基于个人超出值的模糊联盟合作博弈最小二乘预核仁

本文给出了基于个人超出值的无限模糊联盟合作博弈最小二乘预核仁的求解模型,得到该模型的显式解析解,并研究该解的若干重要性质。证明了:本文给出的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的相等解(The equalizer solution),基于个人超出值的字典序解三者相等。进一步证明了:基于Owen线性多维扩展的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的经典合作博弈最小二乘预核仁相等。最后,通过数值实例说明本文提出的无限模糊联盟合作博弈求解模型的实用性与有效性。     

模糊支付合作博弈的可信性准核仁

合作博弈是处理局中人之间协同行为的数学理论。有诸如核心、稳定集、沙普利值、准核仁和核仁等不同的解概念。在很多情形,除了借助专家经验和主观直觉,没有恰当的方式来确定支付函数,由此产生了具有模糊支付的合作博弈模型。准核仁是一种重要的解概念,在模糊支付合作博弈中如何恰当定义准核仁是个重要的问题。本文在可信性理论的框架下研究了这个问题,定义了两类可信性准核仁概念并证明了它们的存在性和唯一性,同时研究了可信性核心、可信性核仁与它们之间的关系。     

The least square nucleolus is a general nucleolus

This short note proves that the least square nucleolus (Ruiz et al. (1996)) and the lexicographical solution (Sakawa and Nishizaki (1994)) select the same imputation in each game with nonempty imputation set. As a consequence the least square nucleolus is a general nucleolus (Maschler et al. (1992)). Received: December 1998/Revised version: July 1999     

The modified nucleolus: Properties and axiomatizations

A new solution concept for cooperative transferable utility games is introduced, which is strongly related to the nucleolus and therefore called modified nucleolus. It takes into account both the “power”, i.e. the worth, and the “blocking power” of a coalition, i.e. the amount which the coalition cannot be prevented from by the complement coalition. It can be shown that the modified nucleolus is reasonable, individually rational for weakly superadditive games, coincides with the prenucleolus for constant-sum games, and is contained in the core for convex games. Finally this paper proposes two axiomatizations of this solution concept on the set of games on an infinite universe of players which are similar to Sobolev's characterization of the prenucleolus.     

A smoothing least square method for nonlinear complementarity problem

By using the smoothing functions and the least square reformulation, in this paper, we present a smoothing least square method for the nonlinear complementarity problem. The method can overcome the difficulty of the non‐smooth method and a major drawback of some existed equation‐based methods. Under the standard assumptions, we obtain the global convergence of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust least square semidefinite programming with applications

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196–1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.     

The simplified modified nucleolus of a cooperative TU-game

In the present paper, we introduce a new solution concept for TU-games, the simplified modified nucleolus or the SM-nucleolus. It is based on the idea of the modified nucleolus (the modiclus) and takes into account both the constructive power and the blocking power of a coalition. The SM-nucleolus inherits this convenient property from the modified nucleolus, but it avoids its high computational complexity. We prove that the SM-nucleolus of an arbitrary n-person TU-game coincides with the prenucleolus of a certain n-person constant-sum game, which is constructed as the average of the game and its dual. Some properties of the new solution are discussed. We show that the SM-nucleolus coincides with the Shapley value for three-person games. However, this does not hold for general n-person cooperative TU-games. To confirm this fact, a counter example is presented in the paper. On top of this, we give several examples that illustrate similarities and differences between the SM-nucleolus and well-known solution concepts for TU-games. Finally, the SM-nucleolus is applied to the weighted voting games.     

The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games

This paper presents the aspiration approach to coalition formation and payoff distribution in games with sidepayments. The approach is based on the idea that players set prices for their participation within coalitions. The solution space which is appropriate for price-setting players is different from that of the usual solution concepts and is called the space of aspirations. Solution concepts defined on the space of aspirations correspond to notions of how players bargain over their prices. Once the players choose a vector of prices, the coalitions which can afford to pay these prices are the coalitions which are predicted to form in the game.     

