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

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

a combined smoothing and regularization method for monotone second order cone complementarity problems the second order cone complementarity problem soccp is a wide class of problems containing the nonlinear complementarity problem ncp and the second order cone programming problem socp recently fukushima luo and tseng siam j optim 12 2001 pp 436 460 extended some merit functions and their smoothing functions for ncp to soccp moreover they derived computable formulas for the jacobians of the smoothing functions and gave conditions for the jacobians to be invertible in this paper we propose a globally and quadratically convergent algorithm which is based on smoothing and regularization methods for solving monotone soccp in particular we study strong semismoothness and jacobian consistency which play an important role in establishing quadratic convergence of the algorithm furthermore we examine the effectiveness of the algorithm by means of numerical experiments

detecting coherent structures using braids abstract the detection of coherent structures is an important problem in fluid dynamics particularly in geophysical applications for instance knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills existing methods detect lagrangian coherent structures which are barriers to transport by examining the stretching field as given by finite time lyapunov exponents these methods are very effective when the velocity field is well determined but in many applications only a small number of flow trajectories are known for example when dealing with oceanic float data we introduce a topological method for detecting invariant regions based on a small set of trajectories in this method we regard the two dimensional trajectory data as a braid in three dimensions with time being the third coordinate invariant regions then correspond to trajectories that travel together and do not entangle other trajectories we detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories and searching for loops that show negligible growth

minimizing the condition number of a gram matrix the condition number of a gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation given a polynomial basis we consider the problem of finding a set of points and or weights which minimizes the condition number of the gram matrix the objective function f in the minimization problem is nonconvex and nonsmooth we present an expression of the clarke generalized gradient of f and show that f is clarke regular and strongly semismooth moreover we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function to illustrate applications of minimizing the condition number we report numerical results for the gram matrix defined by the weighted vandermonde like matrix for least squares approximation on an interval and for the gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

stochastic variational inequalities residual minimization smoothing sample average approximations the stochastic variational inequality vi has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data this paper presents a new expected residual minimization erm formulation for a class of stochastic vi the objective of the erm formulation is lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods we propose a globally convergent a s smoothing sample average approximation ssaa method to minimize the residual function this minimization problem is convex for the linear stochastic vi if the expected matrix is positive semidefinite we show that the erm problem and its ssaa problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the ssaa problems is a minimizer and a stationary point of the erm problem a s our examples come from applications involving traffic flow problems we show that t

a trust region method for minimization of nonsmooth functions with linear constraints we introduce a trust region algorithm for minimization of nonsmooth functions with linear constraints at each iteration the objective function is approximated by a model function that satisfies a set of assumptions stated recently by qi and sun in the context of unconstrained nonsmooth optimization the trust region iteration begins with the resolution of an easy problem as in recent works of martinez and santos and friedlander martinez and santos for smooth constrained optimization in practical implementations we use the infinity norm for defining the trust region which fits well with the domain of the problem we prove global convergence and report numerical experiments related to a parameter estimation problem

optimality conditions and a smoothing trust region newton method for nonlipschitz optimization regularized minimization problems with nonconvex nonsmooth perhaps non lipschitz penalty functions have attracted considerable attention in recent years owing to their wide applications in image restoration signal reconstruction and variable selection in this paper we derive affine scaled second order necessary and sufficient conditions for local minimizers of such minimization problems moreover we propose a global convergent smoothing trust region newton method which can find a point satisfying the affine scaled second order necessary optimality condition from any starting point numerical examples are given to demonstrate the effectiveness of the smoothing trust region newton method

smoothing projected gradient method and its application to stochastic linear complementarity problems a smoothing projected gradient spg method is proposed for the minimization problem on a closed convex set where the objective function is locally lipschitz continuous but nonconvex nondifferentiable we show that any accumulation point generated by the spg method is a stationary point associated with the smoothing function used in the method which is a clarke stationary point in many applications we apply the spg method to the stochastic linear complementarity problem slcp and image restoration problems we study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization erm formulation of slcp preliminary numerical experiments using the spg method for solving randomly generated slcp and image restoration problems of large sizes show that the spg method is promising

