the system under multiple possible scenarios, comparing its e ciency with the optimum one under each feasible scenario. We obtain, as a result, a solution whose e ciency is as close as possible to the optimal one in the set of feasible realizations of the uncertain parameters. In Chapter 2, we study a network design problem in which the costs, the pairs supplier-customer, and the demands can take uncertain values. Furthermore, the uncertainty in the parameters is modeled via polyhedral sets, thereby allowing relationships among the uncertain parameters. In Chapter 3, we propose time-dependent versions of the shortest path and traveling salesman problems in which the costs of traversing an arc depends on the relative position that the arc occupies in the path. Moreover, we assume that some of the parameters de ning these costs can be uncertain. These models can be applied in the context of task sequencing or grid computing. The incorporation of time-dependencies together with uncertainties in the parameters gives rise to dependencies among the uncertain parameters, which require modeling the possible scenarios using more general sets than hypercubes, normally used in this context. In this chapter, we use general polyhedral sets with this purpose. To nalize this rst block of applications, in Chapter 4, we analyze an optimization model in which the set of possible scenarios can be modi ed by making some investments in the system. In the problems studied in this rst block, each feasible decision is evaluated based on the most unfavorable possible reaction of the system. In Chapters 5 and 6, we will still follow this idea, but assuming that the reaction to the initial feasible decision will be held by a follower or an adversary, instead of assuming the most unfavorable one. These two chapters are focused on the study of some bilevel models. Bilevel Optimization addresses optimization problems with multiple decision levels, di erent decision-makers in each level and a hierarchical decision order. In particular, in Chapter 5, we study some price setting problems in the context of portfolio selection. In these problems, the nancial intermediary becomes a decisionmaker and sets the transaction costs for investing in some securities, and the investor chooses her portfolio according to di erent criteria. Finally, in Chapter 6, we study a location problem with several decision-makers and opposite interests, that must set, sequentially, some location points. This bilevel location model can be applied in practical applications such as the location of semi-obnoxious facilities (power or electricity plants, waste dumps, etc.) or interdiction problems. All these models are stated from a Mathematical Programming perspective, analyzing their properties and developing formulations and algorithms, that are tested from a computational point of view. Furthermore, we pay special attention to justifying the validity of the models from the practical applications point of view. The models presented in this thesis share the characteristic of involving the resolution of nested optimization problems.Premio Extraordinario de Doctorado U