Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distributiondriven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty.In this paper we propose an analytic approach to problem-driven scenario generation. This approach applies to stochastic programs where a tail risk measure, such as conditional value-at-risk, is applied to a loss function. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread their scenarios evenly across the support of the solution, struggle to adequately represent tail risk. Our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions. We provide conditions under which our approach is consistent with sampling, and as proof-of-concept demonstrate how our approach could be applied to two classes of problem. Numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard Monte Carlo sampling. arXiv:1511.03074v3 [math.OC] 25 Apr 2018 Tail risk measures and risk regionsIn this section we present the core theory to our scenario generation methodology. Specifically, in Section 3.1 we formally define tail-risk measures of random variables and in Section 3.2 we define risk regions and present some key results related to these.

In the k-partition problem (k-PP), one is given an edge-weighted undirected graph, and one must partition the node set into at most k subsets, in order to minimise (or maximise) the total weight of the edges that have their end-nodes in the same cluster. Various hierarchical variants of this problem have been studied in the context of data mining. We consider a 'two-level' variant that arises in mobile wireless communications. We show that an exact algorithm based on intelligent preprocessing, cutting planes and symmetry-breaking is capable of solving small-and medium-size instances to proven optimality, and providing strong lower bounds for larger instances.the same modulo k, it causes interference. Moreover, some additional interference occurs if they are the same modulo kk . The task is to assign IDs to devices in such a way as to minimise the total interference.The problem turns out to be a generalisation of a well-known N P-hard combinatorial optimisation problem called the k-partition problem or k-PP. For reasons which will become clear later, we call our problem the 2-level partition problem or 2L-PP. Here we develop an exact algorithm for the 2L-PP, which turns out be capable of solving smalland medium-sized instances to proven optimality, and providing strong lower bounds for larger instances.The structure of the paper is as follows. In Section 2, the literature on the k-PP is reviewed. In Section 3, we formulate our problem as an integer program (IP) and derive some valid linear inequalities (i.e. cutting planes). In Section 4, we describe our exact algorithm in detail. In Section 5, we describe some computational experiments and analyse the results. Finally, some concluding remarks are made in Section 6. Literature ReviewSince the 2L-PP is a generalisation of the k-PP, we now review the literature on the k-PP. We define the k-PP in Subsection 2.1. The main IP formulations are presented in Subsection 2.2. The remaining two subsections cover cutting planes and algorithms for generating them, respectively.We remark that some other multilevel graph partitioning problems have been studied in the data mining literature; see, e.g., [CC17,SS11]. In those problems, however, neither the number of clusters nor the number of levels is fixed. For this reason, we do not consider them further. The k-partition problemThe k-PP was first defined in [CN66]. We are given a (simple, loopless) undirected graph G, with vertex set V ={1, . . . , n} and edge set E, a rational weight w e for each edge e∈E, and an integer k with 2 k n. The task is to partition V into k or fewer subsets (called "clusters" or "colours"), such that the sum of the weights of the edges that have both end-vertices in the same cluster is minimised.The k-PP has applications in scheduling, statistical clustering, numerical linear algebra, telecommunications, VLSI layout and statistical physics (see, e.g., [CN66, Eis02, GAL11, Ren12]). It is strongly N P-hard for any fixed k 3, since it includes as a special case the problem of testing whether a graph is ...

Congestion is a problem at airports where capacity does not meet demand. At many such airports, airlines must request time slots for the purpose of landing or take off. Given the imbalance between demand and capacity, slot requests cannot always be scheduled as requested. The difference between the requested and allocated time slots is called displacement. Minimization of the total displacement is a key slot-scheduling objective and expresses the efficiency of the slot-scheduling process. Additionally, fairness has been proposed as a slot-scheduling criterion. Fairness relates to the allocation of the total schedule displacement among the various airlines. Single-and multiobjective models have been proposed for slot scheduling. However, currently the literature lacks models that incorporate the preferences of airlines regarding the allocation of displacement to their flights. This paper proposes a two-stage mechanism for the scheduling of slots at congested airports. The proposed mechanism considers efficiency and fairness objectives and incorporates the preferences of airlines in allocating the total displacement associated with the flights of each airline. The first stage of the mechanism constructs a reference schedule that is fair to the participating airlines. In the second stage, the airlines specify how the displacement allocated to them in the reference schedule should be distributed among their requests. The mechanism then adjusts the fair reference schedule to meet as many of these preferences as possible. The development and implementation of the proposed slotscheduling mechanism is demonstrated using real data from a coordinated airport and simulated displacement preference data. The proposed slot-scheduling mechanism provides useful information to decision makers regarding the equity-efficiency trade-off and enhances the transparency and acceptability of the slot-scheduling outcome.

The k-partition problem is an N P-hard combinatorial optimisation problem with many applications. Chopra and Rao introduced two integer programming formulations of this problem, one having both node and edge variables, and the other having only edge variables. We show that, if we take the polytopes associated with the 'edge-only' formulation, and project them into a suitable subspace, we obtain the polytopes associated with the 'node-and-edge' formulation. This result enables us to derive new valid inequalities and separation algorithms, and also to shed new light on certain SDP relaxations. Computational results are also presented.

Gaussian processes are a class of flexible nonparametric Bayesian tools that are widely used across the sciences, and in industry, to model complex data sources. Key to applying Gaussian process models is the availability of well-developed open source software, which is available in many programming languages. In this paper, we present a tutorial of the GaussianProcesses.jl package that has been developed for the Julia programming language. GaussianProcesses.jl utilizes the inherent computational benefits of the Julia language, including multiple dispatch and just-in-time compilation, to produce a fast, flexible and user-friendly Gaussian processes package. The package provides many mean and kernel functions with supporting inference tools to fit exact Gaussian process models, as well as a range of alternative likelihood functions to handle non-Gaussian data (e.g., binary classification models) and sparse approximations for scalable Gaussian processes. The package makes efficient use of existing Julia packages to provide users with a range of optimization and plotting tools.