A stochastic algorithm using one sample point per iteration and diminishing stepsizes

A stochastic algorithm for finding stationary points of real-valued functions defined on a Euclidean space is analyzed. It is based on the Robbins-Monro stochastic approximation procedure. Gradient evaluations are done by means of Monte Carlo simulations. At each iteratex _i , one sample point is drawn from an underlying probability space, based on which the gradient is approximated. The descent direction is against the approximation of the gradient, and the stepsize is 1/i. It is shown that, under broad conditions, w.p.1 if the sequence of iteratesx ₁,x ₂,...generated by the algorithm is bounded, then all of its accumulation points are stationary.     

Regular population monotonic allocation schemes

The paper introduces a refinement of the notion of population monotonic allocation scheme, called regular population monotonic allocation scheme (regularpmas). This refinement is based on economic situations in which players may have to select new partners from a set of potential players and in which there exist certain capacity constraints. A sufficient condition for the existence of a regularpmas is given. For the class of games with regularpmas, we prove that the core coincides with the Davis and Maschler and the Mas-Colell bargaining sets.     

Index Heuristics for Multiclass M/G/1 Systems with Nonpreemptive Service and Convex Holding Costs

We consider the optimal service control of a multiclass M/G/1 queueing system in which customers are served nonpreemptively and the system cost rate is additive across classes and increasing convex in the numbers present in each class. Following Whittle's approach to a class of restless bandit problems, we develop a Langrangian relaxation of the service control problem which serves to motivate the development of a class of index heuristics. The index for a particular customer class is characterised as a fair charge for service of that class. The paper develops these indices and reports an extensive numerical investigation which exhibits strong performance of the index heuristics for both discounted and average costs.     

On Non-Continuous Dirichlet Processes

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C ¹ transformation.     

NCD系统的数学理论

无索赔折扣系统(No Claim Discount system,简记为NCD系统)是世界各国机动车辆险中广泛采用的一种经验费率厘定机制.本文尝试建立了NCD系统严谨的数学理论, 重点讨论了NCD系统的数学建模和稳态分析.此外,作为本文必要的数学前提,首先在第2节着重探讨了随机矩阵间的随机优序关系,并将所得结论运用至齐次不可约且遍历的马尔科夫链的研究中,这些内容也有其独立的数学上的兴趣.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

On Weak Solutions of an Integral Equation Driven by a p-Semimartingale of Special Type

We consider the integral equation driven by a standard Brownian motion and by a fractional Brownian motion. Sufficient conditions under which the equation has a weak solution are obtained.     

Control Policy for a Manufacturing System with Random Yield and Rework

We develop a production policy that controls work-in-process (WIP) levels and satisfies demand in a multistage manufacturing system with significant uncertainty in yield, rework, and demand. The problem addressed in this paper is more general than those in the literature in three aspects: (i) multiple products are processed at multiple workstations, and the capacity of each workstation is limited and shared by multiple operations; (ii) the behavior of a production policy is investigated over an infinite-time horizon, and thus the system stability can be evaluated; (iii) the representation of yield and rework uncertainty is generalized. Generalizing both the system structure and the nature of uncertainty requires a new mathematical development in the theory of infinite-horizon stochastic dynamic programming. The theoretical contributions of this paper are the existence proofs of the optimal stationary control for a stochastic dynamic programming problem and the finite covariances of WIP and production levels under the general expression of uncertainty. We develop a simple and explicit sufficient condition that guarantees the existence of both the optimal stationary control and the system stability. We describe how a production policy can be constructed for the manufacturing system based on the propositions derived.     

一类It型非线性时滞关联随机大系统的分散鲁棒控制

在非线性项满足全局Lipschitz条件下,本文研究了一类It型非线性时滞关联随机大系统的分散鲁棒控制问题.系统的时滞是关于状态和控制输入的.基于Lyapunov泛函及线性矩阵不等式(LMI)的分析方法,得到了无记忆状态反馈控制器使整个时滞关联随机大系统可镇定的充分条件.     

Creation, Dissipation and Recycling of Resources in Non-Equilibrium Systems

In this article, we define stochastic dynamics for a system coupled to reservoirs. The rules for forward and backward transitions are related by a generalized detailed balance identity involving the system and its reservoirs. We compare the variation of information and of entropy. We define the Carnot dissipation and prove that it can be expressed in terms of cyclic transformations. Lower bounds for partial dissipations are also studied, as well as the effect of switching off certain reservoirs. We also study the near degeneracy of the stochastic matrix, relate it to phase transitions and we show that the reduced dynamics on the set of phases is a permutation. Finally, we relate these concepts to heat, work and more generally to the dissipation and creation of resources, in general systems.