asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

majorizing functions and convergence of the gauss newton method for convex composite optimization we introduce a notion of quasi regularity for points with respect to the inclusion f x in c where f is a nonlinear frechet differentiable function from mathbb r v to mathbb r m when c is the set of minimum points of a convex real valued function h on mathbb r m and f satisfies the l average lipschitz condition of wang we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the gauss newton method with quasi regular initial points for the convex composite function h circ f results are new even when the initial point is regular and f is lipschitz

on the asymptotically well behaved functions and global error bound for convex polynomials in this paper we present some new and tractable sufficient conditions for convex asymptotically well behaved awb functions moreover we establish several lipschitz and holder type global error bound results for a single convex polynomial and for functions which can be expressed as maximum of finitely many nonnegative convex polynomials an advantage of our approach is that the corresponding holder exponent in our holder type global error bound results can be determined explicitly

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

error bounds for piecewise convex quadratic programs and applications in this paper we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program these error estimates provide a unified approach for deriving many old and new error estimates for linear programs linear complementarity problems convex quadratic programs and affine variational inequality problems the approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program in a sense even robinson s result on the upper lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program as an application we derive new global error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

target problems under state constraints for nonlinear controlled impulsive systems we study the problem of reaching a target without leaving a prescribed constraint set for a dynamical system described by a controlled differential equation and a controlled instantaneous reset function we characterize all initial conditions from which the objective can be reached then we characterize the value function associated with the optimal reaching time problem

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

approximate solutions of analytic inequality systems we consider the stability problem for a system of equalities and inequalities we prove that when such a system is satisfied approximately at a certain point it is also satisfied exactly on a nearby point

optimality conditions for non finite valued convex composite functions burke 1987 has recently developed second order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued in this note we present a technique for reducing the infinite valued case to the finite valued one we then use this technique to extend the results in burke 1987 to the case in which the convex function may take infinite values we conclude by comparing these results with those established by rockafellar 1989 for the piecewise linear quadratic case

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

regularities and their relations to error bounds in this paper we mainly study various notions of regularity for a finite collection c1 cm of closed convex subsets of a banach space x and their relations with other fundamental concepts we show that a proper lower semicontinuous function f on x has a lipschitz error bound resp error bound if and only if the pair epi f x 0 of sets in the product space x is linearly regular resp regular similar results for multifunctions are also established next we prove that c1 cm is linearly regular if and only if it has the strong chip and the collection nc1 z ncm z of normal cones at z has property g for each z c i 1mci provided that c1 is a closed convex cone and that c2 y is a closed vector subspace of x we show that c1 y is linearly regular if and only if there exists 0 such that each positive relative to the order induced by c1 linear functional on y of norm one can be extended to a positive linear functional on x with norm bounded by similar characterization is given in terms of normal cones

on convergence of the gauss newton method for convex composite optimization the local quadratic convergence of the gauss newton method for convex composite optimization f h f is established for any convex function h with the minima set c extending burke and ferris results in the case when c is a set of weak sharp minima for h

a gauss newton method for convex composite optimization an extension of the gauss newton method for nonlinear equations to convex composite optimization is described and analyzed local quadratic convergence is established for the minimization ofh f under two conditions namelyh has a set of weak sharp minima c and there is a regular point of the inclusionf x c this result extends a similar convergence result due to womersley this journal 1985 which employs the assumption of a strongly unique solution of the composite functionh f a backtracking line search is proposed as a globalization strategy for this algorithm a global convergence result is established with a quadratic rate under the regularity assumption

on the control of uncertain impulsive systems approximate stabilization and controlled invariance the problems of stabilization and controlled invariance of a fairly wide class of uncertain hybrid systems is considered uncertainty enters in the form of a disturbance input that can affect both the continuous and the discrete dynamics a method for designing piecewise constant feedback controllers for this class of systems is developed in the case of controlled invariance the controller ensures that the state of the system remains arbitrarily close to a desired set over an arbitrarily long time horizon in the case of stabilization the controller ensures approximate exponential convergence of the runs of the closed loop system to the zero level set of a lyapunov function

stabilization of uncertain control systems through piecewise constant feedback abstract this paper investigates the problem of stabilization of nonlinear systems submitted to a deterministic disturbance known by its bounds the stabilization property is considered in terms of exponential decay of a lyapunov like function as such systems cannot be stabilized by continuous state feedback laws we consider an approximate stabilization problem with finite horizon the main result of the paper is the existence of a discontinuous piecewise constant feedback control law achieving the approximate stabilization property moreover in deriving explicitly this feedback law we show that the number of discontinuity jumps can be related to both precision of the approximate stabilization and time horizon of the control problem considered

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

approximate solutions for abstract inequality systems we consider conic inequality systems of the type f x ge k 0 with approximate solution x 0 associated to a parameter tau where f is a twice frechet differentiable function between hilbert spaces x and y and ge k is the partial order in y defined by a nonempty convex not necessarily closed cone k subseteq y we prove that under the suitable conditions the system f x ge k 0 is solvable and the ratio of the distance from x 0 to the solution set s over the distance from f x 0 to the cone k has an upper bound given explicitly in terms of tau and x 0 we show that the upper bound is sharp applications to analytic function inequality equality systems on euclidean spaces are given and the corresponding results of dedieu siam j optim 11 2000 pp 411 425 are extended and significantly improved

local properties of algorithms for minimizing nonsmooth composite functions this paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionf x f x h c x wheref andc are smooth buth c may be nonsmooth local convergence at a second order rate is established for the generalized gauss newton method whenh is convex and globally lipschitz and the minimizer is strongly unique local convergence at a second order rate is established for a generalized newton method when the minimizer satisfies nondegeneracy strict complementarity and second order sufficiency conditions assuming the minimizer satisfies these conditions necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established necessary and sufficient conditions for a two step superlinear rate of convergence are also established when only reduced curvature information is available all these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems

discrete time toda systems in this paper we discuss several concepts of the modern theory of discrete integrable systems including time discretization based on the notion of backlund transformation symplectic realizations of multi hamiltonian structures interrelations between discrete 1d systems and lattice 2d systems multi dimensional consistency as integrability of discrete systems interrelations between integrable systems of quad equations and integrable systems of laplace type pluri lagrangian structure as integrability of discrete variational systems all these concepts are illustrated by the discrete time toda lattices and their relativistic analogs

descent methods for composite nondifferentiable optimization problems we present a framework for the development of globally defined descent algorithms for the minimization of non differentiable objective functionsf h o f withh convex within our structure the global convergence properties of the cauchy modified newton gauss newton and variable metric methods are easily established along with that of several new approaches examples illustrating the calculational techniques are provided

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

extended newton methods for conic inequalities approximate solutions and the extended smale α theory abstract using the convex process theory and the majorizing technique we study the convergence issues of the iterative sequences generated by the extended newton method for solving the conic inequality system f c 0 defined by a cone c and a frechet differentiable function f satisfying the extended weak condition convergence criterion of the extended newton method is presented as an application we establish when f is analytic an extended theory similar to smale s theory for nonlinear analytic equations