a smoothing method for mathematical programs with equilibrium constraints the mathematical program with equilibrium constraints mpec is an optimization problem with variational inequality constraints mpec problems include bilevel programming problems as a particular case and have a wide range of applications mpec problems with strongly monotone variational inequalities are considered in this paper they are transformed into an equivalent one level nonsmooth optimization problem then a sequence of smooth regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced it is shown that the solutions stationary points of the approximate problems converge to a solution stationary point of the original mpec problem numerical results showing viability of the approach are reported

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

a robust direct adaptive controller this paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics the algorithm is designed based on the reduced order plant which is assumed to be minimum phase and of known order and relative degree but is analyzed with respect to the overall plant which due to the unmodeled dynamics may be nonminimum phase and of unknown order and relative degree it is shown that if the unmodeled dynamics are sufficiently small in the low frequency range then the algorithm guarantees boundedness of all signals in the adaptive loop and small residual tracking errors for any bounded initial conditions in the absence of unmodeled dynamics the residual tracking error is shown to be zero

robustness of discrete time direct adaptive controllers the problem of preserving stability of discrete time adaptive controllers in spite of reduced order modeling and output disturbances is addressed in this paper conditions for global stability convergence of the tracking error with bounded signals are derived for a discrete time pole zero placement adaptive controller where the parameter estimator is modified in terms of normalized signals following an input output perpective the overall system is decomposed into two subsystems reflecting the parameter estimation and modeling errors respectively and its stability is studied using the sector stability and passivity theorems first the analysis is carried for the class of disturbances and reference inputs that are either decaying or can be exactly hulled by a linear controller of the chosen structure in this l 2 framework it is shown that the only substantive assumption to assure stability is the existence of a linear controller such that the closed loop transfer function verifies certain conicity conditions the convergence speed and alertness properties of various parameter adaptation algorithms regarding this condition are discussed the results are further extended to a broader class of l infty disturbances and reference inputs

a robust indirect adaptive control approach this paper considers the robust design of an indirect adaptive control approach which is applicable when the unknown parameters of a linear time invariant plant lie in a known convex set throughout which no unstable pole zero cancellation occurs in order to achieve the robustness the use of a relative dead zone in the adaptive law is proposed it is shown that with a suitably designed relative dead zone the adaptive control system is globally stable even in the presence of small unmodeled plant uncertainties

a continuation method for strongly monotone variational inequalities we consider the variational inequality problem denoted by vip x f wheref is a strongly monotone function and the convex setx is described by some inequality and possibly equality constraints this problem is solved by a continuation or interior point method which solves a sequence of certain perturbed variational inequality problems these perturbed problems depend on a parameter 0 it is shown that the perturbed problems have a unique solution for all values of 0 and that any sequence generated by the continuation method converges to the unique solution of vip x f under a well known linear independence constraint qualification licq we also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method 1998 the mathematical programming society inc published by elsevier science b v

a variable dimension solution approach for the general spatial price equilibrium problem algorithms are presented that are specifically designed for solving general nonlinear multicommodity spatial price equilibrium problems i e problems with nonlinear transportation cost functions nonlinear supply and demand functions inter commodity congestion effects intercommodity substitution and complementarity effects and interactions among transportation links and among spatially separated markets the algorithms are specializations of an iterative method for solving nonlinear complementarity problems that requires solving a system of nonlinear equations at each iteration the algorithms exploit the network structure of the problems to reduce the size of the system of equations to be solved at each iteration the decision rules for determining which equations are to be included in the system at each iteration are extremely simple and the remainder of the computational work is carried out by the nonlinear equation solver because of this the algorithms are very easy to implement with readily available software in addition since the decision rules only require sign information only the final system needs to be solved with precision

a class of gap functions for variational inequalities recently auchmuty 1989 has introduced a new class of merit functions or optimization formulations for variational inequalities in finite dimensional space we develop and generalize auchmuty s results and relate his class of merit functions to other works done in this field especially we investigate differentiability and convexity properties and present characterizations of the set of solutions to variational inequalities we then present new descent algorithms for variational inequalities within this framework including approximate solutions of the direction finding and line search problems the new class of merit functions include the primal and dual gap functions introduced by zuhovickii et al 1969a 1969b and the differentiable merit function recently presented by fukushima 1992 also the descent algorithm proposed by fukushima is a special case from the class of descent methods developed in this paper through a generalization of auchmuty s class of merit functions we extend those inherent in the works of dafermos 1983 cohen 1988 and wu et al 1991 new algorithmic equivalence results relating these algorithm classes to each other and to auchmuty s framework are also given

