A Kernel Search Algorithm for Virtual Machine Consolidation Problem

Virtual machine consolidation describes the process of reallocation of virtual machines (VMs) on a set of target servers. It can be formulated as a mixed integer linear programming problem which is proven to be an NP-hard problem. In this paper, we propose a kernel search (KS) heuristic algorithm based on hard variable fixing to quickly obtain a high-quality solution for large-scale virtual machine consolidation problems (VMCPs). Since variable fixing strategies in existing KS works may make VMCP infeasible, our proposed KS algorithm employs a more efficient strategy to choose a set of fixed variables according to the corresponding reduced cost. Numerical results on VMCP instances demonstrate that our proposed KS algorithm significantly outperforms the state-of-the-art mixed integer linear programming solver in terms of CPU time, and our proposed strategy of variable fixing significantly improves the efficiency of the KS algorithm as well as the degradation of solution quality can be negligible.Comment: 17 pages, 3 figure

Exploring Large Feature Spaces with Hierarchical Multiple Kernel Learning

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In this paper, we explore penalizing by sparsity-inducing norms such as the l1-norm or the block l1-norm. We assume that the kernel decomposes into a large sum of individual basis kernels which can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a hierarchical multiple kernel learning framework, in polynomial time in the number of selected kernels. This framework is naturally applied to non linear variable selection; our extensive simulations on synthetic datasets and datasets from the UCI repository show that efficiently exploring the large feature space through sparsity-inducing norms leads to state-of-the-art predictive performance

Solving the European Air Traffic Flow Management Problem with Kernel Search Matheuristics and Machine Learning

The Air Traffic Flow Management (ATFM) problem has the goal of planning flights within a set of constraints representing both capacity limits of the air space and airline company needs, consisting in a delay and a preference assigned to each trajectory; several mathematical linear programming models exist to solve this problem, and the main issue is their size, since they may contain up to millions of variables for real instances. As a consequence, the computational effort required to solve the model to optimality is huge, and not suitable for practical use. This thesis presents a heuristic method based on Kernel Search. The goal of Kernel Search is to solve the model using only an initial subset of variables, called kernel, and dividing the remaining variables into small groups called buckets, ordered by "promising impact on the solution", that is computed from variable information obtained through the resolution of the linear relaxation of the problem. Each iteration of the Kernel Search method consists in solving a small subproblem given by variables from the kernel and from a single bucket, whose size allows to solve it to optimality in a small amount of time; furthermore, in this thesis, Machine Learning techniques have been used in the process of defining the "quality" of each variable, in order to see if such modification in the bucket defining procedure can lead to more efficient or effective methods. The developed algorithms have been implemented and tested on real instances obtained from European data repositories, showing their ability to find optimal or very close to optimal solutions.The Air Traffic Flow Management (ATFM) problem has the goal of planning flights within a set of constraints representing both capacity limits of the air space and airline company needs, consisting in a delay and a preference assigned to each trajectory; several mathematical linear programming models exist to solve this problem, and the main issue is their size, since they may contain up to millions of variables for real instances. As a consequence, the computational effort required to solve the model to optimality is huge, and not suitable for practical use. This thesis presents a heuristic method based on Kernel Search. The goal of Kernel Search is to solve the model using only an initial subset of variables, called kernel, and dividing the remaining variables into small groups called buckets, ordered by "promising impact on the solution", that is computed from variable information obtained through the resolution of the linear relaxation of the problem. Each iteration of the Kernel Search method consists in solving a small subproblem given by variables from the kernel and from a single bucket, whose size allows to solve it to optimality in a small amount of time; furthermore, in this thesis, Machine Learning techniques have been used in the process of defining the "quality" of each variable, in order to see if such modification in the bucket defining procedure can lead to more efficient or effective methods. The developed algorithms have been implemented and tested on real instances obtained from European data repositories, showing their ability to find optimal or very close to optimal solutions

A Hypergraph Multi-Exchange Heuristic for the Single-Source Capacitated Facility Location Problem

