A Parallel Algorithm for Large-Scale Nonconvex Penalized Quantile Regression

Penalized quantile regression (PQR) provides a useful tool for analyzing high-dimensional data with heterogeneity. However, its computation is challenging due to the nonsmoothness and (sometimes) the nonconvexity of the objective function. An iterative coordinate descent algorithm (QICD) was recently proposed to solve PQR with nonconvex penalty. The QICD significantly improves the computational speed but requires a double-loop. In this article, we propose an alternative algorithm based on the alternating direction method of multiplier (ADMM). By writing the PQR into a special ADMM form, we can solve the iterations exactly without using coordinate descent. This results in a new single-loop algorithm, which we refer to as the QPADM algorithm. The QPADM demonstrates favorable performance in both computational speed and statistical accuracy, particularly when the sample size n and/or the number of features p are large. Supplementary material for this article is available online.     

基于Pairwise惩罚方法的面板数据模型结构变点估计

面板数据模型在经济、生物、统计等领域有着广泛的应用。经典的面板数据模型假设解释变量系数不随时间变化。然而在现实中,解释变量系数可能会因多种因素的影响而存在多重未知的结构变点。本文假设交互固定效应面板数据模型中含有多重未知的结构变点。研究发现通过Pairwise惩罚的参数估计方法在目标函数中增加对相邻时间解释变量系数的惩罚项,能够同时进行参数估计和结构变点诊断。蒙特卡洛模拟结果显示,不管是否存在同方差假设,该方法估计的解释变量系数均偏差较小且结构变点诊断错误率低。     

New results on the equivalence between zero-one programming and continuous concave programming

In this work, we study continuous reformulations of zero-one concave programming problems. We introduce new concave penalty functions and we prove, using general equivalence results here derived, that the obtained continuous problems are equivalent to the original combinatorial problem.     

A Simple Finite Cone Covering Algorithm for Concave Minimization

In 1964, in a seminal paper, Tuy proposed a simple algorithm for concave minimization over a polytope. This algorithm was shown to cycle some years later. Recently however it has been shown that despite this possibility of cycling, Tuy's algorithm always finds the optimal solution of the problem. We present a modification of it which simplifies the cycle detection.     

Copula-based dependence between frequency and class in car insurance with excess zeros

A bonus-malus system calculates the premiums for car insurance based on the previous claim experience (class). In this paper, we propose a model that allows dependence between the claim frequency and the class occupied by the insured using a copula function. It also takes into account zero-excess phenomenon. The maximum likelihood method is employed to estimate the model parameters. A small simulation is performed to illustrate the proposed model and method.     

两类逼近精确罚函数法及其数值试验

1 引言 精确罚函数(exact penalty function)的构造主要有两条途径:一是基于Lagrange乘子的乘子罚函数方法,二是直接构造非光滑的精确罚函数。不必进行乘子迭代。本文讨论第三种思路:基于目标函数最优值构造保持光滑性的精确罚函数。某些无参数外点罚函数本应属于此类,但一直仅仅被作为普通外点罚函数的无参数形式。将其与无参 数内点罚函数同等看待,因此基于目标函数最优值构造精确罚函数未得到充分研究。文献[11]给出了初步结果。本文进一步发展了有关理论,导出了两类算法,证明了收敛性,最后给出了数值试验结果。 2 基于目标函数最优值的精确罚函数 考虑如下约束优化问题     

分布式L_(1/2)正则化

研究数据集被分割并存储于不同处理器时的特征提取和变量选择问题,其中处理器通过某种网络结构相互连接.提出分布式L_(1/2)正则化方法,基于ADMM算法给出分布式L_(1/2)正则化算法,证明了算法的收敛性.算法通过相邻处理器之间完成信息交互,其变量选择结果与数据集不分割时利用L_(1/2)正则化相同.实验表明,所提出的新算法有效、实用,适合于分布式存储数据处理.     

An ellipsoidal frontier model: Allocating input via parametric DEA

This paper presents the ellipsoidal frontier model (EFM), a parametric data envelopment analysis (DEA) model for input allocation. EFM addresses the problem of distributing a single total fixed input by assuming the existence of a predefined locus of points that characterizes the DEA frontier. Numeric examples included in the paper show EFM’s capacity to allocate shares of the total fixed input to each DMU so that they will all become efficient. By varying the eccentricities, input distribution can be performed in infinite ways, gaining control over DEA weights assigned to the variables in the model. We also show that EFM assures strong efficiency and behaves coherently within the context of sensitivity analysis, two properties that are not observed in other models found in the technical literature.     

