Singular linear quadratic differential games with bounds on control

A class of two-persons zero-sum differential games with the possibility of singular arcs is considered. Under some assumptions, a sufficient condition for the existence of a saddle-point solution is derived. In this approach, the properties of convexity and concavity of the performance index are used to find the sufficient condition. It is observed that, unlike total singularity, the occurrence of partially singular arcs does not cause any problems for the existence of a unique saddle-point solution.     

On the saddle-point solution of a class of stochastic differential games

This paper deals with the saddle-point solution of a class of stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that both players have access to a common noisy linear measurement of the state and they are permitted to utilize only this information in constructing their controls. The saddle-point solution of such differential game problems has been discussed earlier in Ref. 1, but the conclusions arrived there are incorrect, as is explicitly shown in this paper. We extensively discuss the role of information structure on the saddle-point solution of such stochastic games (specifically within the context of an illustrative discrete-time example) and then obtain the saddle-point solution of the problem originally formulated by employing an indirect approach.This work was done while the author was on sabbatical leave at Twente University of Technology, Department of Applied Mathematics, Enschede, Holland, from Applied Mathematics Division, Marmara Scientific and Industrial Research Institute, Gebze, Kocaeli, Turkey.     

On a class of differential games without saddle-point solutions

The minimax solution of a linear regulator problem is considered. A model representing a game situation in which the first player controls the dynamic system and selects a suitable, minimax control strategy, while the second player selects the aim of the game, is formulated. In general, the resulting differential game does not possess a saddle-point solution. Hence, the minimax solution for the player controlling the dynamic system is sought and obtained by modifying the performance criterion in such a way that (a) the minimax strategy remains unchanged and (b) the modified game possesses a saddle-point solution. The modification is achieved by introducing a regularization procedure which is a generalization of the method used in an earlier paper on the quadratic minimax problem. A numerical algorithm for determining the nonlinear minimax strategy in feedback form, in which Pagurek's result on open-loop and closed-loop sensitivity is used to nontrivially simplify the computational aspects of the problem, is presented and applied on a simple example.     

Game people play: Ayo

An analysis of Ayo is presented in this paper. The game is briefly described and it is shown that myopic decision guarantees solution to the game. The “odu” concept is discussed and it is shown that this strategy does not alter the solution methodology applied to solve the game without it. It is shown that the payoff matrix does not always have a saddle-point. Nevertheless, solution exists.     

Some thoughts on saddle-point conditions and information structures in zero-sum differential games

For a very simple two-stage, linear-quadratic, zero-sum difference game with dynamic information structure, we show that (i) there exist nonlinear saddle-point strategies which require the same existence conditions as the well-known linear, closed-loop, no-memory solution and (ii) there exist both linear and nonlinear saddle-point strategies which require more stringent conditions than the unique open-loop solution. We then discuss the implication of this result with respect to the existence of saddle points in zero-sum differential games for different information patterns.     

Extremum principles in electromagnetic systems

Variational expressions and saddle-point (or “mini-max”) principles for linear problems in electromagnetism are proposed. When conservative conditions are considered, well-known variational expressions for the resonant frequencies of a cavity and the propagation constant of a waveguide are revised directly in terms of electric and magnetic field vectors. In both cases the unknown constants are typefied as stationary (but not extremum) points of some energy-like functionals. On the contrary, if dissipation is involved then variational expressions achieve the extremum property. Indeed, we point out that a saddle-point characterizes the unique solution of Maxwell equations subject to impedancelike dissipative boundary conditions. In particular, we deal with the quasi-static problem and the time-harmonic case.     

Uzawa iteration method for stokes type variational inequality of the second kind

In this paper,the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind.Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity.Moreover,the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ?.Subsequently,the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate.Finally,we give the numerical results to verify the feasibility of the Uzawa algorithm.     

Backward Stability Analysis of Weighted Linear Least-Squares Problems

A provably backward stable algorithm for the solution of weighted linear least-squares problems with indefinite diagonal weighted matrices is presented. However, a similar algorithm is not necessarily backward stable, when the weighted matrices are generalized saddle-point matrices. Thus, conditions are derived under which the algorithm is provably backward stable.     

