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

a new kernel based approach for linear system identification this paper describes a new kernel based approach for linear system identification of stable systems we model the impulse response as the realization of a gaussian process whose statistics differently from previously adopted priors include information not only on smoothness but also on bibo stability the associated autocovariance defines what we call a stable spline kernel the corresponding minimum variance estimate belongs to a reproducing kernel hilbert space which is spectrally characterized compared to parametric identification techniques the impulse response of the system is searched for within an infinite dimensional space dense in the space of continuous functions overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization the proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two step procedure that projects the nonparametric estimate onto the space of nominal models the continuous time derivation immediately extends to the discrete time case on several continuous and discrete time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques

solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization in this paper we consider the stochastic mathematical programs with linear complementarity constraints which include two kinds of models called here and now and lower level wait and see problems we present a combined smoothing implicit programming and penalty method for the problems with a finite sample space then we suggest a quasi monte carlo approximation method for solving a problem with continuous random variables a comprehensive convergence theory is included as well we further report numerical results with the so called picnic vender decision problem

stochastic mathematical programs with hybrid equilibrium constraints this paper considers a stochastic mathematical program with hybrid equilibrium constraints smphec which includes either here and now or wait and see type complementarity constraints an example is given to describe the necessity to study smphec in order to solve the problem the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints limiting behaviors of the proposed approach are discussed preliminary numerical experiments show that the proposed approach is applicable

an approximation scheme for a class of risk averse stochastic equilibrium problems we consider two models for stochastic equilibrium one based on the variational equilibrium of a generalized nash game and the other on the mixed complementarity formulation each agent in the market solves a single stage risk averse optimization problem with both here and now investment variables and production wait and see variables a shared constraint couples almost surely the wait and see decisions of all the agents an important characteristic of our approach is that the agents hedge risk in the objective functions on costs or profits of their optimization problems which has a clear economic interpretation this feature is obviously desirable but in the risk averse case it leads to variational inequalities with set valued operators a class of problems for which no established software is currently available to overcome this difficulty we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents problems such as average or conditional value at risk the smoothed variational inequalities can be tackled by the path solver for example the approximation scheme is shown to converge including the case when smoothed problems are solved approximately an interesting by product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem to assess the proposed technique numerical results are presented the first set of experiments is on randomly generated equilibrium problems for which we compare the proposed methodology with the standard smooth reformulation of average value at risk minimization using additional variables to rewrite the associated max function the second set of experiments deals with a part of the real life european gas network for which dantzig wolfe decomposition can be combined with the smoothing approach

entropic approximation for mathematical programs with robust equilibrium constraints in this paper we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi infinite complementarity constraints sicc we propose a numerical scheme for tackling sicc specifically by relaxing the complementarity constraints and then randomizing the index set of sicc we employ the well known entropic risk measure to approximate the semi infinite constraints with a finite number of stochastic inequality constraints under some moderate conditions we quantify the approximation in terms of the feasible set and the optimal value the approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules finally we extend the discussion to a mathematical program with distributionally robust equilibrium constraints which is essentially a one stage stochastic program with semi infinite stochastic constraints indexed by some probability measures from

model structure selection by cross validation two criteria for choosing between different model structures are proposed their derivation is within a natural cross validatory assessment context and is fairly assumption free in particular the two criteria can be used for discriminating between non nested model structures and more importantly the true system is not required to belong to the considered set of models should the true system belong to the model set the two proposed criteria will asymptotically reduce to some well known structure selection criteria this is believed to be a desirable feature of our proposals on the other hand it provides a nice cross validation interpretation of some well known model structure selection rules also the cross validation interpretation helps one to choose which criteria to use in a given application the paper also has a second purpose somewhat decoupled from that mentioned above it contains an extensive survey of the literature that is useful in its own right

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

on tikhonov regularization bias and variance in nonlinear system identification abstract regularization is a general method for solving ill posed and ill conditioned problems traditionally ill conditioning in system identification problems is approached using regularization methods such as ridge regression and principal component regression in this work it is argued that the tikhonov regularization method is a powerful alternative for regularization of nonlinear system identification problems by introducing smoothness of the model as a prior its properties is discussed in terms of an analysis of bias and variance and illustrated by a semirealistic simulation example

asymptotic properties of least squares estimates of hammerstein wiener models this paper investigates the asymptotic properties of least squares estimates of hammerstein wiener model structures and in doing so establishes consistency and asymptotic normality under fairly mild conditions on the additive noise process the inputs and the static non linearities in relation to the asymptotic distributional results a consistent procedure for the estimation of the asymptotic variance of the parameter estimates is provided a key theme of this paper is to demonstrate how recent results from the econometrics literature may be employed in an engineering setting in this respect the hammerstein wiener model structure serves as a demonstration example a simulation study complements the theoretical findings