Minimum concave cost production system: A further generalization of multi-echelon model

We will consider a concave minimization problem associated with a series production system in which raw material is processed inm consecutive facilities. The products at some facility are either sent to the next facility or stocked in the warehouse. The amount of demand for the final products during periodi, i = 1,,n, are known in advance. Our problem is to minimize the sum of processing, holding and backlogging cost, all of which are assumed to be concave.The origin of this model is the classical economic lot size problem of Wagner and Whitin and was extensively studied by Zangwill. This model is very important from the theoretical as well as practical point of view and this is one of the very rare instances in which polynomial time algorithm has been constructed for concave minimization problems.The purpose of this paper is to extend the model further to the situation in which time lag is associated with processing at each facility. We will propose an efficient O(n ⁴ m) algorithm for this class of problems.     

关于判断矩阵一致性检验与调整的一个注记

首先分析了判断矩阵不一致形成的原因,认为一个判断矩阵中的不一致是由强矛盾判断,弱矛盾判断,标度离散性,标度有限性共同作用的结果,并通过两个例子指出现有一致性检验与调整方法中存在的问题,最后在已有研究基础上给出了判断矩阵一致性调整的新步骤.     

A continuous approach for the concave cost supply problem via DC programming and DCA

The paper addresses an important but difficult class of concave cost supply management problems which consist in minimizing a separable increasing concave objective function subject to linear and disjunctive constraints. We first recast these problems into mixed zero-one nondifferentiable concave minimization over linear constraints problems and then apply exact penalty techniques to state equivalent nondifferentiable polyhedral DC (Difference of Convex functions) programs. A new deterministic approach based on DC programming and DCA (DC Algorithms) is investigated to solve the latter ones. Finally numerical simulations are reported which show the efficiency, the robustness and the globality of our approach.     

一个超前有奖迟后受罚的排序问题

本文考虑货物装卸管理中船主和港口之间的下述相互制约关系；有n条船在同一时刻到达同一港口，因而也希望在同一时刻完成装卸货物。如某船的货物不能如期装卸完，船主会向港方索取赔偿，反之，如货物提前装卸完，则船主会向港方付取奖金，因此从港方来说是适当考虑n条船的一个装卸程序以使总费用最少。对这样一个NP－困难的排序问题，本文给出了一个动态规划解法，且在逆一致性条件下给出了一伪多项式时间的动态规划算法。     

一类带约束min-max-min问题的区间算法

构造了求解一类带不等式约束的min-max-min问题的区间算法,其中目标函数和约束函数都是一阶连续可微函数,证明了方法的收敛性,给出了数值算例.该方法可以同时求出问题的最优值和全部全局最优解,是有效和可靠的.     

货物装卸中的一个排序问题

本文考虑货物装卸管理中船主和港口之间的下述相互制约关系：有n条船在时刻零同时抵达同一码头装卸货物,因而也希望在同一时刻守成装卸货物。如某船的货物不能如期装卸守而延误了该船的离港,船主会向港方索取赔偿,反之如货物提前装卸完而使该船河提前投入运输,则船主会向港方付取奖金,加上正常装卸费用,从港方来说要适当考虑n条船的一个装卸顺序,使总费用减少,对这一NP-困难的排序问题,文中给出了几个多项式可解的特殊情形,一般情况下的一个快速下界估计方法以及相应的分支定界算法。     

A simplicial branch-and-bound algorithm conscious of special structures in concave minimization problems

In this paper, we develop a simplicial branch-and-bound algorithm for generating globally optimal solutions to concave minimization problems with low rank nonconvex structures. We propose to remove all additional constraints imposed on the usual linear programming relaxed problem. Therefore, in each bounding operation, we solve a linear programming problem whose constraints are exactly the same as the target problem. Although the lower bound worsens as a natural consequence, we offset this weakness by using an inexpensive bound tightening procedure based on Lagrangian relaxation. After giving a proof of the convergence, we report a numerical comparison with existing algorithms. T. Kuno was partially supported by the Grand-in-Aid for Scientific Research (C) 17560050 from the Japan Society for the Promotion of Sciences.     

Normal conical algorithm for concave minimization over polytopes

A new conical algorithm is developed for finding the global minimum of a concave function over a polytope. To ensure faster convergence and overcome some major drawbacks of previous conical algorithms, a normal (rather than exhaustive) subdivision process is used.