求解线性二层规划问题的多表旋转算法

多表旋转算法是一种基于旋转算法来求解线性二层规划问题的方法,通过表格组合还可以求解线性多层规划、以及线性一主多从有关联的stackelberg-nash均衡等问题,求解的思想是使用旋转算法,在多个主体间通过约束传递达到均衡。通过算例显示该方法可以迅速地算出局部最优解,如果问题的诱导域是连通的,还可以计算出全局最优解。     

Stackelberg–Nash exact controllability for the Kuramoto–Sivashinsky equation

This paper deals with a hierarchical control problem for the Kuramoto–Sivashinsky equation following a Stackelberg–Nash strategy. We assume that there is a main control, called the leader, and two secondary controls, called the followers. The leader tries to drive the solution to a prescribed target and the followers intend to be a Nash equilibrium for given functionals. It is known that this problem is equivalent to a null controllability result for an optimality system consisting of three non-linear equations. One of the novelties is a new Carleman estimate for a fourth-order equation with right-hand sides in Sobolev spaces of negative order, which allows to relax some geometric conditions for the observation sets for the followers.     

基于动机的联合风险投资收益分配契约设计(A2016-0158)

从联合风险投资的领导者采取联合投资的动机出发,考虑分配给跟随者的收益份额是否能达到目的,运用优化理论建立了联合投资双方的收益分配模型,提出了联合风险投资机构之间的收益分配契约设计的一种可供参考的方法,并用实例说明了模型方法的应用。研究还表明,潜在的市场竞争可能对领导者造成的损失越大,领导者越愿意给予跟随者更多的收益份额;当项目质量信息的不确定性越高,单独投资将存在较大风险的时候,为了获得项目质量信息的补充,也使得领导者愿意给予跟随者更多的收益份额。     

Systems of nonlinear Volterra integro-differential equations

An efficient method based on operational Tau matrix is developed, to solve a type of system of nonlinear Volterra integro-differential equations (IDEs). The presented method is also modified for the problems with separable kernel. Error estimation of the new schemes are analyzed and discussed. The advantages of this approach and its modification is that, the solution can be expressed as a truncated Taylor series, and the error function at any stage can be estimated. Methods are applied on the four problems with separable kernel to show the applicability and efficiency of our schemes, specially for those problems at broad intervals.     

Linear bilevel multi-follower programming with independent followers

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader’s variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563–578 (2005)] are based on a misconstruction of the inducible region.     

An inexact parallel splitting augmented Lagrangian method for large system of linear equations

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n²). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.     

Time-axis decomposition of large-scale optimal control problems

Continuous-time optimal control problems can rarely be solved directly but have to be approximated with discrete analogues. Shorter time steps lead to more accurate approximations, but result in formulations that are often too big for computer memory. This paper presents a technique for decomposing the problem along the time axis and iterating toward a solution in a leader-follower framework.In the model, the leader controls a set of coordination parameters, which he passes to the followers, who then solve their individual subproblems. State and sensitivity information is returned to the leader, who attempts to minimize an unconstrained problem in the coordination space. Parameters are updated and the process continues until improvement ceases. Two advantages of this technique are that feasible solutions to the original problem are available at each iteration and that the optimal coordination parameters obtained provide some measure of feedback control. Computational results are presented for a comprehensive set of test problems.This work was supported by a grant from the Advanced Research Program of the Texas Higher Education Coordinating Board.     

多寡头市场下领先者、追随者和竞争者的市场绩效比较分析

根据"结构-行为-绩效"的SCP分析框架,分析了企业在多寡头产量竞争的Cournot市场结构、多寡头价格竞争的Bertrand市场结构、1个领先者和多个追随者的Stackberg市场结构下,分别采取自主创新、跟踪新产品开发和引进模仿等不同的产品开发战略的市场绩效.结果表明,在同质产品多寡头市场上的产量竞争中,企业采取领先者、竞争者和追随者3种行为的企业均衡产量和企业利润依次递减;多寡头Stackberg市场结构在总产量、消费者剩余和社会福利上表现更佳;多寡头Cournot市场结构在市场价格和行业总利润上更高.在异质产品多寡头市场上的Bertrand价格竞争中,互补品市场的均衡价格和均衡产量相对于替代品都提高;当替代程度较大时,寡头数目较少,同时每个寡头的均衡产量和均衡价格都上升.     

Equilibrium selection in multi-leader-follower games with vertical information

We consider two-stage multi-leader-follower games, called multi-leader-follower games with vertical information, where leaders in the first stage and followers in the second stage choose simultaneously an action, but those chosen by any leader are observed by only one “exclusive” follower. This partial unobservability leads to extensive form games that have no proper subgames but may have an infinity of Nash equilibria. So it is not possible to refine using the concept of subgame perfect Nash equilibrium and, moreover, the concept of weak perfect Bayesian equilibrium could be not useful since it does not prescribe limitations on the beliefs out of the equilibrium path. This has motivated the introduction of a selection concept for Nash equilibria based on a specific class of beliefs, called passive beliefs, that each follower has about the actions chosen by the leaders rivals of his own leader. In this paper, we illustrate the effectiveness of this concept and we investigate the existence of such a selection for significant classes of problems satisfying generalized concavity properties and conditions of minimal character on possibly discontinuous data.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Fuzzy bilevel programming with multiple objectives and cooperative multiple followers

Classic bilevel programming deals with two level hierarchical optimization problems in which the leader attempts to optimize his/her objective, subject to a set of constraints and his/her follower’s solution. In modelling a real-world bilevel decision problem, some uncertain coefficients often appear in the objective functions and/or constraints of the leader and/or the follower. Also, the leader and the follower may have multiple conflicting objectives that should be optimized simultaneously. Furthermore, multiple followers may be involved in a decision problem and work cooperatively according to each of the possible decisions made by the leader, but with different objectives and/or constraints. Following our previous work, this study proposes a set of models to describe such fuzzy multi-objective, multi-follower (cooperative) bilevel programming problems. We then develop an approximation Kth-best algorithm to solve the problems.     