lipschitz behavior of the robust regularization to minimize or upper bound the value of a function robustly we might instead minimize or upper bound the epsilon robust regularization defined as the map from a point to the maximum value of the function within an epsilon radius this regularization may be easy to compute convex quadratics lead to semidefinite representable regularizations for example and the spectral radius of a matrix leads to pseudospectral computations for favorable classes of functions we show that the robust regularization is lipschitz around any given point for all small epsilon 0 even if the original function is non lipschitz like the spectral radius one such favorable class consists of the semi algebraic functions such functions have graphs that are finite unions of sets defined by finitely many polynomial inequalities and are commonly encountered in applications

a superlinearly convergent method for a class of complementarity problems with non lipschitzian functions we consider a class of complementarity problems involving functions which are not lipschitz continuous in this paper we reformulate this class of non lipschitzian complementarity problems into a lipschitzian complementarity problem then we propose an inexact smoothing newton method to solve this lipschitzian complementarity problem we prove that our proposed method converges quadratically and globally under a mild condition numerical results show that this method is promising this method can solve these kinds of complementarity problems with one million variables in reasonable time on a pc with 1 gb of ram

optimizing condition numbers in this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite n times n matrices we show that the condition number is a clarke regular strongly pseudoconvex function we prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program this asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

first order conditions for nonsmooth discretized constrained optimal control problems this paper studies first order conditions karush kuhn tucker conditions for discretized optimal control problems with nonsmooth constraints we present a simple condition which can be used to verify that a local optimal point satisfies the first order conditions and that a point satisfying the first order conditions is a global or local optimal solution of the optimal control problem

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

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

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

smoothing neural network for constrained non lipschitz optimization with applications in this paper a smoothing neural network snn is proposed for a class of constrained non lipschitz optimization problems where the objective function is the sum of a nonsmooth nonconvex function and a non lipschitz function and the feasible set is a closed convex subset of using the smoothing approximate techniques the proposed neural network is modeled by a differential equation which can be implemented easily under the level bounded condition on the objective function in the feasible set we prove the global existence and uniform boundedness of the solutions of the snn with any initial point in the feasible set the uniqueness of the solution of the snn is provided under the lipschitz property of smoothing functions we show that any accumulation point of the solutions of the snn is a stationary point of the optimization problem numerical results including image restoration blind source separation variable selection and minimizing condition number are presented to illustrate the theoretical results and show the efficiency of the snn comparisons with some existing algorithms show the advantages of the snn

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

on the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming we estimate the worst case complexity of minimizing an unconstrained nonconvex composite objective with a structured nonsmooth term by means of some first order methods we find that it is unaffected by the nonsmoothness of the objective in that a first order trust region or quadratic regularization method applied to it takes at most mathcal o epsilon 2 function evaluations to reduce the size of a first order criticality measure below epsilon specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective and constraint evaluation worst case complexity of nonconvex equality constrained optimization when the solution is computed using a first order exact penalty method we obtain that in the reasonable case when the penalty parameters are bounded the complexity of reaching within epsilon of a kkt point is at most mathcal o epsilon 2 problem evaluations which is the same in order as the function evaluation complexity of steepest descent methods applied to unconstrained nonconvex smooth optimization

well posedness and a numerical study of a regularization model with adaptive nonlinear filtering for incompressible fluid flow we give a detailed analytical study of a leray model of incompressible flow that uses nonlinear filtering based on indicator functions the indicator functions allow for local regularization instead of global regularization which can over smooth and dampen out important flow structures the key to the analysis is the identification of the indicator function as a nemyskii operator after proving well posedness we provide a numerical study which includes proving optimal convergence of finite element method for the model as well as several numerical experiments