Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes

In this paper, the authors first study two kinds of stochastic differential equations (SDEs) with Lévy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations (BSDEs) driven by Lévy processes, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Lévy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results. This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904, the Natural Science Foundation of China under Grant No. 10671112 and Shandong Province under Grant No. Z2006A01, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018.     

Global convergence of a class of smooth penalty methods for semi-infinite programming

For the semi-infinite programming (SIP) problem, the authors first convert it into an equivalent nonlinear programming problem with only one inequality constraint by using an integral function, and then propose a smooth penalty method based on a class of smooth functions. The main feature of this method is that the global solution of the penalty function is not necessarily solved at each iteration, and under mild assumptions, the method is always feasible and efficient when the evaluation of the integral function is not very expensive. The global convergence property is obtained in the absence of any constraint qualifications, that is, any accumulation point of the sequence generated by the algorithm is the solution of the SIP. Moreover, the authors show a perturbation theorem of the method and obtain several interesting results. Furthermore, the authors show that all iterative points remain feasible after a finite number of iterations under the Mangasarian-Fromovitz constraint qualification. Finally, numerical results are given.     

关于动态用户最优模型的几点分析

Ran等人(1996)构造了动态用户最优的变分不等式(VI)模型，并利用变分不等式问题与NLP之间的等价关系设计了松弛的优化算法。在深入分析Ran等(1996)的模型及求解算法的基础土，对其中的一些概念、理论推导以及算法过程进行了一定的改进。在此基础上，提出一个改进的动态用户最优(DUO)条件下的变分不等式(VI)模型。     

A Nonlinear Age-Structured Forest EvolutionSystem

ＡＮｏｎｌｉｎｅａｒＡｇｅ－ＳｔｒｕｃｔｕｒｅｄＦｏｒｅｓｔＥｖｏｌｕｔｉｏｎＳｙｓｔｅｍＡＮｏｎｌｉｎｅａｒＡｇｅ－ＳｔｒｕｃｔｕｒｅｄＦｏｒｅｓｔＥｖｏｌｕｔｉｏｎＳｙｓｔｅｍ￥ＷＡＮＧＤｉｎｇｊｉａｎｇ（ＰｉｎｇｄｉｎｇｓｈａｎＴｅａｃｈｅｒ＇...     

THE POPULATION DYNAMICS WITH N-INTERSECTIONS

ＴＨＥＰＯＰＵＬＡＴＩＯＮＤＹＮＡＭＩＣＳＷＩＴＨＮ－ＩＮＴＥＲＳＥＣＴＩＯＮＳ￥ＧＵＯＢａｏｚｈｕ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓＢｅｉｊｉｎｇＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，Ｂｅｉｊｉｎｇ１０００８１，Ｃ...     

混合交通方式划分与交通分配联合模型

目前我国很多城市各种类型车辆混合行驶，本文重点研究对称混合交通条件下方式划分与交通分配联合模型。描述了UE原则的变分不等式问题。在对称混合交通条件下，证明该联合模型的最优性条件与Wardrop用户均衡原则等价，以及路段流量最优解的唯一性，得到了Loglt形式的分担率模型。对实现该联合模型算法，提高算法的效率具有指导作用。     

Adaptive Numerical Approach for Optimal Control of a Single Train

This paper considers the optimal control problem of a single train, which is formulated as an optimal control problem of nonlinear systems with switching controller. The switching sequence and the switching time are decision variables to be chosen optimally. Generally speaking, it is very difficult to solve this problem analytically due to its nonlinear nature, the complexity of the controller,and the existence of system state and control input constraints. To obtain the numerical solution, by introducing binary functions for every value of the control input, relaxing the binary functions, and imposing a penalty function on the relaxation, the problem is transformed into a parameter optimization problem, which can be efficiently solved by using any gradient-based numerical approach. Then, the authors propose an adaptive numerical approach to solve this problem. Convergence results indicate that any optimal solution of the parameter optimization problem is also an optimal solution of the original problem. Finally, an optimal control problem of a single train illustrates that the adaptive numerical approach proposed by us is less time-consuming and obtains a better cost function value than the existing approaches.     

基于限价订单簿的最优高频做市商策略研究

建立具有成交风险和存货风险的价差过程模型,在引入存货惩罚函数的同时将策略的目标确定为效用最大化.将策略求解的过程看成是随机最优控制问题,并通过动态规划求解,离散模型框架下采用有限差分的方法对每个时间点不同存货及市场价差下的下单策略进行求解.该策略满足了模型定义之初对于成交强度,市场价差及存货量对下单行为影响的假设,而策略的实证及可靠性检验进一步表明了该策略具有较为稳定的收益.     

A quadratic objective penalty function for bilevel programming

The bilevel programming is applied to solve hierarchical intelligence control problems in such fields as industry, agriculture, transportation, military, and so on. This paper presents a quadratic objective penalty function with two penalty parameters for inequality constrained bilevel programming. Under some conditions, the optimal solution to the bilevel programming defined by the quadratic objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the quadratic objective penalty function, an algorithm is developed to find an optimal solution to the original bilevel programming, and its convergence proved under some conditions. Furthermore, under the assumption of convexity at lower level problems, a quadratic objective penalty function without lower level problems is defined and is proved equal to the original bilevel programming.     

