The vehicle routing and scheduling problem has been studied with much interest within the last four decades. In this paper, some of the existing literature dealing with routing and scheduling problems with environmental issues is reviewed, and a description is provided of the problems that have been investigated and how they are treated using combinatorial optimization tools.
As the risk is a part of the objective function, it is quantified with a path evaluation function @cite_89 . This function is not additive since the probability of a release accident on a link depends on the probability of a release accident on the traveled links of the path. This important property leads to non-linear integer formulations which can not optimized using a classical shortest path algorithm. Generally approximations are needed by considering additive functions (Usually considering independent release accident probabilities on links) for obtaining tractable models.
@cite_89
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Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process theory implies uniform convergence of the empirical risk to the population risk. While uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. In order to capture the complexity of computing M-estimators, we propose to study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we can establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as non-convex binary classification, robust regression, and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provide a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the unknown parameters vector (modulo logarithmic factors), then a suitable uniform convergence result takes place. We apply this result to non-convex binary classification and robust regression in very high-dimension. Many object detectors, such as AdaBoost, SVM and deformable part-based models (DPM), compute additive scoring functions at a large number of windows scanned over image pyramid, thus computational efficiency is an important consideration beside accuracy performance. In this paper, we present a framework of learning cost-sensitive decision policy which is a sequence of two-sided thresholds to execute early rejection or early acceptance based on the accumulative scores at each step. A decision policy is said to be optimal if it minimizes an empirical global risk function that sums over the loss of false negatives (FN) and false positives (FP), and the cost of computation. While the risk function is very complex due to high-order connections among the two-sided thresholds, we find its upper bound can be optimized by dynamic programming (DP) efficiently and thus say the learned policy is near-optimal. Given the loss of FN and FP and the cost in three numbers, our method can produce a policy on-the-fly for Adaboost, SVM and DPM. In experiments, we show that our decision policy outperforms state-of-the-art cascade methods significantly in terms of speed with similar accuracy performance. We present a convex framework to learn sequential decisions and apply it to the problem of learning under a budget. We consider the structure proposed in [1], where sensor measurements are acquired in a sequence. The goal after acquiring each new measurement is to make a decision whether to stop and classify or to pay the cost of using the next sensor in the sequence. We introduce a novel formulation of an empirical risk objective for the multi stage sequential decision problem. This objective naturally lends itself to a non-convex multilinear formulation. Nevertheless, we derive a novel perspective that leads to a tight convex objective. This is accomplished by expressing the empirical risk in terms of linear superposition of indicator functions. We then derive an LP formulation by utilizing hinge loss surrogates. Our LP achieves or exceeds the empirical performance of the nonconvex alternating algorithm that requires a large number of random initializations. Consequently, the LP has the advantage of guaranteed convergence, global optimality, repeatability and computation eciency. We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate.