Approximation of some stochastic differential equations by the splitting up method

In this paper we deal with the convergence of some iterative schemes suggested by Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic equation is split into two problems which are simpler for numerical computations, as already shown, for example, for the Zakaï equation. An estimate of the approximation error is given in a particular case.The work of A. Bensoussan and R. Glowinski was supported by the U.S. Army Research Office under Contract DAAL03-86-K-0138. Additional support was given by NSF via Grant INT-8612680.     

Barycentric scenario trees in convex multistage stochastic programming

This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined. This work has been supported by Schweizerischen Nationalfonds Grant Nr. 21-39 575.93.     

Inexact subgradient methods with applications in stochastic programming

In many instances, the exact evaluation of an objective function and its subgradients can be computationally demanding. By way of example, we cite problems that arise within the context of stochastic optimization, where the objective function is typically defined via multi-dimensional integration. In this paper, we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods. We refer to this new class of methods as inexact subgradient algorithms. With relatively mild conditions imposed on the approximations, we show that the inexact subgradient algorithms inherit properties associated with their traditional (i.e., exact) counterparts. Within the context of stochastic optimization, the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi-gradient methods. Additionally, we study methods in which steplengths may be defined adaptively, in a manner that reflects the improvement in the objective function approximations as the iterations proceed. We illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs.This work was supported by Grant Nos. NSF-DDM-89-10046 and NSF-DDM-9114352 from the National Science Foundation.     

Approximation Methods in Multiobjective Programming

Approaches to approximate the efficient set and Pareto set of multiobjective programs are reviewed. Special attention is given to approximating structures, methods generating Pareto points, and approximation quality. The survey covers more than 50 articles published since 1975.His work was supported by Deutsche Forschungsgemeinschaft, Grant HA 1795/7-2.Her work was done while on a sabbatical leave at the University of Kaiserslautern with support of Deutsche Forschungsgemeinschaft, Grant Ka 477/24-1.     

Finite master programs in regularized stochastic decomposition

Stochastic decomposition is a stochastic analog of Benders' decomposition in which randomly generated observations of random variables are used to construct statistical estimates of supports of the objective function. In contrast to deterministic Benders' decomposition for two stage stochastic programs, the stochastic version requires infinitely many inequalities to ensure convergence. We show that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included. Our computational results suggest that the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution.This work was supported in part by Grant No. AFOSR-88-0076 from the Air Force Office of Scientific Research and Grant Nos. DDM-89-10046, DDM-9114352 from the National Science Foundation.Corresponding author.     

Approximating Stationary Points of Stochastic Mathematical Programs with Equilibrium Constraints via Sample Averaging

We investigate sample average approximation of a general class of one-stage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show the convergence of M(Mordukhovich)-stationary point and C(Clarke)-stationary point of the sample average approximation problem to those of the true problem. The research complements the existing work in the literature by considering a general constraint to be represented by a stochastic generalized equation and exploiting graphical convergence of coderivative mappings.     

Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints

We study the quantitative stability of the solution sets, optimal value and M-stationary points of one stage stochastic mathematical programs with complementarity constraints when the underlying probability measure varies in some metric probability space. We show under moderate conditions that the optimal solution set mapping is upper semi-continuous and the optimal value function is Lipschitz continuous with respect to probability measure. We also show that the set of M-stationary points as a mapping is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that optimal value and M-stationary points of SAA programs converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.     

Suggested research topics in sensitivity and stability analysis for semi-infinite programming problems

We suggest several important research topics for semi-infinite programs whose problem functions and index sets contain parameters that are subject to perturbation. These include optimal value and optimal solution sensitivity and stability properties and penalty function approximation techniques. The approaches proposed are a natural carryover from parametric nonlinear programming, with emphasis on practical applicability and computability.Research supported by National Science Foundation Grant SES 8722504 and Grant ECS-86-19859 and Grant N00014-89-J-1537, Office of Naval Research.     

Approximation and support theorems in modulus spaces

Summary We prove an approximation theorem for stochastic differential equations, under rather weak smoothness conditions on the coefficients, when the driving semimartingales are approximated by continuous semimartingales, in probability, and the solutions are considered in several Banach spaces, defined in terms of different types of the modulus of continuity. Hence Stroock-Varadhan's support theorem is obtained in these spaces, in particular, in appropriate Besov and Hölder spaces.Partially supported by the Foundation of National Research n° 2290Partially supported by the DGICYT grant n^o PB 90-0452     

