Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

An Integer Programming approach to Bayesian Network Structure Learning

We study the problem of learning a Bayesian Network structure from data using an Integer Programming approach. We study the existing approaches, an in particular some recent works that formulate the problem as an Integer Programming model. By discussing some weaknesses of the existing approaches, we propose an alternative solution, based on a statistical sparsification of the search space. Results show how our approach can lead to promising results, especially for large network

Learning to compare nodes in branch and bound with graph neural networks

En informatique, la résolution de problèmes NP-difficiles en un temps raisonnable est d’une grande importance : optimisation de la chaîne d’approvisionnement, planification, routage, alignement de séquences biologiques multiples, inference dans les modèles graphiques pro- babilistes, et même certains problèmes de cryptographie sont tous des examples de la classe NP-complet. En pratique, nous modélisons beaucoup d’entre eux comme un problème d’op- timisation en nombre entier, que nous résolvons à l’aide de la méthodologie séparation et évaluation. Un algorithme de ce style divise un espace de recherche pour l’explorer récursi- vement (séparation), et obtient des bornes d’optimalité en résolvant des relaxations linéaires sur les sous-espaces (évaluation). Pour spécifier un algorithme, il faut définir plusieurs pa- ramètres, tel que la manière d’explorer les espaces de recherche, de diviser une recherche l’espace une fois exploré, ou de renforcer les relaxations linéaires. Ces politiques peuvent influencer considérablement la performance de résolution. Ce travail se concentre sur une nouvelle manière de dériver politique de recherche, c’est à dire le choix du prochain sous-espace à séparer étant donné une partition en cours, en nous servant de l’apprentissage automatique profond. Premièrement, nous collectons des données résumant, sur une collection de problèmes donnés, quels sous-espaces contiennent l’optimum et quels ne le contiennent pas. En représentant ces sous-espaces sous forme de graphes bipartis qui capturent leurs caractéristiques, nous entraînons un réseau de neurones graphiques à déterminer la probabilité qu’un sous-espace contienne la solution optimale par apprentissage supervisé. Le choix d’un tel modèle est particulièrement utile car il peut s’adapter à des problèmes de différente taille sans modifications. Nous montrons que notre approche bat celle de nos concurrents, consistant à des modèles d’apprentissage automatique plus simples entraînés à partir des statistiques du solveur, ainsi que la politique par défaut de SCIP, un solveur open-source compétitif, sur trois familles NP-dures: des problèmes de recherche de stables de taille maximum, de flots de réseau multicommodité à charge fixe, et de satisfiabilité maximum.In computer science, solving NP-hard problems in a reasonable time is of great importance, such as in supply chain optimization, scheduling, routing, multiple biological sequence align- ment, inference in probabilistic graphical models, and even some problems in cryptography. In practice, we model many of them as a mixed integer linear optimization problem, which we solve using the branch and bound framework. An algorithm of this style divides a search space to explore it recursively (branch) and obtains optimality bounds by solving linear relaxations in such sub-spaces (bound). To specify an algorithm, one must set several pa- rameters, such as how to explore search spaces, how to divide a search space once it has been explored, or how to tighten these linear relaxations. These policies can significantly influence resolution performance. This work focuses on a novel method for deriving a search policy, that is, a rule for select- ing the next sub-space to explore given a current partitioning, using deep machine learning. First, we collect data summarizing which subspaces contain the optimum, and which do not. By representing these sub-spaces as bipartite graphs encoding their characteristics, we train a graph neural network to determine the probability that a subspace contains the optimal so- lution by supervised learning. The choice of such design is particularly useful as the machine learning model can automatically adapt to problems of different sizes without modifications. We show that our approach beats the one of our competitors, consisting of simpler machine learning models trained from solver statistics, as well as the default policy of SCIP, a state- of-the-art open-source solver, on three NP-hard benchmarks: generalized independent set, fixed-charge multicommodity network flow, and maximum satisfiability problems

DASH: Dynamic Approach for Switching Heuristics

Complete tree search is a highly effective method for tackling MIP problems, and over the years, a plethora of branching heuristics have been introduced. Recently, portfolio algorithms have taken the process a step further, trying to predict the best heuristic for each instance at hand. This thesis identifies a method which decides the best time to switch the branching heuristic and it is shown how such\na system can be trained efficientl

