Real-world extensions to scheduling algorithms based on lagrangian relaxation

Recently, efficient scheduling algorithms based on Lagrangian relaxation have been proposed for scheduling parallel machine systems and job shops. In this article, we develop real-world extensions to these scheduling methods. In the first part of the paper, we consider the problem of scheduling single operation jobs on parallel identical machines and extend the methodology to handle multiple classes of jobs, taking into account setup times and setup costs. The proposed methodology uses Lagrangian relaxation and simulated annealing in a hybrid framework. In the second part of the paper, we consider a Lagrangian relaxation based method for scheduling job shops and extend it to obtain a scheduling methodology for a real-world flexible manufacturing system with centralized material handling. This research was supported in part by the Office of Naval Research and the Department of Science and Technology grant N00014-93-1017.     

无线传感网络中拥塞控制与路由的跨层设计:分布式牛顿法

无线传感网络应用广泛, 其性能与路由选择和拥塞控制密切相关. 致力于拥塞控制与多径路由的跨层优化, 以实现在链路容量受限和节点能量受限情况下的无线传感网络效用最大化. 针对对偶次梯度算法具有收敛速度慢与信息交互量大等缺陷, 设计了具有二阶收敛性能的分布式牛顿算法来实现网络效用最大化. 通过矩阵分裂技术, 实现了只需单跳信息交互的牛顿对偶方向的分布式求解方法. 仿真结果表明, 分布式牛顿算法的收敛性能显著优于对偶次梯度算法.     

Double auction mechanisms on Markovian networks

This paper studies the double auction (DA) mechanism in Ma and Li (2011) for a class of exchange economies. We extend their results to more general cases where sellers and buyers each form a complex time non-homogeneous Markovian chain, as specified in Ram et al. (2009), in the communication of their private information. A numerical example is also provided. Both bubbles and crashes are observed in the example, consistent with results of our theorems. Our example and theoretical results provide new evidence that a DA mechanism, widely utilized in real exchange markets, may contribute to the excess volatility identified in Shiller (1981) and LeRoy and Porter (1981).     

局部Lipschitz连续函数差的刻画

在Banach空间中利用广义方向导数和Clarke次微分的定义,指出两个局部Lipschitz连续函数差与Clarke次微分之间的关系.在此基础上,指出如果两个局部Lipschitz连续函数f,g：X→R是Clarke正则的,那么结果退化到经典意义下ε次微分与局部Lipschitz连续函数差的关系,并指出了当函数h是可微偶凸函数时,在定理1的条件下两个局部Lipschitz连续函数的Clarke次微分之间的关系,最后指出当两个局部Lipschitz连续函数差为常数时,两个函数的Clarke次微分之间的关系.     

Solving linear generalized Nash equilibrium problems numerically

This paper considers the numerical solution of linear generalized Nash equilibrium problems (LGNEPs). Since many methods for nonlinear problems require the nonsingularity of some second-order derivative, standard convergence conditions are not satisfied in our linear case. We provide new convergence criteria for a potential reduction algorithm (PRA) that allow its application to LGNEPs. Furthermore, we discuss a projected subgradient method (PSM) and a penalty method that exploit some known Nikaido–Isoda function-based constrained and unconstrained optimization reformulations of the LGNEP. Moreover, it is shown that normalized Nash equilibria of an LGNEP can be obtained by solving a single linear program. All proposed algorithms are tested on randomly generated instances of economic market models that are introduced and analysed in this paper and that lead to LGNEPs with shared and with non-shared constraints. It is shown that these problems have some favourable properties that can be exploited to obtain their solutions. With the PRA and in particular with the PSM we are able to compute solutions with satisfying precision even for problems with up 10,000 variables.     

Incremental subgradient method for nonsmooth convex optimization with fixed point constraints

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.     

Implementation of a subgradient projection algorithm

This paper discusses the implementation of an algorithm due to Sreedharan [8] for the minimization, subject to linear constraints, of an objective function composed of the sum of a piecewise-affine, convex function with a smooth, strictly convex function. Successful techniques for two subproblems arising in the algorithm, a projection problem and a line search problem, are described in detail. Computational experience with the algorithm on several test problems is presented.     

离散时间Hopfield网络的动力系统分析

离散时间的Hopfield网络模型是一个非线性动力系统．对网络的状态变量引入新的能量函数，利用凸函数次梯度性质可以得到网络状态能量单调减少的条件．对于神经元的连接权值且激活函数单调非减(不一定严格单调增加)的Hopfield网络，若神经元激活函数的增益大于权值矩阵的最小特征值，则全并行时渐进收敛；而当网络串行时，只要网络中每个神经元激活函数的增益与该神经元的自反馈连接权值的和大于零即可．同时，若神经元激活函数单调，网络连接权值对称，利用凸函数次梯度的性质，证明了离散时间的Hopfield网络模型全并行时收敛到周期不大于2的极限环．     

A multi-item transportation problem with fuzzy tolerance

This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermann's method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermann's methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.