带交易费用的离散多因素投资组合最优化

研究带有凹的交易费函数的离散多因素投资组合模型．与传统的投资组合模型不同的是,该模型中投资组合的决策变量是交易手数（整数）,其最优化模型是一个非线性整数规划问题．为此本文提出了一个基于拉格朗日松弛和连续松弛的混合分枝定界算法,为测试算法的有效性,我们分别采用美国股票市场真实数据和随机产生的数据,数值结果表明该算法是有效的．     

离散单因素投资组合模型的对偶算法

本文研究金融优化中的离散单因素投资组合问题,该问题与传统投资组合模型的不同之处是决策变量为整数(交易手数),从而导致要求解一个二次整数规划问题．针对该模型的可分离性结构,我们提出了一种基于拉格朗日对偶和连续松弛的分枝定界算法。我们分别用美国股票市场的交易数据和随机产生的数据对算法进行了测试．数值结果表明该算法是有效的,可以求解多达150个风险证券的离散投资组合问题．     

离散投资组合问题的一种基于Bundle对偶搜索的精确算法

本文提出了离散均值-方差投资组合模型的一种新的精确算法.该算法是一个基于拉格朗日松弛和Bundle对偶搜索的分枝定界算法.我们分别用随机产生的数据和美国股票市场的真实数据进行了数值实验,并与传统次梯度对偶搜索进行了比较,数值结果表明本文提出的算法对解决中小规模的离散投资组合问题是有效的.     

手数约束和凹交易费下的离散投资组合模型及算法

本文建立带手数约束和凹交易费的离散投资组合模型,给出求解该模型的一种精确算法。该算法是一个基于拉格朗日松弛和次梯度对偶搜索的分枝定界算法。为测试算法的有效性,用随机产生的数据对模型进行数值实验。作为其应用,用沪深300指数的真实数据实证检验该模型,并与不含交易费用的离散投资组合模型进行数值比较分析。数值分析表明算法能在合理的时间内给出模型的投资组合策略, 对解决中小规模的离散投资组合问题是有效的。     

整数线性规划投资组合模型的直接搜索算法求解

利用松弛最优邻近解临域整数点搜索法作过滤条件,建立求解整数规划的新方法——直接搜索算法,利用直接搜索算法并借助Matlab软件求解整数线性规划投资组合模型.数值结果表明了模型的建立与提出方法的有效性.     

带最小交易单位的Markowitz模型及算法

在证券交易市场中,交易规则要求购买的股票数量为整数.基于这种情况,将Markowitz模型中资产的投资比例改进为资产的投资数量,构造了一个二次整数规划模型.设计了求解该模型的算法,经过实证分析,算法是有效的.     

离散线性投资组合模型的分枝定界算法

本文提出了一类新的带整数交易手数和凹型交易费用的均值绝对偏差模型(MAD)和极大极小投资组合模型(Minmax),并给出了离散模型的分枝定界算法.我们分别用随机产生的数据和Nasdaq股票市场的真实数据进行了数值实验,数值分析表明在一定的收益水平下均值绝对偏差离散模型风险控制上优于极大极小投资组合离散模型,而计算效率上极大极小投资组合离散模型优于期望绝对偏差离散模型.     

企业家人力资源培训问题的整数规划模型

本文强调了企业家人力资源在企业核心能力和可持续成长能力生成中的重要作用和地位。基于时间和费用指标,研究了企业家人力资源培训问题,利用0-1线性整数规划建立了一个带有培训时间约束的最小培训费用模型,给出了求解模型的基于Lagrange松弛的分解算法,计算实例表明给出的算法是有效的。     

稠密k-子图问题的双非负松弛

稠密k-子图问题是组合优化里面一类经典的优化问题,其在通常情况下是非凸且NP-难的。本文给出了求解该问题的一个新凸松弛方法-双非负松弛方法,并建立了问题的相应双非负松弛模型,而且证明了其在一定的条件下等价于一个新的半定松弛模型。最后,我们使用一些随机例子对这些模型进行了数值测试,测试的结果表明双非负松弛的计算效果要优于等价的半定松弛。     