Promoting Generalization for Exact Solvers via Adversarial Instance Augmentation

Machine learning has been successfully applied to improve the efficiency of Mixed-Integer Linear Programming (MILP) solvers. However, the learning-based solvers often suffer from severe performance degradation on unseen MILP instances -- especially on large-scale instances from a perturbed environment -- due to the limited diversity of training distributions. To tackle this problem, we propose a novel approach, which is called Adversarial Instance Augmentation and does not require to know the problem type for new instance generation, to promote data diversity for learning-based branching modules in the branch-and-bound (B&B) Solvers (AdaSolver). We use the bipartite graph representations for MILP instances and obtain various perturbed instances to regularize the solver by augmenting the graph structures with a learned augmentation policy. The major technical contribution of AdaSolver is that we formulate the non-differentiable instance augmentation as a contextual bandit problem and adversarially train the learning-based solver and augmentation policy, enabling efficient gradient-based training of the augmentation policy. To the best of our knowledge, AdaSolver is the first general and effective framework for understanding and improving the generalization of both imitation-learning-based (IL-based) and reinforcement-learning-based (RL-based) B&B solvers. Extensive experiments demonstrate that by producing various augmented instances, AdaSolver leads to a remarkable efficiency improvement across various distributions

CONSTRAINED MULTI-GROUP PROJECT ALLOCATION USING MAHALANOBIS DISTANCE

Optimal allocation is one of the most active research areas in operation research using binary integer variables. The allocation of multi constrained projects among several options available along a given planning horizon is an especially significant problem in the general area of item classification. The main goal of this dissertation is to develop an analytical approach for selecting projects that would be most attractive from an economic point of view to be developed or allocated among several options, such as in-house engineers and private contractors (in transportation projects). A relevant limiting resource in addition to the availability of funds is the in-house manpower availability. In this thesis, the concept of Mahalanobis distance (MD) will be used as the classification criterion. This is a generalization of the Euclidean distance that takes into account the correlation of the characteristics defining the scope of a project. The desirability of a given project to be allocated to an option is defined in terms of its MD to that particular option. Ideally, each project should be allocated to its closest option. This, however, may not be possible because of the available levels of each relevant resource. The allocation process is formulated mathematically using two Binary Integer Programming (BIP) models. The first formulation maximizes the dollar value of benefits derived by the traveling public from those projects being implemented subject to a budget, total sum of MD, and in-house manpower constraints. The second formulation minimizes the total sum of MD subject to a budget and the in-house manpower constraints. The proposed solution methodology for the BIP models is based on the branchand- bound method. In particular, one of the contributions of this dissertation is the development of a strategy for branching variables and node selection that is consistent with allocation priorities based on MD to improve the branch-and-bound performance level as well as handle a large scale application. The suggested allocation process includes: (a) multiple allocation groups; (b) multiple constraints; (c) different BIP models. Numerical experiments with different projects and options are considered to illustrate the application of the proposed approach

Learning to Branch in Combinatorial Optimization with Graph Pointer Networks

Branch-and-bound is a typical way to solve combinatorial optimization problems. This paper proposes a graph pointer network model for learning the variable selection policy in the branch-and-bound. We extract the graph features, global features and historical features to represent the solver state. The proposed model, which combines the graph neural network and the pointer mechanism, can effectively map from the solver state to the branching variable decisions. The model is trained to imitate the classic strong branching expert rule by a designed top-k Kullback-Leibler divergence loss function. Experiments on a series of benchmark problems demonstrate that the proposed approach significantly outperforms the widely used expert-designed branching rules. Our approach also outperforms the state-of-the-art machine-learning-based branch-and-bound methods in terms of solving speed and search tree size on all the test instances. In addition, the model can generalize to unseen instances and scale to larger instances

Mathematical Optimization of the Tactical Allocation of Machining Resources in Aerospace Industry

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers (which includes, measures of quality and production lead-times) and to maintain control of the tied-up working capital. We introduce a new multi-item, multi-level capacitated planning model with a medium-to-long term planning horizon. The model can be used by most companies having functional workshops where costly and/or time- and resource demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimizes the maximum excess resource loading above a given loading threshold, while incurring as low qualification costs as possible. In Paper I (Bi-objective optimization of the tactical allocation of jobtypes to machines), we propose a new bi-objective mathematical optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. Another contribution is a modified version of the bi-directional $\epsilon$ -constraint method especially tailored for our problem. We perform numerical tests on industrial test cases generated for our class of problem which indicates computational superiority of our method over conventional solution approaches. In Paper II (Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs), we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. We also suggest a solution approach for identifying all the relevant robust efficient (RE) solutions. Our proposed approach is significantly faster than an existing approach for robust bi-objective optimization problems