Core,least core and nucleolus for multiple scenario cooperative games

Multiple scenario cooperative games model situations where the worth of the coalitions is valued in different scenarios simultaneously or under different states of nature. In this paper we analyze solution concepts for this class of games keeping the multidimensional nature of the characteristic function. We obtain extensions of the notions of core, least core and nucleolus, and explore the relationship among these solution concepts.     

On the computation of the nucleolus of a cooperative game

We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel. Received February 2000/Final version April 2001     

Error estimates for moving least square approximations

In this paper we obtain error estimates for moving least square approximations for the function and its derivatives. We introduce, at every point of the domain, condition numbers of the star of nodes in the normal equation, which are practically computable and are closely related to the approximating power of the method.     

Learning rates for least square regressions with coefficient regularization

We analyze the learning rates for the least square regression with data dependent hypothesis spaces and coefficient regularization algorithms based on general kernels. Under a very mild regularity condition on the regression function, we obtain a bound for the approximation error by estimating the corresponding K-functional. Combining this estimate with the previous result of the sample error, we derive a dimensional free learning rate by the proper choice of the regularization parameter.     

四元数体上一类矩阵方程的极小范数最小二乘解

借助于四元数体上自共轭矩阵的奇异值分解,给出了四元数矩阵方程AX＋XB＋CXD=F的极小范数最小二乘解.同时,在有解的条件下给出了Hermite最小二乘解及其通解的表达形式.     

Consistent and covariant solutions for TU games

One of the important properties characterizing cooperative game solutions is consistency. This notion establishes connections between the solution vectors of a cooperative game and those of its reduced game. The last one is obtained from the initial game by removing one or more players and by giving them the payoffs according to a specific principle (e.g. a proposed payoff vector). Consistency of a solution means that the restriction of a solution payoff vector of the initial game to any coalition belongs to the solution set of the corresponding reduced game. There are several definitions of the reduced games (cf., e.g., the survey of T. Driessen [2]) based on some intuitively acceptable characteristics. In the paper some natural properties of reduced games are formulated, and general forms of the reduced games possessing some of them are given. The efficient, anonymous, covariant TU cooperative game solutions satisfying the consistency property with respect to any reduced game are described.The research was supported by the NWO grant 047-008-010 which is gratefully acknowledgedReceived: October 2001     

不定最小二乘问题的改进的不完全双曲Gram-Schmidt预处理算法

应用改进的不完全双曲Gram-Schmidt(IHMGS)方法预处理不定最小二乘问题的共轭梯度法(CGILS)、正交分解法(ILSQR)与广义的最小剩余法(GMRES)等迭代算法来求解大型稀疏的不定最小二乘问题.数值实验表明,IHMGS预处理方法可有效提高相应算法的迭代速度,且当矩阵的条件数比较大时,效果更加显著.     

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoffs is introduced. It is based on extensions of the vector order in \(\mathbb {R}^d\) to order relations in the power set of \(\mathbb {R}^d\), so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified.     

Random reduction consistency of the Weber set,the core and the anti-core

In this paper we introduce a new consistency condition and provide characterizations for several solution concepts in TU cooperative game theory. Our new consistency condition, which we call the random reduction consistency, requires the consistency of payoff vectors assigned by a solution concept when one of the players is removed with some probability. We show that the random reduction consistency and other standard properties characterize the Weber set, the convex hull of the marginal contribution vectors. Another salient feature of random reduction consistency is that, by slightly changing its definition, we can characterize the core and the anti-core in a parallel manner. Our result enables us to compare the difference between the three solution concepts from the viewpoint of consistency.     

Some new results on least square values for TU games

In this paper we present some new results concerning the least square family of values for TU games. We first examine the stability of these values from a dynamic point of view and propose a reinteropretation of them from the standpoint of the propernsity to disrupt approach. In the second part the family of individually rational least square (IRLS) values is introduced and an alternative kernel-like formulation of them is provided. Finally we describe a natural and simple algorithm for calculating any IRLS value of a game. We gratefully acknowledge financial support from the Basque Government (Project PI95/101) and from the D.G.E.S. of the Ministerio de Educatión y Cultura (Project PB96-0247).     

On equilibria on the square

We construct a non-zero sum game on the square, with separately continuous payoff functions, and which has no correlated equilibrium. This answers a recent question of Nowak.