Saddle point methods,and alogorithms,for non-symmetric linear equations ∗

This paper describes methods for solving non-singular, non-symmetric linear equations whose symmetric part is positive definite. First, the solutions are characterized as saddle points of a convex-concave function. The associated primal and dual variational principles provide quadratic, strictly convex, functions whose minima are the solutions of the original equation and which generalize the energy function for symmetric problems.

Direct iterative methods for finding the saddle point are then developed and analyzed. A globally convergent algorithm for finding the saddle points is described. We show that requiring conjugacy of successive search directions with respect to the symmetric part of the equation is a poor strategy.     

Feedback saddle point solution of counterterror measures and economic growth game

In this paper, the feedback Nash equilibrium solutions of the differential game between counterterror measures and economic growth are investigated. The Hamilton–Jacobi–Isaacs (HJI) equation is used to obtain the feedback saddle point of this zero-sum game. Moreover, the characteristics of the feedback strategies for the government and terrorist organization as well as their relationships with both resource states are analyzed.     

Saddle point equilibrium under uncertain environment

The saddle point equilibrium problem is of great importance in differential game theory. In this paper, an optimistic value model for the saddle point equilibrium problem under uncertain environment is investigated. The equilibrium equation for the proposed model is presented. Then a linear quadratic model is discussed. Finally, a counter terror problem is analyzed by the results obtained in the paper.     

Phase boundary solutions to model kinetic equations via the Conley index theory. Part I

In this paper, we consider phase boundary solutions to a four-velocity kinetic model of a kinetic equation governing the motion of van der Waals fluids. These solutions connect such equilibrium states, which are saddle critical points of the suitable dynamic system. Solutions of this type can be interpreted as dynamic phase transition. The mathematical apparatus is that of the Coney index theory.     

Slow Passage Through the Nonhyperbolic Homoclinic Orbit of the Saddle-Center Hamiltonian Bifurcation

Slowly varying Hamiltonian systems, for which action is a well-known adiabatic invariant, are considered in the case where the system undergoes a saddle center bifurcation. We analyze the situation in which the solution slowly passes through the nonhyperbolic homoclinic orbit created at the saddle-center bifurcation. The solution near this homoclinic orbit consists of a large sequence of homoclinic orbits surrounded by near approaches to the autonomous nonlinear nonhyperbolic saddle point. By matching this solution to the strongly nonlinear oscillations obtained by averaging before and after crossing the homoclinic orbit, we determine the change in the action. If one orbit comes sufficiently close to the nonlinear saddle point, then that one saddle approach instead satisfies the nonautonomous first Painlevé equation, whose stable manifold of the unstable saddle (created in the saddle-center bifurcation) separates solutions approaching the stable center from those involving sequences of nearly homoclinic orbits.     

Nonlinear behavior for nanoscale electrostatic actuators with Casimir force

The influence of Casimir force on the nonlinear behavior of nanoscale electrostatic actuators is studied in this paper. A one degree of freedom mass-spring model is adopted and the bifurcation properties of the actuators are obtained. With the change of the geometrical dimensions, the number of equilibrium point varies from zero to two. Stability analysis shows that one equilibrium point is Hopf point and the other is unstable saddle point when there are two equilibrium points. We also obtain the phase portraits, in which the periodic orbits exist around the Hopf point, and a homoclinic orbit passes through the unstable saddle point.     

Competitive facility location under attrition

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

     

一类四阶差分系统的多重周期解（英文）

By using critical point theory,we study periodic solutions for a class of fourthorder difference systems with partially periodic potential and linear nonlinearity.Some sufficient conditions for the existence of multiplicity of periodic solutions are obtained via generalized saddle point theorem.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue

Some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory.     

Multicriteria equilibrium programming: Extragradient method

Multicriteria equilibrium programming includes as its particular cases mathematical programming, saddle point calculation, the multicriteria search for Pareto solutions, minimization with an equilibrium choice of the feasible set, etc. An extragradient method is proposed for the numerical solution of the multicriteria equilibrium programming problem, and the convergence of this method is examined.     

Zero-sum risk-sensitive stochastic games for continuous time Markov chains

We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game, we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game, we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.     

Inf–sup conditions for twofold saddle point problems

Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.     

序线性空间中向量优化问题的K-T型定理

本文在广义凸性条件下,研究实线性空间中一类向量优化问题的最优性条件,我们引入F ritz-John鞍点,、“K-T鞍点”讨论它们与有效解、弱有效解之间的关系.     

Existence of periodic solutions of second order nonlinear p-Laplacian difference equations

By using the critical point theory, the existence of periodic solutions to second order nonlinear p-Laplacian difference equations is obtained. The main approach used is a variational technique and the saddle point theorem. The problem is to solve the existence of periodic solutions of second order nonlinear p-Laplacian difference equations.     

Variational approach for a p-Laplacian boundary value problem on time scales

In this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem.     

A Class of Solvable Stopping Games

We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.     

向量映射的鞍点和Lagrange对偶问题

本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。     

集值映射多目标半定规划的Kuhn-Tucker鞍点最优性条件

首先在序拓扑线性空间中定义了集值映射多目标半定规划问题的KuhnTucker鞍点,在广义锥-次类凸条件下,讨论了此集值优化问题的弱有效解和Benson真有效性解与Kuhn-Tucker鞍点之间的关系.     

Partially observed semi-Markov zero-sum games with average payoff

We study a zero-sum partially observed semi-Markov game with average payoff on a countable state space. Under certain ergodicity conditions we show that a saddle point equilibrium exists. We achieve this by solving the corresponding average cost optimality equation using a span contraction method. The average value is shown to be the unique zero of a Lipschitz continuous function. A value iteration scheme is developed to compute the value.