次模函数最大化的流算法综述

次模函数优化在计算机科学、数学、经济学等学科得到广泛研究.大数据环境下的次模优化是相对较新的研究领域,受到更多关注.特别地,考虑基于流模型的次模最大化问题.在该问题中,数据以流的形式呈现,其目的是从数据流中抽取满足某些特性的稀疏子集,最大化次模收益函数值.介绍了基于流模型的次模最大化问题的阈值和优先权方法,同时也介绍了若干次模最大化变形的流算法进展.     

基于Logit离散选择模型的品类优化问题综述

品类优化问题(Assortment Optimization Problem)是收益管理的经典问题.它研究零售商在满足运营约束的前提下,应如何从给定产品集合中选择一个子集提供给消费者,以最大化预期收益.该问题的核心在于如何准确地刻画消费者在面对细分产品时的选择行为、建立相应的优化模型并设计高效率的求解算法.基于Logit离散选择模型的品类优化问题:首先,介绍了基于Multinomial Logit模型的品类优化问题.然后介绍了两个更复杂的变种:第一个是基于两层以及多层Nested Logit模型的品类优化问题,这类问题可合理刻画细分产品之间的"替代效应";第二个是基于Mixtures of Multinomial Logits模型的品类优化问题,这类问题可充分考虑消费者群体的异质性.随后,介绍了数据驱动的品类优化问题的相关进展.最后,指出该问题未来可能的若干研究方向.     

带基数约束的次模+超模(BP)函数最大化问题的流算法

本文研究在基数约束下具有单调性的次模+超模函数最大化问题的流模型。该问题在数据处理、机器学习和人工智能等方面都有广泛应用。借助于目标函数的收益递减率（$\gamma$）,我们设计了单轮读取数据的过滤-流算法,并结合次模、超模函数的全局曲率（$\kappa^{g}$）得到算法的近似比为$\min\left\{\frac{（1-\varepsilon）\gamma}{2^{\gamma}},1-\frac{\gamma}{2^{\gamma}（1-\kappa^{g}）^{2}}\right\}$。数值实验验证了过滤-流算法对BP最大化问题的有效性并且得出：次模函数和超模函数在同量级条件下,能保证在较少的时间内得到与贪婪算法相同的最优值。     

带基数约束的次模+超模(BP)函数最大化问题的流算法

本文研究在基数约束下具有单调性的次模+超模函数最大化问题的流模型。该问题在数据处理、机器学习和人工智能等方面都有广泛应用。借助于目标函数的收益递减率（$\gamma$）,我们设计了单轮读取数据的过滤-流算法,并结合次模、超模函数的全局曲率（$\kappa^{g}$）得到算法的近似比为$\min\left\{\frac{（1-\varepsilon）\gamma}{2^{\gamma}},1-\frac{\gamma}{2^{\gamma}（1-\kappa^{g}）^{2}}\right\}$。数值实验验证了过滤-流算法对BP最大化问题的有效性并且得出：次模函数和超模函数在同量级条件下,能保证在较少的时间内得到与贪婪算法相同的最优值。     

基于固定效应的纵向数据分位点回归模型的参数估计及CDM和MSOM的等价性

研究了基于固定效应的纵向数据模分位点回归模型的参数估计及统计诊断问题.首先给出了参数估计的MM迭代算法,然后讨论了统计诊断中数据删除模型(CDM)和均值移模型(MSOM)的等价性问题,最后利用消炎镇痛药数据说明了方法的应用.     

求解稀疏逻辑回归问题的嵌套BB算法的分裂增广拉格朗日算法

逻辑回归是经典的分类方法,广泛应用于数据挖掘、机器学习和计算机视觉.现研究带有程。模约束的逻辑回归问题.这类问题广泛用于分类问题中的特征提取,且一般是NP-难的.为了求解这类问题,提出了嵌套BB(Barzilai and Borwein)算法的分裂增广拉格朗日算法(SALM-BB).该算法在迭代中交替地求解一个无约束凸优化问题和一个带程。模约束的二次优化问题.然后借助BB算法求解无约束凸优化问题.通过简单的等价变形直接得到带程。模约束二次优化问题的精确解,并且给出了算法的收敛性定理.最后通过数值实验来测试SALM-BB算法对稀疏逻辑回归问题的计算精确性.数据来源包括真实的UCI数据和模拟数据.数值实验表明,相对于一阶算法SLEP,SALM-BB能够得到更低的平均逻辑损失和错分率.     

