一种多目标条件风险值数学模型

研究了一种多目标条件风险值(CVaR)数学模型理论.先定义了一种多目标损失函数下的α-VaR和α-CVaR值,给出了多目标CVaR最优化模型.然后证明了多目标意义下的α-VaR和α-CVaR值的等价定理,并且给出了对于多目标损失函数的条件风险值的一致性度量性质.最后,给出了多目标CVaR模型的近似求解模型.     

Asymptotic representations for importance-sampling estimators of value-at-risk and conditional value-at-risk

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are important risk measures. They are often estimated by using importance-sampling (IS) techniques. In this paper, we derive the asymptotic representations for IS estimators of VaR and CVaR. Based on these representations, we are able to prove the consistency and asymptotic normality of the estimators and to provide simple conditions under which the IS estimators have smaller asymptotic variances than the ordinary Monte Carlo estimators.     

多损失条件风险值的多层规划模型

对于多个损失函数,在给定的置信水平下,首先定义了不超过给定损失值的最小风险值(即Va R值)和基于权值的累积期望损失值(即CVa R损失值)概念,然后建立了一个多损失条件风险值的多层规划模型.该模型的目标是求各层多损失CVa R值达最小的最优策略,并证明了它等价于另一个较容易求解的多层规划模型.最后,给出了三级供应链中多产品的定价与订购的条件风险值模型(三层线性规划模型).     

Multi-objective evacuation routing in transportation networks

In this paper, the optimal design and analysis of evacuation routes in transportation networks is examined. An methodology for optimal egress route assignment is suggested. An integer programming (IP) formulation for optimal route assignment is presented, which utilizes M/G/c/c state dependent queueing models to cope with congestion and time delays on road links. M/G/c/c simulation software is used to evaluate performance measures of the evacuation plan: clearance time, total travelled distance and blocking probabilities. Extensive experimental results are included.     

Large deviations bounds for estimating conditional value-at-risk

In this paper, we prove an exponential rate of convergence result for a common estimator of conditional value-at-risk for bounded random variables. The bound on optimistic deviations is tighter while the bound on pessimistic deviations is more general and applies to a broader class of convex risk measures.     

Computational aspects of minimizing conditional value-at-risk

We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers. János Mayer: Financial support by the national center of competence in research "Financial Valuation and Risk Management" is gratefully acknowledged. The national centers in research are managed by the Swiss National Science Foundation on behalf of the federal authorities.     

Deviation inequalities for an estimator of the conditional value-at-risk

In this paper, we present a deviation inequality for a common estimator of the conditional value-at-risk for bounded random variables. The result improves a deviation inequality which is obtained by Brown [D.B. Brown, Large deviations bounds for estimating conditional value-at-risk, Operations Research Letters 35 (2007) 722-730].     

Asymptotic behavior of the empirical conditional value-at-risk

We study asymptotic behavior of the empirical conditional value-at-risk (CVaR). In particular, the Berry–Essen bound, the law of iterated logarithm, the moderate deviation principle and the large deviation principle for the empirical CVaR are obtained. We also give some numerical examples.     

Asymptotic behavior of the empirical conditional value-at-risk

We study asymptotic behavior of the empirical conditional value-at-risk (CVaR). In particular, the Berry–Essen bound, the law of iterated logarithm, the moderate deviation principle and the large deviation principle for the empirical CVaR are obtained. We also give some numerical examples.     

A multiobjective programming model for locating treatment sites and routing hazardous wastes

In this paper, a multiobjective model for locating disposal or treatment facilities and transporting hazardous waste along the links of a transportation network are presented. Some of the nodes of this network may be population centres generating hazardous waste which must be transported to the treatment facilities. Four objectives are considered: (1) minimisation of total operating cost, (2) minimisation of total perceived risk, (3) equitable distribution of risk among population centres and (4) equitable distribution of the disutility caused by the operation of the treatment facilities. A goal programming model to solve the problem is developed and a small hypothetical example is presented to illustrate how penalty functions can be used to obtain more satisfactory solutions in real life applications.     