基于松弛策略解半无限规划模型的显式修正算法

讨论了一类线性半无限最优规划模型的求解算法.采用松弛方法解其系列子问题LP(T_k)及DLP(T_k),基于松弛策略和在适当的假设条件下,提出了一个我们称之为显式算法的新型算法.新算法的主要改进之处是算法在每一步迭代计算时,允许丢弃一些不必要的约束.在这种方式下,算法避免了求解系列太大规模的子问题.最后,基于提出的显式修正算法,并与传统割平面方法和已有文献中的松弛修正算法、对同一问题作了初步的数值比较实验.     

求不定二次规划全局解的一个新算法

本文提出了一个求不定二次规划问题全局最优解的新算法.首先,给出了三种计算下界的方法:线性逼近法、凸松弛法和拉格朗日松弛法;并且证明了拉格朗日对偶界与通过凸松弛得到的下界是相等的;然后建立了基于拉格朗日对偶界和矩形两分法的分枝定界算法,并给出了初步的数值试验结果.     

New Lagrangian Relaxation Based Algorithm for Resource Scheduling with Homogeneous Subproblems

Solution oscillations, often caused by identical solutions to the homogeneous subproblems, constitute a severe and inherent disadvantage in applying Lagrangian relaxation based methods to resource scheduling problems with discrete decision variables. In this paper, the solution oscillations caused by homogeneous subproblems in the Lagrangian relaxation framework are identified and analyzed. Based on this analysis, the key idea to alleviate the homogeneous oscillations is to differentiate the homogeneous subproblems. A new algorithm is developed to solve the problem under the Lagrangian relaxation framework. The basic idea is to introduce a second-order penalty term in the Lagrangian. Since the dual cost function is no longer decomposable, a surrogate subgradient is used to update the multiplier at the high level. The homogeneous subproblems are not solved simultaneously, and the oscillations can be avoided or at least alleviated. Convergence proofs and properties of the new dual cost function are presented in the paper. Numerical testing for a short-term generation scheduling problem with two groups of identical units demonstrates that solution oscillations are greatly reduced and thus the generation schedule is significantly improved.     

Convex relaxation and Lagrangian decomposition for indefinite integer quadratic programming

We study lower bounding methods for indefinite integer quadratic programming problems. We first construct convex relaxations by D.C. (difference of convex functions) decomposition and linear underestimation. Lagrangian bounds are then derived by applying dual decomposition schemes to separable relaxations. Relationships between the convex relaxation and Lagrangian dual are established. Finally, we prove that the lower bound provided by the convex relaxation coincides with the Lagrangian bound of the orthogonally transformed problem.     

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.     

A Lagrangian relaxation approach to large-scale flow interception problems

The paper presents a tight Lagrangian bound and an efficient dual heuristic for the flow interception problem. The proposed Lagrangian relaxation decomposes the problem into two subproblems that are easy to solve. Information from one of the subproblems is used within a dual heuristic to construct feasible solutions and is used to generate valid cuts that strengthen the relaxation. Both the heuristic and the relaxation are integrated into a cutting plane method where the Lagrangian bound is calculated using a subgradient algorithm. In the course of the algorithm, a valid cut is added and integrated efficiently in the second subproblem and is updated whenever the heuristic solution improves. The algorithm is tested on randomly generated test problems with up to 500 vertices, 12,483 paths, and 43 facilities. The algorithm finds a proven optimal solution in more than 75% of the cases, while the feasible solution is on average within 0.06% from the upper bound.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

0-1多项式规划问题的SDP松弛方法(英)

本文提出了一类新的构造0-1多项式规划的半定规划(SDP)松弛方法. 我们首先利用矩阵分解和分片线性逼近给出一种新的SDP松弛, 该 松弛产生的界比标准线性松弛产生的界更紧. 我们还利用 拉格朗日松弛和平方和(SOS)松弛方法给出了一种构造Lasserre的SDP 松弛的新方法.