On saddle-point optimality in differential games

A family of two-person, zero-sum differential games in which the admissible strategies are Borel measurable is defined, and two types of saddle-point conditions are introduced as optimality criteria. In one, saddle-point candidates are compared at each point of the state space with all playable pairs at that point; and, in the other, they are compared only with strategy pairs playable on the entire state space. As a theorem, these two types of optimality are shown to be equivalent for the defined family of games. Also, it is shown that a certain closure property is sufficient for this equivalence. A game having admissible strategies everywhere constant, in which the two types of saddle-point candidates are not equivalent, is discussed.This paper is based on research supported by ONR.     

Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics

The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.     

Minimax design of two-state k-out-of-n systems

The optimal size of k is specified for two-state k-out-of-n systems that may be functioning or fail in either state. It is assumed that the steady-state, success and failure probabilities are not known exactly. The problem is reduced to finding the saddle-point solution to a minimax optimization problem. An example shows that the minimax design is robust with regard to uncertainty.     

A near-optimal closed-loop solution method for nonsingular zero-sum differential games

This paper presents a method for generating nearoptimal closed-loop solutions to zero-sum perfect information differential games with and without the final time explicitly specified, and with and without control constraints. This near-optimal closed-loop solution is generated by periodically updating the solution to the two-point boundary-value problem obtained by the application of the necessary conditions for a saddle-point solution. The resulting updated open-loop control is then used between updating intervals. Three examples are presented to illustrate the application of this method.     

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR SADDLE-POINT PROBLEM

1. IntroductionThere are many work to investigate the stability of the mired finite element methodfor the saddle-point problems, i.e., to construct the finite element spaces, such that theso-called discrete BB-codition is satisfied (c.f. [1],[21,[7],[81 and the references therein).To circumvent the discrete BB-conditon, recently there has been an increased interest inuse of least-squares approach for the solution of the mixed finite element approximationof the saddel-point problem (c.f.[3]--[…     

Singular surfaces in a linear pursuit-evasion game with elliptical vectograms

In a linear pursuit-evasion game with elliptical vectograms, singular surfaces appear in the plane of symmetry containing the ellipses minor axes and, as a consequence, locally nonsmooth isocost tubes are generated. Both dispersal and attractive focal surfaces are encountered. The dispersal surface is a zone of initial conditions for trajectories leaving the plane of symmetry. Optimal trajectories attracted by the focal surface merge tangentially to the plane of symmetry and remain there until the boundary of the dispersal zone is reached. Determination of the saddle-point strategies in the focal surface leads to constructing the isocost surfaces in the entire game space.     

Modified parameterized inexact Uzawa method for singular saddle-point problems

As well known, each of the consistent singular saddle-point (CSSP) problems has more than one solutions, and most of the iteration methods can only be proved to converge to one of the solutions of the CSSP problem. However, we do not know which solution it is and whether this solution depends on the initial iteration guesses. In this work, we introduce a new iteration method by slightly modifying the parameterized inexact Uzawa (PIU) iteration scheme. Theoretical analysis shows that, under suitable restrictions on the involved iteration parameters, the iteration sequence produced by the new method converges to the solution \(\mathcal {A}^{\dag }b\) for any initial guess, no matter the singular saddle-point system \(\mathcal {A}~x=b\) is consistent or inconsistent, where \(\mathcal {A}^{\dag }\) denotes the Moore-Penrose inverse of matrix \(\mathcal {A}\). In addition, the quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor are determined. Numerical examples are given to verify the correctness of the theoretical results and the effectiveness of our new method.     

ON AUGMENTED LAGRANGIAN METHODS FOR SADDLE-POINT LINEAR SYSTEMS WITH SINGULAR OR SEMIDEFINITE （1, 1） BLOCKS

An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite （1, 1） blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.     

Equivalence of saddle-points and optima for non-concave programmes

A basic result of optimisation theory is that a saddle-point of the Lagrangian is an optimum of the associated programming problem, independently of any concavity assumptions. It is also well known that under concavity assumptions the two are equivalent; i.e., an optimum is always a saddle-point. It is demonstrated that this basic equivalence of saddle-points and optima in fact holds for a much larger class of problems, which are not necessarily concave, but are equivalent to concave programmes up to a diffeomorphism. This class generalises the class of geometric programmes.     

Fast and reliable high-accuracy computation of Gauss–Jacobi quadrature

Numerical Algorithms - There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of...