An M-Objective Penalty Function Algorithm Under Big Penalty Parameters

Some classical penalty function algorithms may not always be convergent under big penalty parameters in Matlab software, which makes them impossible to find out an optimal solution to constrained optimization problems. In this paper, a novel penalty function (called M-objective penalty function) with one penalty parameter added to both objective and constrained functions of inequality constrained optimization problems is proposed. Based on the M-objective penalty function, an algorithm is developed to solve an optimal solution to the inequality constrained optimization problems, with its convergence proved under some conditions. Furthermore, numerical results show that the proposed algorithm has a much better convergence than the classical penalty function algorithms under big penalty parameters, and is efficient in choosing a penalty parameter in a large range in Matlab software.     

弹性需求用户平衡分配模型及其应用

对具有弹性需求的交通网络随机用户平衡分配问题进行了分析研究,将其描述为一个等价的变分不等式。最后将模型应用于拥护公交网络系统的平衡分配问题,并针对公交网络系统的特殊性,提出相应的求解算法 法。实例表明了模型的合理性和算法的有效性。     

Existence and Uniqueness Results for Quaternion-Valued Nonlinear Impulsive Differential Systems

This paper studies the existence and uniqueness conditions for the quaternion-valued nonlinear impulsive system. Firstly, a space of quaternion-valued piecewise functions is constructed and completeness of the space is also proved. Then by taking advantage of the Bielecki norm and fixed point theorem, existence and uniqueness criteria of quaternion-valued nonlinear impulsive system are obtained. At last, an example is given to illustrate our theoretical results.     

VIBRATIONAL SYSTEMS OF A CLASS OF THIN CYLINDRICAL SHELLS

The vibration of thin circular cylindrical shells has been studied extensively.In this paper, the existence and uniqueness of the solution of this system are proved bymeans of semigroup theory on the Hilbert space, and the eigenvalue problem is discussed.This offers a theoretical basis for solving this kind of engineering problems.     

用投影收缩算法求解考虑排放的交通分配模型

给出一种考虑排放约束条件下的交通分配均衡模型,在所定义的广义出行费用中明确考虑排放因素,通过将变分不等式问题转换成等价的非线性互补问题,采用一种自适应的投影收缩算法求解该模型,所给出的数值算例得到广义费用下的UE最优解,在同样收敛精度的要求下,和其他投影算法相比,采用该算法收敛速度较快.     

Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings

This paper introduces a three-step iteration for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping by viscosity approximation methods in a Hilbert space. The authors show that the iterative sequence converges strongly to a common element of the two sets, which solves some variational inequality. Subsequently, the authors consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the corresponding results announced by Nakajo, Takahashi, and Toyoda. This research is supported by the National Natural Science Foundation of China under Grant No. 10771050.     

EQUILIBRIUM PROBLEMS ON A NETWORK WITH VECTOR-VALUED COST FUNCTIONS

ＥＱＵＩＬＩＢＲＩＵＭＰＲＯＢＬＥＭＳＯＮＡＮＥＴＷＯＲＫＷＩＴＨＶＥＣＴＯＲ－ＶＡＬＵＥＤＣＯＳＴＦＵＮＣＴＩＯＮＳＣＨＥＮＧｕａｎｇｙａ（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ１０００８０...     

利用IC 卡数据估计公交OD 矩阵的模型及算法

利用公交IC卡数据建立了估计公交系统OD矩阵的双层数学规划模型,上层问题为广义最小二乘模型,下层问题为以变分不等式表示的基于超路径的公交网络均衡配流模型.采用启发式算法求解该双层数学规划模型,由于非对称性,运用了自适应的投影方法来求解下层的变分不等式问题,算例表明了模型的可行性和算法的有效性.     

New Results on Interval General Cohen-Grossberg BAM Neural Networks

This paper is concerned with an interval general Cohen-Grossberg bidirectional associative memory neural networks with mixed delays. Under proper conditions, the authors studied the existence,the uniqueness and the global exponential stability of almost automorphic solutions for the suggested system. The proposed method was mainly based on the exponential dichotomy of linear differential equation, the Banach's fixed point principle and the differential inequality techniques. The authors illustrate with an example to demonstrate the effectiveness of the proposed findings.     

纳什均衡问题解的特征

将纳什均衡问题转化为一个求解变分不等式问题 .在满足 nested $monotone^+$的条件下 ,给出求纳什均衡的思想 ,并对纳什均衡解的特征作了描述 .     

NONLINEAR GALERKIN METHOD FOR STEADYNONLINEAR DIFFERENTIAL EQUATIONS

1.IntroductionNonlinearGalerkinmethodsarenumericalschemesforthedissipativeevolutionpartialdifferelltialequationswherethespatialdiscretizationreliesonanonlinearmanifoldinsteadofalinearspaceasintheclassicalGalerkinmethod.Moreprecisely,oneconsidersafinitedim…