上三角算子矩阵的本征函数展开法在应力形式的二维弹性问题中的应用

深入研究了求解基于应力形式的二维弹性问题的本征函数展开法．根据已有的研究结果,将基于应力形式的二维弹性问题的基本偏微分方程组等价地转化为上三角微分系统,并导出了相应的上三角算子矩阵．通过深入研究,分别获得了该算子矩阵的两个对角块算子更为简洁的正交本征函数系,并证明了它们在相应空间中的完备性,进而应用本征函数展开法给出了该二维弹性问题的更为简洁实用的一般解．此外,对该二维弹性问题,还指出了什么样的边界条件可以应用此方法求解．最后应用具体的算例验证了所得结论的合理性．     

A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

In this paper, we propose a general iterative scheme based on CQ projection method for finding a common solution of system of equilibrium problems and the fixed point set of a finite family of demicontractive mappings. We also prove strong convergence of the scheme to a common element of the two above-described sets. We then give a numerical example to justify our main result. An example is given in an infinite dimensional space for supporting our main result. Moreover, we apply our main result to solve the unconstrained image restoration problems with a finite family of blurring operators. Our results extend and improve some existing results in the literature.     

Hierarchic Control for the Wave Equation

This paper deals with the hierarchical control of the wave equation. We use Stackelberg–Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves an exact controllability problem. We consider linear and semilinear equations.     

Controlled predatory pricing in a multiperiod Stackelberg game: an MPEC approach

We analyze a multiperiod oligopolistic market where each period is a Stackelberg game between a leader firm and multiple follower firms. The leader chooses his production level first, taking into account the reaction of the followers. Then, the follower firms decide their production levels after observing the leader’s decision. The difference between the proposed model and other models discussed in literature is that the leader firm has the power to force the follower firms out of business by preventing them from achieving a target sales level in a given time period. The leader firm has an incentive to lower the market prices possibly lower than the Stackelberg equilibrium in order to push the followers to sell less and eventually go out of business. Intentionally lowering the market prices to force competitors to fail is known as predatory pricing, and is illegal under antitrust laws since it negatively affects consumer welfare. In this work, we show that there exists a predatory pricing strategy where the market price is above the average cost and consumer welfare is preserved. We develop a mixed integer nonlinear problem (MINLP) that models the multiperiod Stackelberg game. The MINLP problem is transformed to a mixed integer linear problem (MILP) by using binary variables and piecewise linearization. A cutting plane algorithm is used to solve the resulting MILP. The results show that firms can engage in predatory pricing even if the average market price is forced to remain higher than the average cost. Furthermore, we show that in order to protect the consumers, antitrust laws can control predatory pricing by setting rules on consumer welfare.     

解一类结构变分不等式问题的非精确并行交替方向法

带线性约束的具有两分块结构的单调变分不等式问题, 出现在许多现代应用中, 如交通和经济问题等. 基于该问题良好的可分结构, 分裂型算法被广泛研究用于其求解. 提出新的带回代的非精确并行交替方向法解该类问题, 在每一步迭代中,首先以并行模式通过投影得到预测点, 然后对其校正得到下一步的迭代点. 在压缩型算法的理论框架下, 在适当条件下证明了所提算法的全局收敛性. 数值结果表明了算法的有效性. 此外, 该算法可推广到求解具有多分块结构的问题.     

Phenomena in Inverse Stackelberg Games,Part 1: Static Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

Variational Inequality Formulation of a Class of Multi-Leader-Follower Games

The multi-leader-follower game can be looked on as a generalization of the Nash equilibrium problem and the Stackelberg game, which contains several leaders and a number of followers. Recently, the multi-leader-follower game has been drawing more and more attention, for example, in electricity power markets. However, when we formulate a general multi-leader-follower game as a single-level game, it will give rise to a lot of problems, such as the lack of convexity and the failure of constraint qualifications. In this paper, to get rid of these difficulties, we focus on a class of multi-leader-follower games that satisfy some particular, but still reasonable assumptions, and show that these games can be formulated as ordinary Nash equilibrium problems, and then as variational inequalities. We establish some results on the existence and uniqueness of a leader-follower Nash equilibrium. We also present illustrative numerical examples from an electricity power market model.     

部分信息下一类一主多从激励策略

本文研究在部分信息下,一主多从且从者之间非合作的激励问题,给出了寻找主者的激励策略的一般方法;并对线性二次情形提出了仿射型激励策略所应满足的充分条件.     

EQUILIBRIUM EXISTENCE FOR MULTI-LEADER-FOLLOWER GENERALIZED CONSTRAINED MULTIOBJECTIVE GAMES IN LOCALLY FC-UNIFORM SPACES

In this article,we introduce and study some new classes of multi-leader-follower generalized constrained multiobjective games in locally FC-uniform spaces where the number of leaders and followers may be finite or infinite and the objective functions of the followers obtain their values in infinite-dimensional spaces.Each leader has a constrained correspondence.By using a collective fixed point theorem in locally FC-uniform spaces due to author,some existence theorems of equilibrium points for the multi-leader-follower generalized constrained multiobjective games are established under nonconvex settings.These results generalize some corresponding results in recent literature.