5,572 research outputs found
The error and perturbation bounds for the absolute value equations with some applications
To our knowledge, so far, the error and perturbation bounds for the general
absolute value equations are not discussed. In order to fill in this study gap,
in this paper, by introducing a class of absolute value functions, we study the
error bounds and perturbation bounds for two types of absolute value equations
(AVEs): Ax-B|x|=b and Ax-|Bx|=b. Some useful error bounds and perturbation
bounds for the above two types of absolute value equations are presented. By
applying the absolute value equations, we also obtain the error and
perturbation bounds for the horizontal linear complementarity problem (HLCP).
In addition, a new perturbation bound for the LCP without constraint conditions
is given as well, which are weaker than the presented work in [SIAM J. Optim.,
2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some
computable estimates for the above upper bounds are given, which are sharper
than some existing results under certain conditions. Some numerical examples
for the AVEs from the LCP are given to show the feasibility of the perturbation
bounds
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Extensive-Form Perfect Equilibrium Computation in Two-Player Games
We study the problem of computing an Extensive-Form Perfect Equilibrium
(EFPE) in 2-player games. This equilibrium concept refines the Nash equilibrium
requiring resilience w.r.t. a specific vanishing perturbation (representing
mistakes of the players at each decision node). The scientific challenge is
intrinsic to the EFPE definition: it requires a perturbation over the agent
form, but the agent form is computationally inefficient, due to the presence of
highly nonlinear constraints. We show that the sequence form can be exploited
in a non-trivial way and that, for general-sum games, finding an EFPE is
equivalent to solving a suitably perturbed linear complementarity problem. We
prove that Lemke's algorithm can be applied, showing that computing an EFPE is
-complete. In the notable case of zero-sum games, the problem is
in and can be solved by linear programming. Our algorithms also
allow one to find a Nash equilibrium when players cannot perfectly control
their moves, being subject to a given execution uncertainty, as is the case in
most realistic physical settings.Comment: To appear in AAAI 1
A sequential semidefinite programming method and an application in passive reduced-order modeling
We consider the solution of nonlinear programs with nonlinear
semidefiniteness constraints. The need for an efficient exploitation of the
cone of positive semidefinite matrices makes the solution of such nonlinear
semidefinite programs more complicated than the solution of standard nonlinear
programs. In particular, a suitable symmetrization procedure needs to be chosen
for the linearization of the complementarity condition. The choice of the
symmetrization procedure can be shifted in a very natural way to certain linear
semidefinite subproblems, and can thus be reduced to a well-studied problem.
The resulting sequential semidefinite programming (SSP) method is a
generalization of the well-known SQP method for standard nonlinear programs. We
present a sensitivity result for nonlinear semidefinite programs, and then
based on this result, we give a self-contained proof of local quadratic
convergence of the SSP method. We also describe a class of nonlinear
semidefinite programs that arise in passive reduced-order modeling, and we
report results of some numerical experiments with the SSP method applied to
problems in that class
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