In this paper, we introduce a large-scale neighborhood search procedure for solving the single-source capacitated facility location problem (SSCFLP). The neighborhood structures are induced by innovative split multi-customer multi-exchanges, where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa. To represent these exchanges, we use two types of improvement hypergraphs. The improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed-integer program. We develop a hypergraph search framework, including forward and backward procedures, to identify improving solutions efficiently. Our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi-exchange heuristic (Ahuja et al., 2004). In addition, when compared with the Kernel Search algorithm (Guastaroba and Speranza, 2014), which at present is the most effective for solving SSCFLP, our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

Mixed Integer Linear Programming for Feature Selection in Support Vector Machine

This work focuses on support vector machine (SVM) with feature selection. A MILP formulation is proposed for the problem. The choice of suitable features to construct the separating hyperplanes has been modelled in this formulation by including a budget constraint that sets in advance a limit on the number of features to be used in the classification process. We propose both an exact and a heuristic procedure to solve this formulation in an efficient way. Finally, the validation of the model is done by checking it with some well-known data sets and comparing it with classical classification methods.Comment: 37 pages, 20 figure

Models and Heuristics for the Flow-Refuelling Location Problem

Purpose of this paper: Firstly, the paper serves as an overview of the emerging field of flow-refuelling location, which mainly occurs in the context of locating alternative-fuel (hydrogen, electric, liquefied natural gas and hybrid) vehicle refuelling stations. We aim to review and explain models and solution approaches, with a particular focus on mathematical programming formulations. Secondly, we propose a new heuristic for this problem and investigate its performance. Design/methodology/approach: The subject scope of this paper is the flow-refuelling location model (FRLM). While in most location problems demand arises at customer locations, in so-called flow-capturing models it is associated with journeys (origin-destination pairs). What makes the FRLM even more challenging is that due to the limited driving range of alternative-fuel vehicles, more than one facility may be required to satisfy the demand of a journey. There are currently very few such refuelling stations, but ambitious plans exist for massive development – making this an especially ripe time for researchers to investigate this problem. There already exists a body of work on this problem; however different authors make different model assumptions, making comparison difficult. For example, in some models facilities must lie on the shortest route from origin to destination, while in others detours are allowed. We aim to highlight difference in models in our review. Our proposed methodology is built on the idea of solving the relaxation of the mixed-integer linear programming formulation of the problem, identifying promising variables, fixing their values and solving the resulting (so-called restricted) problems optimally. It is somewhat similar to Kernel Search which has recently gained popularity. We also use a parallel computing strategy to simultaneously solve a number of restricted problems with less computation effort for large-sized instances. Findings: Our experimental results show that the proposed heuristic can find optimal solutions in a reasonable amount of time, outperforming other heuristics from the literature. Value: We believe the paper is of value to both academics and practitioners. The review should help researchers new to this field to orient themselves in the maze of different problem versions, while helping practitioners identify models and approaches applicable to their particular problem. The heuristic proposed can be directly used by practitioners; we hope it will spark further works on this area of logistics but also on other optimisation problems where Kernel Search type methods can be applied. Research limitations: This being the first paper applying a restricted-subproblem approach to this problem it is necessarily limited in scope. Applying a traditional Kernel Search method would be an interesting next step. The proposed heuristic should also be extended to cover for more than just one FRLM model: certainly the capacitated FRLM, the FRLM with deviation, the fixed-charge FRLM and the multi-period FRLM should be investigated. Practical implications: Our work adds to a body of research that can inform decisionmakers at governmental or international level on strategic decisions relating to the establishment or development of alternative-fuel refuelling station networks

A heuristic framework for the bi-objective enhanced index tracking problem

The index tracking problem is the problem of determining a portfolio of assets whose performance replicates, as closely as possible, that of a financial market index chosen as benchmark. In the enhanced index tracking problem the portfolio is expected to outperform the benchmark with minimal additional risk. In this paper, we study the bi-objective enhanced index tracking problem where two competing objectives, i.e., the expected excess return of the portfolio over the benchmark and the tracking error, are taken into consideration. A bi-objective Mixed Integer Linear Programming formulation for the problem is proposed. Computational results on a set of benchmark instances are given, along with a detailed out-of-sample analysis of the performance of the optimal portfolios selected by the proposed model. Then, a heuristic procedure is designed to build an approximation of the set of Pareto optimal solutions. We test the proposed procedure on a reference set of Pareto optimal solutions. Computational results show that the procedure is significantly faster than the exact computation and provides an extremely accurate approximation