the minimum cost flow problem a unifying approach to dual algorithms and a new tree search algorithm this paper is concerned with the minimum cost flow problem it is shown that the class of dual algorithms which solve this problem consists of different variants of a common general algorithm we develop a new variant which is in fact a new form of the primal dual algorithm and which has several interesting properties it uses explicitly only dual variables the slope of the change in the dual objective is monotone the bound on the maximum number of iterations to solve a problem with integral bounds on the flow is better than bounds for other algorithms

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

robust optimization for unconstrained simulation based problems in engineering design an optimized solution often turns out to be suboptimal when errors are encountered although the theory of robust convex optimization has taken significant strides over the past decade all approaches fail if the underlying cost function is not explicitly given it is even worse if the cost function is nonconvex in this work we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large partial differential equations pde solver response surface and kriging metamodels moreover this technique can be employed for most real world problems because it operates directly on the response surface and does not assume any specific structure of the problem we present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics relevant to nanophotonic design the corresponding objective function is highly nonconvex and resides in a 100 dimensional design space starting from an optimized design we report a robust solution with a significantly lower worst case cost while maintaining optimality we further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

fast kalman like optimal unbiased fir filtering with applications in this paper an optimal unbiased finite impulse response oufir filter is proposed as a linking solution between the unbiased fir ufir filter and the kalman filter kf we first derive the batch oufir estimator to minimize the mean square error mse subject to the unbiasedness constraint and then find its fast iterative form it is shown that the oufir filter is full horizon fh and that its estimate converges to the kf estimate by increasing the horizon length n as a special feature we note that the fh oufir filter operates almost as fast as the kf several other critical properties of the oufir filter are illustrated based on simulations and practical applications similar to the ufir filter and contrary to the kf the oufir filter is highly insensitive to the initial conditions it has much better robustness than kf against temporary model uncertainties finally the oufir filter allows for ignoring system noise which is typically not well known to the engineer

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

an approximation scheme for distributionally robust nonlinear optimization we consider distributionally robust optimization problems drops with nonlinear and nonconcave dependence on uncertain parameters the drop can be written as a nonsmooth nonlinear program with a

dijkstra s algorithm and l concave function maximization dijkstra s algorithm is a well known algorithm for the single source shortest path problem in a directed graph with nonnegative edge length we discuss dijkstra s algorithm from the viewpoint of discrete convex analysis where the concept of discrete convexity called l convexity plays a central role we observe first that the dual of the linear programming lp formulation of the shortest path problem can be seen as a special case of l concave function maximization we then point out that the steepest ascent algorithm for l concave function maximization when applied to the lp dual of the shortest path problem and implemented with some auxiliary variables coincides exactly with dijkstra s algorithm

bayesian state estimation on finite horizons the case of linear state space model abstract the finite impulse response fir filter and infinite impulse response filter including the kalman filter kf are generally considered as two different types of state estimation methods in this paper the sequential bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements with the aim of exploiting the fir structure and unifying some basic fir filters with the kf specifically the conditional mean and covariance of the posterior probability density functions are first derived to show the fir counterpart of the kf to remove the dependence on initial states the corresponding likelihood is further maximized and realized iteratively it shows that the maximum likelihood modification unifies the existing unbiased fir filters by tuning a weighting matrix moreover it converges to the kalman estimate with the increase of horizon length and can thus be considered as a link between the fir filtering and the kf several important properties including stability and robustness against errors of noise statistics are illustrated finally a moving target tracking example and an experiment with a three degrees of freedom helicopter system are introduced to demonstrate effectiveness

improved robust h2 and h filtering for uncertain discrete time systems this paper is concerned with the robust h 2 and h filtering problems for linear discrete time systems with polytopic parameter uncertainty we aim to derive a less conservative design than existing linear matrix inequality lmi based sufficient conditions it is shown that a more efficient evaluation of robust h 2 or h performance can be obtained by a matrix inequality condition which contains additional free parameters as compared to existing characterizations when applying this new matrix inequality condition to the robust filter design these parameters provide extra degrees of freedom in optimizing the guaranteed h 2 or h performance and lead to a less conservative design

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

on steepest descent algorithms for discrete convex functions this paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions m convex functions and l convex functions simple tie breaking rules yield complexity bounds that are polynomials in the dimension of the variables and the size of the effective domain combining the present results with a standard scaling approach leads to an efficient algorithm for l convex function minimization

an iterative kalman like algorithm ignoring noise and initial conditions we address a p shift finite impulse response fir unbiased estimator ue for linear discrete time varying filtering p 0 p step prediction p 0 and p lag smoothing p 0 in state space with no requirements for initial conditions and zero mean noise a solution is found in a batch form and represented in a computationally efficient iterative kalman like one it is shown that the kalman like fir ue is able to outperform the kalman filter if the noise covariances and initial conditions are not known exactly noise is not white and both the system and measurement noise components need to be filtered out otherwise the errors are similar extensive numerical studies of the fir ue are provided in gaussian and non gaussian environments with outliers and temporary uncertainties