Optimization architecture for joint multi-path routing and scheduling in wireless mesh networks

In Wireless Mesh Networks (WMN), the optimal routing of data depends on the link capacities which are determined by link scheduling. The optimal performance of the network, therefore, can only be achieved by joint routing and scheduling optimization. Although the joint single-path routing and scheduling optimization problem has been extensively studied, its multi-path counterpart within wireless mesh networks has not yet been fully investigated. In this paper, we present an optimization architecture for joint multi-path QoS routing and the underlying wireless link scheduling in wireless mesh networks. By employing the contention matrix to represent the wireless link interference, we formulate a utility maximization problem for the joint multi-path routing and MAC scheduling and resolve it using the primal–dual method. Since the multi-path routing usually results in the non-strict concavity of the primal objective function, we first introduce the Proximal Optimization Algorithm to get around such difficulty. We then propose an algorithm to solve the routing subproblem and the scheduling subproblem via the dual decomposition. Simulations demonstrate the efficiency and correctness of our algorithm.     

Concentration bounds for empirical conditional value-at-risk: The unbounded case

Conditional Value-at-Risk (CVaR) is a popular risk measure for modelling losses in the case of a rare but extreme event. We consider the problem of estimating CVaR from i.i.d. samples of an unbounded random variable, which is either sub-Gaussian or sub-exponential. We derive a novel one-sided concentration bound for a natural sample-based CVaR estimator in this setting. Our bound relies on a concentration result for a quantile-based estimator for Value-at-Risk (VaR), which may be of independent interest.     

A planning and routing model for patient transportation in health care

In this paper, a problem concerning both the planning of health care services and the routing of vehicles, for patients transportation is addressed. An integrated approach, based on the column generation technique, is proposed to solve the planning and routing problem. Preliminary results on real data show the effectiveness of the proposed approach.     

Iterative estimation maximization for stochastic linear programs with conditional value-at-risk constraints

We present a new algorithm, iterative estimation maximization (IEM), for stochastic linear programs with conditional value-at-risk constraints. IEM iteratively constructs a sequence of linear optimization problems, and solves them sequentially to find the optimal solution. The size of the problem that IEM solves in each iteration is unaffected by the size of random sample points, which makes it extremely efficient for real-world, large-scale problems. We prove the convergence of IEM, and give a lower bound on the number of sample points required to probabilistically bound the solution error. We also present computational performance on large problem instances and a financial portfolio optimization example using an S&P 500 data set.     

A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs

Recent years have seen growing interest in coherent risk measures, especially in Conditional Value-at-Risk ( \(\mathrm {CVaR}\) ). Since \(\mathrm {CVaR}\) is a convex function, it is suitable as an objective for optimization problems when we desire to minimize risk. In the case that the underlying distribution has discrete support, this problem can be formulated as a linear programming (LP) problem. Over more general distributions, recent techniques, such as the sample average approximation method, allow to approximate the solution by solving a series of sampled problems, although the latter approach may require a large number of samples when the risk measures concentrate on the tail of the underlying distributions. In this paper we propose an automatic primal-dual aggregation scheme to exactly solve these special structured LPs with a very large number of scenarios. The algorithm aggregates scenarios and constraints in order to solve a smaller problem, which is automatically disaggregated using the information of its dual variables. We compare this algorithm with other common approaches found in related literature, such as an improved formulation of the full problem, cut-generation schemes and other problem-specific approaches available in commercial software. Extensive computational experiments are performed on portfolio and general LP instances.     