robustness of model reference adaptive controllers an input output approach a general input output approach is employed to study the direct model reference adaptive control problem in the presence of parametric and unstructured uncertainty a sufficient condition for the boundedness of the closed loop signals is developed relating an appropriately defined size of the unstructured uncertainty to the size of the parametric uncertainty this condition involves the intuitively appealing gains of various sensitivity operators and allows for the study of the effect of the design parameters on the boundedness properties of the closed loop system the results of this work indicate a direction towards a unified and systematic theory in adaptive control

a general descent framework for the monotone variational inequality problem we present a framework for descent algorithms that solve the monotone variational inequality problem vip v which consists in finding a solutionv v satisfyings v t v v 0 for allv v this unified framework includes as special cases some well known iterative methods and equivalent optimization formulations a descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given based on this unified logarithmic framework we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

a noninterior continuation method for quadratic and linear programming the noninterior point path following algorithm presented by the authors in 1990 is specialized to the mixed linear complementarity problem and its special cases quadratic and linear programming the new algorithm is related to but has several advantages over the interior point path following algorithms

accelerating the convergence of the diagonalization and projection algorithms for finite dimensional variational inequalities this paper presents an acceleration step for the linearly convergent diagonalization and projection algorithms for finite dimensional variational inequalities which is reminiscent of a partan step in nonlinear programming after establishing the convergence of this technique for both algorithms several numerical examples are presented to illustrate the sometimes dramatic savings in computation time which this simple acceleration step yields

multiple model adaptive mixing control the discrete time case most of the work in multiple model adaptive control with various forms of switching focused on continuous time systems the purpose of this technical note is to extend the results of one approach the adaptive mixing control amc to discrete time systems further the technical note solves the tracking problem which has not been addressed in most schemes of this class stability and robustness properties of the amc scheme for discrete time systems are analyzed it is shown that in the ideal case when no disturbances or unmodeled dynamics are present the tracking error converges to zero in the non ideal case the mean square tracking error is of the order of magnitude of the modeling error provided the unmodeled dynamics satisfy a norm bound condition while these robustness results are conceptually similar to those of traditional robust adaptive control the proposed scheme does not suffer from the drawback of stabilizability of the estimated plant and in addition performs much better in simulation studies furthermore it allows well developed results from robust control to be incorporated in the design

vanishing viscosity limit for a coupled navier stokes allen cahn system abstract in this paper we study the vanishing viscosity limit for a coupled navier stokes allen cahn system in a bounded domain we first show the local existence of smooth solutions of the euler allen cahn equations by modified galerkin method then using the boundary layer function to deal with the mismatch of the boundary conditions between navier stokes and euler equations and assuming that the energy dissipation for navier stokes equation in the boundary layer goes to zero as the viscosity tends to zero we prove that the solutions of the navier stokes allen cahn system converge to that of the euler allen cahn system in a proper small time interval in addition for strong solutions of the navier stokes allen cahn system in 2d the convergence rate is c 1 2

compressible fluids driven by stochastic forcing the relative energy inequality and applications we show the relative energy inequality for the compressible navier stokes system driven by a stochastic forcing as a corollary we prove the weak strong uniqueness property pathwise and in law and convergence of weak solutions in the inviscid incompressible limit in particular we establish a yamada watanabe type result in the context of the compressible navier stokes system that is pathwise weak strong uniqueness implies weak strong uniqueness in law

on the stochastic cahn hilliard equation with a singular double well potential abstract we prove well posedness and regularity for the stochastic pure cahn hilliard equation under homogeneous neumann boundary conditions with both additive and multiplicative wiener noise in contrast with great part of the literature the double well potential is treated as generally as possible its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed the regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem the proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework

a new merit function and a successive quadratic programming algorithm for variational inequality problems recently various merit functions for variational inequality problems have been proposed and their properties have been studied unfortunately these functions may not be easy to evaluate unless the constraints of the problem have a relatively simple structure in this paper a new merit function for variational inequality problems with general convex constraints is proposed at each point the proposed function is defined as an optimal value of a quadratic programming problem whose constraints consist of a linear approximation of the given nonlinear constraints it is shown that the set of constrained minima of the proposed merit function coincides with the set of solutions to the original variational inequality problem it is also shown that this function is directionally differentiable in all directions and under suitable assumptions any stationary point of the function over the constraint set actually solves the original variational inequality problem finally a descent method for solving the variational inequality problem is proposed and its convergence is proved the method is closely related to a successive quadratic programming method for solving nonlinear programming problems

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

solving variational inequalities by a modified projection method with an effective step size in this paper we propose a new projection method for the solution of variational inequality problems the method is simple which uses only function evaluations and projections onto the feasible set we adopt a new step size rule and a new search direction in the new method under the mild conditions we prove the proposed method is globally convergent preliminary numerical results are reported