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

l_ 2 e fir smoothers for deterministic discrete time state space signal models in this note a new type of l2 e performance criterion for a fixed lag smoother is introduced which is given by a gain between the energy of the external disturbances during the recent time horizon and the estimation error at the fixed delayed time from the current one by minimizing the maximum value of the l2 e performance criterion the l2 e finite impulse response fir smoother lefs is obtained for a deterministic discrete time state space signal model the lets is designed to meet requirements such as linearity and fir structure with respect to measured inputs and outputs unbiased property in the deterministic sense and independence of any state information simultaneously the proposed lets is represented in a batch form for simplicity the lets is compared with a conventional h infin infinite impulse response iir smoother via simulation

epi convergent smoothing with applications to convex composite functions smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems in this note we synthesize and extend recent results due to beck and teboulle on infimal convolution smoothing for convex functions with those of x chen on gradient consistency for nonconvex functions we use epi convergence techniques to define a notion of epi smoothing that allows us to tap into the rich variational structure of the subdifferential calculus for nonsmooth nonconvex and nonfinite valued functions as an illustration of the versatility and range of epi smoothing techniques the results are applied to the general constrained optimization for which nonlinear programming is a special case

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

minimum variance distortionless response estimators for linear discrete state space models for linear discrete state space models under certain conditions the linear least mean squares filter estimate has a convenient recursive predictor corrector format aka the kalman filter the purpose of this paper is to show that the linear minimum variance distortionless response mvdr filter shares exactly the same recursion except for the initialization which is based on a weighted least squares estimator if the mvdr filter is suboptimal in mean squared error sense it is an infinite impulse response distortionless filter a deconvolver which does not depend on the prior knowledge first and second order statistics on the initial state in other words the mvdr filter can be pre computed and its behaviour can be assessed in advance independently of the prior knowledge on the initial state

robust discrete time minimum variance filtering the bounded variance filtered estimation of the state of an uncertain linear discrete time system with an unknown norm bounded parameter matrix is considered an upper bound on the variance of the estimation error is found for all admissible systems and estimators are derived that minimize the latter bound we treat the finite horizon time varying case and the infinite time case where the nominal system model is time invariant in the special stationary case where it is known that the uncertain system is time invariant we provide a robust filter for all uncertainties that still keep the system asymptotically stable

fir smoothing of discrete time polynomial signals in state space we address a smoothing finite impulse response fir filtering solution for deterministic discrete time signals represented in state space with finite degree polynomials the optimal smoothing fir filter is derived in an exact matrix form requiring the initial state and the measurement noise covariance function the relevant unbiased solution is represented both in the matrix and polynomial forms that do not involve any knowledge about measurement noise and initial state the unique l degree unbiased gain and the noise power gain are derived for a general case the widely used low degree gains are investigated in detail as an example the best linear fit is provided for a two state clock error model

duality and sensitivity in nonconvex quadratic optimization over an ellipsoid abstract in this paper a duality framework is discussed for the problem of optimizing a nonconvex quadratic function over an ellipsoid additional insight is obtained from the observation that this nonconvex problem is in a sense equivalent to a convex problem of the same type from which known necessary and sufficient conditions for optimality readily follow based on the duality results some existing solution procedures are interpreted as in fact solving the dual the duality relations are also shown to provide a natural framework for sensitivity analysis

unified forms for kalman and finite impulse response filtering and smoothing the kalman filter and smoother are optimal state estimators under certain conditions the kalman filter is typically presented in a predictor corrector format but the kalman smoother has never been derived in that format we derive the kalman smoother in a predictor corrector format thus providing a unified form for the kalman filter and smoother we also discuss unbiased finite impulse response ufir filters and smoothers which can provide a suboptimal but robust alternative to kalman estimators we derive two unified forms for ufir filters and smoothers and we derive lower and upper bounds for their estimation error covariances

minimum variance unbiased fir filter for discrete time variant systems this paper is concerned with the minimum variance unbiased mvu finite impulse response fir filtering problem for linear system described by discrete time variant state space models an mvu fir filter is derived by minimizing the variance from the unbiased fir ufir filter the relationship between the filter gains of mvu fir ufir and optimal fir ofir filters is derived analytically and the mean square errors mses of different fir filters are compared to provide an insight into the estimation performance simulations provided verify that errors in the mvu fir filter are in between the ufir and ofir filters it is also shown that the mvu fir filter can offer optimal estimates without a prior knowledge of the initial state and exhibits better robustness against temporary modeling uncertainties than the kalman filter

a note on two filter smoothing formulas in this note the two filter smoothing formulas are obtained from general optimal smoothers in consideration of the control inputs and the correlation between system and measurement noises the inverse of the system matrix and any backwards time model for a backwards filter are not required besides we do not take heuristic approaches such as infinite covariances we first propose a new batch form of the general optimal smoother and then represent it with two filters without any assumptions and heuristic approaches even in the case that the lag size is small the proposed formulas in this note still work without any singularity problem and thus are guaranteed to be optimal even for a singular system matrix a recursive form is introduced for efficient calculation and is shown to be reduced to the existing two filter formulas if some condition is satisfied