Portfolio optimization with a copula-based extension of conditional value-at-risk

The paper presents a copula-based extension of Conditional Value-at-Risk and its application to portfolio optimization. Copula-based conditional value-at-risk (CCVaR) is a scalar risk measure for multivariate risks modeled by multivariate random variables. It is assumed that the univariate risk components are perfect substitutes, i.e., they are expressed in the same units. CCVaR is a quantile risk measure that allows one to emphasize the consequences of more pessimistic scenarios. By changing the level of a quantile, the measure permits to parameterize prudent attitudes toward risk ranging from the extreme risk aversion to the risk neutrality. In terms of definition, CCVaR is slightly different from popular and well-researched CVaR. Nevertheless, this small difference allows one to efficiently solve CCVaR portfolio optimization problems based on the full information carried by a multivariate random variable by employing column generation algorithm.     

Optimization of discrete broadcast under uncertainty using conditional value-at-risk

The paper considers the problem of scheduling packets in wireless broadcast systems under uncertainty with the conditional value-at-risk (CVaR) constraints. In such systems, a server periodically transmits a stream of packets over a broadcast channel. The client that needs to access the data, tunes in to the channel and waits for the next packet. This allows one to serve a large number of clients in an efficient way and also keep clients’ location secret. We formulate and solve two alternative stochastic optimization problems that minimize the transmission time subject to CVaR constraints. Our results indicate that it is possible to derive an analytical solution to the problems in certain cases of practical interest. We also propose a methodology to obtain numerical solutions for the general case.     

Truck routing and scheduling

The problem is part of a complex software solution for truck itinerary construction for one of the largest public road transportation companies in the EU. In practice a minor improvement on the operational cost per tour can decide whether a freight services company is profitable or not. Thus the optimization of routes has key importance in the operation of such companies. Given an initial location and an asset state one must be able to calculate a cost optimal itinerary containing all Point of Interests. Such an itinerary is an executable plan which exactly specifies the location and activity of an asset during the whole timespan of the itinerary. If parking places and gas stations are included in the planning then it is NP hard to find an optimal solution. This means that for long range tours an approximately optimal solution for refueling has to be given within an acceptable running time. Also the corridoring of the trucks is an important problem so that we try to optimize the performance, hence tours cannot be recalculated at each data arrival. The vehicle assignment part of this work is already finished and applied with very good results. The remaining part is subject of an ongoing research which started at January 2014. The company started to apply and test our product in the beginning of 2015 under increased human supervision. As a consequence of the project a large cost saving is anticipated by the company.     

Dynamic mortality factor model with conditional heteroskedasticity

In most methods for modeling mortality rates, the idiosyncratic shocks are assumed to be homoskedastic. This study investigates the conditional heteroskedasticity of mortality in terms of statistical time series. We start from testing the conditional heteroskedasticity of the period effect in the naïve Lee–Carter model for some mortality data. Then we introduce the Generalized Dynamic Factor method and the multivariate BEKK GARCH model to describe mortality dynamics and the conditional heteroskedasticity of mortality. After specifying the number of static factors and dynamic factors by several variants of information criterion, we compare our model with other two models, namely, the Lee–Carter model and the state space model. Based on several error-based measures of performance, our results indicate that if the number of static factors and dynamic factors is properly determined, the method proposed dominates other methods. Finally, we use our method combined with Kalman filter to forecast the mortality rates of Iceland and period life expectancies of Denmark, Finland, Italy and Netherlands.     

Dynamic mortality factor model with conditional heteroskedasticity

In most methods for modeling mortality rates, the idiosyncratic shocks are assumed to be homoskedastic. This study investigates the conditional heteroskedasticity of mortality in terms of statistical time series. We start from testing the conditional heteroskedasticity of the period effect in the naïve Lee-Carter model for some mortality data. Then we introduce the Generalized Dynamic Factor method and the multivariate BEKK GARCH model to describe mortality dynamics and the conditional heteroskedasticity of mortality. After specifying the number of static factors and dynamic factors by several variants of information criterion, we compare our model with other two models, namely, the Lee-Carter model and the state space model. Based on several error-based measures of performance, our results indicate that if the number of static factors and dynamic factors is properly determined, the method proposed dominates other methods. Finally, we use our method combined with Kalman filter to forecast the mortality rates of Iceland and period life expectancies of Denmark, Finland, Italy and Netherlands.