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

limiting behavior of trajectories generated by a continuation method for monotone complementarity problems defining the mapping f from the 2n dimensional euclidean space r2n into itself by fx y x1y1 xn yn y1 f1x yn fnx for every x y r2n we write the cp the complementarity problem with a mapping f rn rn as the system of equations fx y 0 and x y r 2n where r 2n denotes the nonnegative orthant of r2n under the assumption that f is a monotone function on r n we show that f maps the positive orthant r 2n of r2n homeomorphically this result then ensures the existence of a trajectory consisting of the solutions x y t of the system of equations fx y wt and x y t r 2n 0 1 where w is a continuous mapping from the unit interval 0 1 into fr 2n such that w0 0 if t 0 the system coincides with the cp we study limiting behavior of the trajectory as t 0 and give some sufficient conditions for the trajectory to lead to a solution of the cp this approach is an extension of the one used in the polynomially bounded algorithm recently given by kojima mizuno and yoshise for solving positive semidefinite linear complementarity problems

the stampacchia maximum principle for stochastic partial differential equations and applications abstract stochastic partial differential equations spdes are considered linear and nonlinear for which we establish comparison theorems for the solutions or positivity results a e and a s for suitable data comparison theorems for spdes are available in the literature the originality of our approach is that it is based on the use of truncations following the stampacchia approach to maximum principle we believe that our method which does not rely too much on probability considerations is simpler than the existing approaches and to a certain extent more directly applicable to concrete situations among the applications boundedness results and positivity results are respectively proved for the solutions of a stochastic boussinesq temperature equation and of reaction diffusion equations perturbed by a non lipschitz nonlinear noise stabilization results to a chafee infante equation perturbed by a nonlinear noise are also derived

design issues in adaptive control an integrated approach to the design of practical adaptive control algorithms is presented many existing ideas are brought together and the effect of various design parameters available to a user is explored the theory is extended by showing how the problem of stabilizability of the estimated model can be overcome by running parallel estimators it is shown how asymptotic tracking of deterministic set points can be achieved in the presence of unmodeled dynamics

robustness of discrete time adaptive controllers an input output approach a general input output approach that can be used for the design and analysis of robust discrete time adaptive control schemes is presented in particular a model reference adaptive controller is considered the design approach involves the use of the certainty equivalence principle to combine a robust control structure with a robust adaptive law using exponentially weighted i sub 2 norms and exploiting the properties of the adaptive law and the control structure robust stability is established for the closed loop adaptive control scheme

on finite termination of an iterative method for linear complementarity problems based on a well known reformulation of the linear complementarity problem lcp as a nondifferentiable system of nonlinear equations a newton type method will be described for the solution of lcps under certain assumptions it will be shown that this method has a finite termination property i e if an iterate is sufficiently close to a solution of lcp the method finds this solution in one step this result will be applied to a recently proposed algorithm by harker and pang in order to prove that their algorithm also has the finite termination property

theory and design of robust direct and indirect adaptive control schemes a general approach for designing and the theory for analysing robust direct and indirect adaptive control schemes for continuous time plants is presented the design approach involves the development of a general robust adaptive law and the use of the certainty equivalence principle to combine it with robust model reference and pole placement control structures the global stability properties and robustness of the developed adaptive control schemes are established by using a general theory which relates the properties of signals in the mean sense over intervals of time the developed theory and design approach are used to analyse and compare the robustness properties and performance of a wide class of robust adaptive laws which employ a dead zone fixed e1 and a switching modification as well as their variations

stability of strong solutions for a model of incompressible two phase flow under thermal fluctuations abstract we consider a model of a two phase flow based on the phase field approach where the fluid bulk velocity obeys the standard navier stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the allen cahn equations possible thermal fluctuations are incorporated through a random forcing term in the allen cahn equation we show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality this fact is used for showing the weak strong uniqueness principle both pathwise and in law

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

multiple model self tuning control for a class of nonlinear systems this study develops a novel nonlinear multiple model self tuning control method for a class of nonlinear discrete time systems an increment system model and a modified robust adaptive law are proposed to expand the application range thus eliminating the assumption that either the nonlinear term of the nonlinear system or its differential term is global bounded the nonlinear self tuning control method can address the situation wherein the nonlinear system is not subject to a globally uniformly asymptotically stable zero dynamics by incorporating the pole placement scheme a novel nonlinear control structure based on this scheme is presented to improve control precision stability and convergence can be confirmed when the proposed multiple model self tuning control method is applied furthermore simulation results demonstrate the effectiveness of the proposed method

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness