on relaxation algorithms in computation of noncooperative equilibria this paper considers a special class of numerical algorithms the so called relaxation algorithm for nash equilibrium points in noncooperative games the relaxation algorithms have been studied by various authors for the deterministic case convergence conditions of this algorithm are based on fixed point theorems for example basar 1987 and li and basar 1987 have proved its convergence for a two player game via the contraction mapping theorem for the quadratic case these conditions can be easily checked for other nonlinear payoff functions it is sometimes difficult to check these convergence conditions in this paper the authors propose an alternative approach using the residual terms of the nikaido isoda function the convergence theorem is proved for nonsmooth weakly convex concave nikaido isoda functions the family of weakly convex concave functions is broad enough for applications since if includes the family of smooth functions when the payoff functions are twice continuously differentiable the condition for the residual terms is reduced to strict positiveness of a matrix representing the difference of the hessians of the nikaido isoda function with respect to the first and second groups of variables an analogous condition was used by uryas ev 1988 to prove convergence of the gradient type algorithm for the nash equilibrium problem in this paper the authors discuss only the deterministic case nevertheless this approach can be generalized for the stochastic nash equilibrium problems with uncertainties in parameters

elastic mode algorithms for mathematical programs with equilibrium constraints global convergence and stationarity properties the elastic mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that for a finite value of the penalty parameter local solutions satisfying first and second order necessary optimality conditions for the original problem are also first and second order points of the elastic mode formulation here we study global convergence properties of methods based on this formulation which involve generating an exact or inexact first or second order point of the formulation for nondecreasing values of the penalty parameter under certain regularity conditions on the active constraints we establish finite or asymptotic convergence to points having a certain stationarity property such as strong stationarity m stationarity or c stationarity numerical experience with these approaches is discussed in particular our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic mode formulation is solved inexactly

maximin joint optimization of transmitting code and receiving filter in radar and communications in this paper we conduct the joint design of transmitting sequence s and receiving filters subject to the peak to average ratio par constraint in radar and communications applications we consider optimizing the worst case performance and the resulting optimization problem takes a maximin format we propose two algorithms based on the mm majorization minimization or minorization maximization method as opposed to the traditional epigraph based smooth reformulation on top of that both algorithms are guaranteed to converge to a b oulingand stationary solution and b stationarity is the appropriate stationarity condition for problems with a nonconvex constraint set the proposed algorithms successively solve a series of simple convex problems that enjoy low computational complexity numerical simulations have shown that the proposed algorithms empirically achieve slightly higher objective values and converge faster in terms of cpu time than the existing methods

a unified framework for low autocorrelation sequence design via majorization minimization in this paper we consider the low autocorrelation sequence design problem we optimize a unified metric over a general constraint set the unified metric includes the integrated sidelobe level isl and the peak sidelobe level psl as special cases and the general constraint set contains the unimodular constraint peak to average ratio par constraint and similarity constraint to name a few the optimization technique we employ is the majorization minimization mm method which is iterative and enjoys guaranteed convergence to a stationary solution we carry out the mm method in two stages in the majorization stage we propose three majorizing functions two for the unified metric and one for the isl metric in the minimization stage we give closed form solutions for algorithmic updates under different constraints the update step can be implemented with a few fast fourier transformations ffts and or inverse ffts iffts we also show the connections between the mm and gradient projection method under our algorithmic scheme numerical simulations have shown that the proposed mm based algorithms can produce sequences with low autocorrelation and converge faster than the traditional gradient projection method and the state of the art algorithms

computing b stationary points of nonsmooth dc programs motivated by a class of applied problems arising from physical layer based security in a digital communication system in particular by a secrecy sum rate maximization problem this paper studies a nonsmooth difference of convex dc minimization problem the contributions of this paper are i clarify several kinds of stationary solutions and their relations ii develop and establish the convergence of a novel algorithm for computing a d stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions iii extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm a distributed penalty approach for problems whose objective functions are sums of dc functions and problems with a specially structured nonconvex dc constraint for the latter class of problems a pointwise slater constraint qualification is introduced that facilitates the ver

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

generalized nash games and quasi variational inequalities abstract a generalized nash game is an n person noncooperative game with nondisjoint strategy sets other names for this game form include social equilibria and pseudo nash games this paper explores both the qualitative and quantitative properties of such games through the use of quasi variational inequality theory several interesting relationships between the variational and quasi variational inequality forms of this class of games are described and the practical implementation of generalized nash games are explored at length

partially b regular optimization and equilibrium problems this paper introduces a concept termed partial b regularity for a feasible solution to a bivariate constraint system and shows that this condition leads to the equivalence between the b stationarity of a pair of lifted and unlifted programs in particular for an optimization problem with a univariate pseudoconvex objective function constrained by such a nonconvex bivariate system partial b regularity provides a sufficient condition for a b stationary point to be globally optimal applications of partial b regularity to several classes of optimization and equilibrium problems are presented these include a lexicographic optimization problem a nonconvex mathematical program with equilibrium constraints mpec that arises from a convex implicit value function optimization problem and a nash equilibrium program with equilibrium constraints

a family of implicit chebyshev methods for the numerical integration of second order differential equations abstract a family of implicit methods based on intra step chebyshev interpolation is developed for the solution of initial value problems whose differential equations are of the special second order form y f y x x the general procedure allows stepsizes which are considerably larger than commonly used in conventional methods computation overhead is comparable to that required by high order single or multistep procedures in addition the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth

analysis of a family of chebyshev methods for y f x y abstract methods proposed by panovsky and richardson this journal 1988 are interpreted as a family of symmetric two step hybrid methods each method is based on a polynomial interpolant of degree n for y for which the nodes are determined by the extrema of the chebyshev polynomial of degree n it is shown that the order of accuracy is n 1 for odd n and n 2 for even n and expressions for the local error constants are given the stability properties of each method are determined by the roots of a quadratic equation 2 2 n 1 0 where n 2 is a rational approximation for cos the intervals of periodicity which are also intervals of absolute stability are tabulated for n 10 numerical examples provide comparisons with other methods

an explicit runge kutta nystro m method is canonical if and only if its adjoint is explicit canonical runge kutta nystrom methods for hamiltonian dynamical systems are considered these systems arise in various areas in the physical sciences canonical methods preserve certain properties of the system in this paper it is shown that an explicit runge kutta nystrom method is canonical if and only its adjoint is explicit the adjoint of a method is obtained by time reversal one application for this result and other results in this paper is that if an explicit canonical runge kutta nystrom method of odd order is concatenated with its adjoint the result is an explicit canonical method of one order higher

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

variable step size does not harm second order integrators for hamiltonian systems abstract experiments have shown that variable step size degrades some symplectic integrators for certain hamiltonian systems in particular for the two body problem in this paper we show that the error growth in time for variable step size symplectic is not in general quadratic we demonstrate this by applying several integrators to the two body problem the error growth in time in the case of the second order integrators symplectic or nonsymplectic turns out to be approximately linear

the chebyshev methods of panovsky and richardson as runge kutta nystro m methods abstract the two step methods proposed by panovsky and richardson this journal 1988 and analysed by coleman and booth this journal 1992 are shown to be equivalent to certain runge kutta nystrom methods for differential equations of the form y f x y an important advantage of the new formulation is that it is no longer necessary to provide additional starting values at the beginning of a calculation or when the steplength or the order is changed the new formulation is also more stable with respect to the propagation of rounding errors

constraints on the cutoff frequencies of mth band linear phase fir filters constraints are derived for the cutoff frequencies of linear phase fir mth band filters such that the filters have good passband and stopband characteristics i e ones that very closely approximate an ordinary non mth band filter designed using some optimal method constraints on lowpass filters are first considered and the results are extended to multiband filters

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

an implementable active set algorithm for computing a b stationary point of a mathematical program with linear complementarity constraints erratum in m fukushima and p tseng siam j optim 12 2002 pp 724 739 an active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality complementarity constraints it is asserted therein that under a uniform licq on the feasible set this algorithm generates iterates whose cluster points are b stationary points of the problem however the proof has a gap and shows only that each cluster point is an m stationary point we discuss this gap and show that b stationarity can be achieved if the algorithm is modified and an additional error bound condition holds

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

modeling and computing two settlement oligopolistic equilibrium in a congested electricity network a model of two settlement electricity markets is introduced which accounts for flow congestion demand uncertainty system contingencies and market power we formulate the subgame perfect nash equilibrium for this model as an equilibrium problem with equilibrium constraints epec in which each firm solves a mathematical program with equilibrium constraints mpec the model assumes linear demand functions quadratic generation cost functions and a lossless dc network resulting in equilibrium constraints as a parametric linear complementarity problem lcp we introduce an iterative procedure for solving this epec through repeated application of an mpec algorithm this mpec algorithm is based on solving quadratic programming subproblems and on parametric lcp pivoting numerical examples demonstrate the effectiveness of the mpec and epec algorithms and the tractability of the model for realistic size power systems