identification of non linear systems using empirical data and prior knowledge an optimization approach abstract the choice of a parametric model structure in empirical and semi empirical non linear modeling is usually viewed as an important and critical step however it is known that by augmenting the least squares identification criterion with a term that imposes a penalty on the non smoothness of the model an optimal non parametric model can be found explicitly the optimal non parametric model will depend on the particular form of the penalty which can be looked upon as a priori knowledge or the desired properties of the model in this paper these results are extended in several directions i we show how useful types of prior knowledge other than smoothness can be included as a term in the criterion or as a constraint and how this influences the optimal model ii dynamic models and a general prediction error penalty are considered iii we present a practical numerical procedure for the identification of a close to optimal semi parametric model the numerical approach is motivated by the difficulty of deriving the optimal non parametric model if there are complicated constraints or penalty terms in the criterion and finally iv we discuss determination of the appropriate model complexity through selecting weights on the different penalties on the basis of empirical data since the optimal non parametric model is identified using an augmented optimization criterion and all prior knowledge and desired model properties may be specified through the augmented optimization criterion it is the choice of this criterion that is critical in this approach this elevates the empirical and semi empirical modeling problems to a more transparent and engineering friendly level than is achieved by directly specifying a parametric model structure hence our results shed some more light on modern non linear empirical modeling approaches like radial basis functions splines and neural networks in addition to providing a fundamental insight into the role of different kinds of prior knowledge this high level formulation is also attractive from a practical point of view as it lies closer to engineering thinking and less guesswork is required the flexibility and power of this approach are illustrated with a semi realistic simulation example

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

on using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints we investigate the possibility of solving mathematical programs with complementarity constraints mpccs using algorithms and procedures of smooth nonlinear programming although mpccs do not satisfy a constraint qualification we establish sufficient conditions for their lagrange multiplier set to be nonempty mpccs that have nonempty lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter in this context the elastic mode transforms mpcc into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming sqp algorithms one such algorithm is shown to achieve local linear convergence once the problem is relaxed under stronger conditions we also prove superlinear convergence to the solution of an mpcc using an adaptive elastic mode approach for an sqp algorithm recently analyzed in an mpcc context in r fletcher s leyffer s sholtes and d ralph local convergence of sqp methods for mathematical programs with equilibrium constraints tech report na 210 university of dundee dundee uk 2002 our assumptions are more general since we do not use a critical assumption from that reference in addition we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen

convergence properties of a regularization scheme for mathematical programs with complementarity constraints we study the convergence behavior of a sequence of stationary points of a parametric nlp which regularizes a mathematical program with equilibrium constraints mpec in the form of complementarity conditions accumulation points are feasible points of the mpec they are c stationary if the mpec linear independence constraint qualification holds they are m stationary if in addition an approaching subsequence satisfies second order necessary conditions and they are b stationary if in addition an upper level strict complementarity condition holds these results complement recent results of fukushima and pang convergence of a smoothing continuation method for mathematical programs with equilibrium constraints in ill posed variational problems and regularization techniques springer verlag new york 1999 we further show that every local minimizer of the mpec which satisfies the linear independence upper level strict complementarity and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric nlp

on the relationship of approximation algorithms for the minimum and the maximum traveling salesman problem abstract we consider the relationship between the minimum and the maximum traveling salesman problem the paper is based on the idea of applying heuristics for the maximum traveling salesman problem to the minimum traveling salesman problem numerical results confirm the efficiency of the proposed method

a class of smoothing saa methods for a stochastic mathematical program with complementarity constraints abstract a class of smoothing sample average approximation saa methods is proposed for solving the stochastic mathematical program with complementarity constraints smpcc considered by birbil et al s i birbil g gurkan o listes solving stochastic mathematical programs with complementarity constraints using simulation math oper res 31 2006 739 760 the almost sure convergence of optimal solutions of the smoothed saa problem to that of the true problem is established by the notion of epi convergence in variational analysis it is demonstrated that under suitable conditions any accumulation point of karash kuhn tucker points of the smoothed saa problem is almost surely a kind of stationary point of smpcc as the sample size tends to infinity moreover under a strong second order sufficient condition for smpcc the exponential convergence rate of the sequence of karash kuhn tucker points of the smoothed saa problem is investigated through an application of robinson s stability theory some preliminary numerical results are reported to show the efficiency of proposed method

smooth sqp methods for mathematical programs with nonlinear complementarity constraints mathematical programs with nonlinear complementarity constraints are reformulated using better posed but nonsmooth constraints we introduce a class of functions parameterized by a real scalar to approximate these nonsmooth problems by smooth nonlinear programs this smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations we present two globally convergent algorithms based on sequential quadratic programming sqp as applied in exact penalty methods for nonlinear programs global convergence of the implicit smooth sqp method depends on existence of a lower level nondegenerate strictly complementary limit point of the iteration sequence global convergence of the explicit smooth sqp method depends on a weaker property i e existence of a limit point at which a generalized constraint qualification holds we also discuss some practical matters relating to computer implementations

combined monte carlo sampling and penalty method for stochastic nonlinear complementarity problems in this paper we consider a new formulation with recourse for a class of stochastic nonlinear complementarity problems we show that the new formulation is equivalent to a smooth semi infinite program that no longer contains recourse variables we then propose a combined monte carlo sampling and penalty method for solving the problem in which the underlying sample space is assumed to be compact furthermore we suggest a compact approximation approach for the case where the sample space is unbounded two preliminary numerical examples are included as well

regularized mathematical programs with stochastic equilibrium constraints estimating structural demand models the article considers a particular class of optimization problems involving set valued stochastic equilibrium constraints we develop a solution procedure that relies on an approxima tion scheme for the equilibrium constraints based on regularization we replace the approximated equilibrium constraints by those involving only single valued lipschitz continuous functions in ad dition sampling has the further effect of replacing the simplified equilibrium constraints by more manageable ones obtained by implicitly discretizing the given probability measure so as to render the problem computationally tractable convergence is obtained by relying in particular on the graphical convergence of the approximated equilibrium constraints the problem of estimating the characteristics of a demand model a widely studied problem in microeconometrics serves both as motivation and illustration of the regularization and sampling procedure

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation

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