生态工业共生网络中的鲁棒优化模型

生态工业共生网络(EISN)的稳定可持续运行是生态工业园中各企业实现经济和环境利益最大化的基本保障。在考虑政府激励因素的情况下,建立一个以n个生产者企业和m个分解者企业为核心的EISN利润最大化问题的最优化模型。在此基础上,为更真实反映EISN的运行情况,考虑到分解者企业利用次级原材料的单位成本及次级原材料转化率的不确定性,构建了EISN利润最大化问题的离散鲁棒优化模型,提出了基于网络最小费用流的模型求解方法,证明了模型解的存在性,并给出了相应的案例。     

不确定并串联系统的贮备冗余优化

在可靠性工程中,可靠性优化设计是系统设计阶段需要解决的一个重要问题.文章针对不确定并串联系统,研究了具有不确定寿命和不确定费用的贮备冗余优化问题.以最大化系统可靠性,最大化系统寿命和最小化系统费用为目标,构建了3种不同的贮备冗余优化模型.在不确定理论框架下,给出了3种模型的等价模型,且根据决策者的偏好构建了3种带有优先级的模型.此外,给出的数值算例说明了所构建模型的合理性.     

缺失数据下含几何分布的对数线性模型的EM算法

本文研究缺失数据下对数线性模型参数的极大似然估计问题.通过Monte-Carlo EM算法去拟合所提出的模型.其中,在期望步中利用Metropolis-Hastings算法产生一个缺失数据的样本,在最大化步中利用Newton-Raphson迭代使似然函数最大化.最后,利用观测数据的Fisher信息得到参数极大似然估计的渐近方差和标准误差.     

基于Hash建立索引和Kmp快速匹配算法的DNA序列查找方法

研究了DNA序列片段的查找问题,针对DNA数据量大和DNA序列碱基排列的特点提出了DNA序列检索的问题.在对DNA序列检索中,基于Hash建立了索引表以提高在大数据中检索的速度和效率,同时在平衡树的数据存储模型上使用了改进的Kmp快速匹配算法,提高了在索引上的检索效率.介绍了Hash索引的建立、Kmp的优化以及平衡树的再平衡.利用软件评估实验得出的实验结果表明了该算法的有效性.     

Maximizing a class of submodular utility functions

Given a finite ground set N and a value vector ${a \in \mathbb{R}^N}$ , we consider optimization problems involving maximization of a submodular set utility function of the form ${h(S)= f \left(\sum_{i \in S} a_i \right ), S \subseteq N}$ , where f is a strictly concave, increasing, differentiable function. This utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences, for instance, in risk-averse capital budgeting under uncertainty, competitive facility location, and combinatorial auctions. These problems can be formulated as linear mixed 0-1 programs. However, the standard formulation of these problems using submodular inequalities is ineffective for their solution, except for very small instances. In this paper, we perform a polyhedral analysis of a relevant mixed-integer set and, by exploiting the structure of the utility function h, strengthen the standard submodular formulation significantly. We show the lifting problem of the submodular inequalities to be a submodular maximization problem with a special structure solvable by a greedy algorithm, which leads to an easily-computable strengthening by subadditive lifting of the inequalities. Computational experiments on expected utility maximization in capital budgeting show the effectiveness of the new formulation.     

An exact method for constrained maximization of the conditional value-at-risk of a class of stochastic submodular functions

We consider a class of risk-averse submodular maximization problems (RASM) where the objective is the conditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We propose valid inequalities and an exact general method for solving RASM under the assumption that we have an efficient oracle that computes the CVaR of the random function. We demonstrate the proposed method on a stochastic set covering problem that admits an efficient CVaR oracle for the random coverage function.     

A note on submodular set cover on matroids

We consider a problem related to the submodular set cover on polymatroids, when the ground set is the family of independent sets of a matroid. The achievement here is getting a strongly polynomial running time with respect to the ground set of the matroid even though the family of independent sets has exponential size. We also address the optimization problem of the maximization of submodular set functions on the independent sets of a matroid.     

可替代产品库存模型的研究

以零售商的角度,讨论了在允许进货的情况下可替代产品的库存问题,建立了这类问题利润最大化的库存模型,讨论了各参数对利润和库存的影响.然后证明了问题的解是存在的,利润函数是子模的,并给出了最优解的一阶必要性条件,同时探讨了目标函数的凹性问题.通过理论分析和数值试验证明了在一定条件下替代和重新进货都能提高利润,并能降低总的库存水平.     

Generalized roof duality

The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds.We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables.     

Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions

We consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints. We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-type min-max theorem for submodular and supermodular functions. We also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal, which is a counterpart of the Karush-Kuhn-Tucker condition for convex programs. This work is supported by the Alexander von Humboldt fellowship (1982/83), West Germany.     

Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint

Journal of Global Optimization - Arising from practical problems such as in sensor placement and influence maximization in social network, submodular and non-submodular maximization on the integer...     

An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.