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

nonsmooth optimization via quasi newton methods we investigate the behavior of quasi newton algorithms applied to minimize a nonsmooth function f not necessarily convex we introduce an inex act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the armijo and wolfe conditions if f is absolutely continuous along the line furthermore the line search is guaranteed to terminate if f is semi algebraic it seems quite difficult to establish a convergence theorem for quasi newton methods applied to such general classes of functions so we give a care ful analysis of a special but illuminating case the euclidean norm in one variable using the inexact line search and in two variables assuming that the line search is exact in practice we find that when f is locally lipschitz and semi algebraic with bounded sublevel sets the bfgs broyden fletcher goldfarb shanno method with the inexact line search almost always generates sequences whose cluster points are clarke stationary and with function values converging r linearly to a clarke station ary value we give references documenting the successful use of bfgs in a variety of nonsmooth applications particularly the design of low order controllers for linear dynamical systems we conclude with a challenging open question

smoothing technique and its applications in semidefinite optimization in this paper we extend the smoothing technique nesterov in math program 103 1 127 152 2005 nesterov in unconstrained convex mimimization in relative scale 2003 onto the problems of semidefinite optimization for that we develop a simple framework for estimating a lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices using this technique we can justify the lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices we analyze the efficiency of the special gradient type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix which depends linearly on the design variables we show that in the first case the number of iterations of the method is bounded by o 1 epsilon where epsilon is the required absolute accuracy of the problem in the second case the number of iterations is bounded by 4 delta sqrt 1 delta r ln r where is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality thus the latter method is a fully polynomial approximation scheme

linear optimal fir estimation of discrete time invariant state space models this paper addresses a general p shift linear optimal finite impulse response fir estimator intended for solving universally the problems of filtering p 0 smoothing p 0 of discrete time invariant models in state space an optimal solution is found in the batch form with the initial mean square state function self determined by solving the discrete algebraic riccati equation an unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state the mean square errors in both the optimal and unbiased estimates are found via the noise power gain npg and a recursive algorithm for fast computation of the npg is supplied applications are given for fir filtering with fixed receding and full averaging horizons

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

suboptimal fir filtering of nonlinear models in additive white gaussian noise the first and second order extended finite impulse response efir1 and efir2 respectively filters are addressed for suboptimal estimation of nonlinear discrete time state space models with additive white gaussian noise it is shown that unlike the extended kalman filter ekf and efir2 filter the efir1 one does not require noise statistics and initial errors only within a narrow region around actual noise covariances efir filters fall a bit short of ekf and they demonstrate better performance otherwise it is shown that the optimal averaging interval for efir filters can be determined via measurement without a reference model in a learning cycle we also notice that the second order approximation can improve the local performance but it can also deteriorate it we thus have no recommendations about its use at least for tracking considered as an example of applications

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

hidden convexity in some nonconvex quadratically constrained quadratic programming we consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints we then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints which is also shown to be equivalent to a minimax convex problem in both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts special cases and applications are also discussed we outline interior point polynomial time algorithms for the solution of the equivalent convex programs

an approximation technique for robust nonlinear optimization nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst case formulation that is however difficult to treat computationally in this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set in case of any norm bounded parameter uncertainty this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints the main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems one similar to the forward mode the other similar to the reverse mode of automatic differentiation we show how to generalize the techniques to optimal control problems and discuss how even infinite dimensional uncertainties can be treated efficiently finally we present optimization results for an example from process engineering a batch distillation

nfir nonlinear filter a nonlinear filter is introduced whose output is nearest in distance to the output of a fir finite impulse response linear phase filter hence it is called the nfir filter after choosing the impulse response of the fir filter properly the nfir filter can clean impulsive noise and preserve edges of signals as a special case of the nfir filter a nearest mean neme filter is derived theoretical analysis shows that the neme filter is better than the median filter in smoothing additive gaussian noise in addition based on the definition of the neme filter a new interpretation of the low pass property of the median filter with window width 3 is described also the performance of the nearest mean filter on a noisy edge is quantitatively compared with that of the median and mean filters

order recursive fir smoothers this paper introduces order recursive fir smoothers and shows that order recursive fir filters are special forms that occur when no future data values are used to estimate the signal the formulation leads naturally to generalizations of the concepts of prediction error basis and cholesky factorization which are well known in fir filter design

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