Stochastic programming approaches to stochastic scheduling

Practical scheduling problems typically require decisions without full information about the outcomes of those decisions. Yields, resource availability, performance, demand, costs, and revenues may all vary. Incorporating these quantities into stochastic scheduling models often produces diffculties in analysis that may be addressed in a variety of ways. In this paper, we present results based on stochastic programming approaches to the hierarchy of decisions in typical stochastic scheduling situations. Our unifying framework allows us to treat all aspects of a decision in a similar framework. We show how views from different levels enable approximations that can overcome nonconvexities and duality gaps that appear in deterministic formulations. In particular, we show that the stochastic program structure leads to a vanishing Lagrangian duality gap in stochastic integer programs as the number of scenarios increases.This author's work was supported in part by the National Science Foundation under Grants ECS 88-15101, ECS 92-16819, and SES 92-11937.This author's work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-5489 and by the UK Engineering and Physical Sciences Research Council under Grants J90855 and K17897.     

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed methods. The work of S. Wu was partially supported by the National Science Council, Taiwan, ROC (Grant No. 19731001). S.-C. Fang’s research has been supported by the US Army Research Office (Grant No. W911NF-04-D-0003) and National Science Foundation (Grant No. DMI-0553310).     

Rate of convergence of transport processes with an application to stochastic differential equations

Summary We obtain a rate of convergence of uniform transport processes to Brownian motion, which we apply to the Wong and Zakai approximation of stochastic integrals.The research of both authors was supported by a NSERC Canada Grant and by an EMR Canada Grant of M. Csörgö at Carleton University, Ottawa     

A note on the effect of numerical quadrature in finite element eigenvalue approximation

Summary In a recent work by the author and J.E. Osborn, it was shown that the finite element approximation of the eigenpairs of differential operators, when the elements of the underlying matrices are approximated by numerical quadrature, yield optimal order of convergence when the numerical quadrature satisfies a certain precision requirement. In this note we show that this requirement is indeed sharp for eigenvalue approximation. We also show that the optimal order of convergence for approximate eigenvectors can be obtained, using numerical quadrature with less precision.The author would like to thank Prof. I. Babuka for several helpful discussions. This work was done during the author's visit to the Institute of Physical Sciences and Technology and the Department of Mathematics of University of Maryland, College Park, MD 20742, USA, and was supported in part by the Office of Naval Research under Naval Research Grant N0001490-J-1030     

Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example. The first author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science and SRF for ROCS, SEM. The second author’s work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council grant. The third author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

Numerical solution of quasi-variational inequalities arising in stochastic game theory

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes.     

Continuous weak approximation for stochastic differential equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented.     

Global Convergence of a Robust Smoothing SQP Method for Semi-Infinite Programming

The semi-infinite programming (SIP) problem is a program with infinitely many constraints. It can be reformulated as a nonsmooth nonlinear programming problem with finite constraints by using an integral function. Due to the nondifferentiability of the integral function, gradient-based algorithms cannot be used to solve this nonsmooth nonlinear programming problem. To overcome this difficulty, we present a robust smoothing sequential quadratic programming (SQP) algorithm for solving the nonsmooth nonlinear programming problem. At each iteration of the algorthm, we need to solve only a quadratic program that is always feasible and solvable. The global convergence of the algorithm is established under mild conditions. Numerical results are given. Communicated by F. Giannessi His work was supported by the Hong Kong Research Grant Council His work was supported by the Australian Research Council.     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

Suboptimal shape control for quasi-static distributed-parameter systems

We propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed-parameter system theory and allows refocusing using a limited number of control actuators and sensors.The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. The controller gain calculation is computationally simpler than that proposed in a companion paper. The Galerkin (finite element) approximation method is used for model reduction. We prove that both gain and state convergence can be achieved by using the proposed approximation scheme.This work was partially supported by the Air Force Office of Scientific Research, Grant No. AFOSR 83-0124, and by the National Aeronautics and Space Administration, Grant No. NAG-1-515.     

An analysis of reduced Hessian methods for constrained optimization

We study the convergence properties of reduced Hessian successive quadratic programming for equality constrained optimization. The method uses a backtracking line search, and updates an approximation to the reduced Hessian of the Lagrangian by means of the BFGS formula. Two merit functions are considered for the line search: the ₁ function and the Fletcher exact penalty function. We give conditions under which local and superlinear convergence is obtained, and also prove a global convergence result. The analysis allows the initial reduced Hessian approximation to be any positive definite matrix, and does not assume that the iterates converge, or that the matrices are bounded. The effects of a second order correction step, a watchdog procedure and of the choice of null space basis are considered. This work can be seen as an extension to reduced Hessian methods of the well known results of Powell (1976) for unconstrained optimization.This author was supported, in part, by National Science Foundation grant CCR-8702403, Air Force Office of Scientific Research grant AFOSR-85-0251, and Army Research Office contract DAAL03-88-K-0086.This author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contracts W-31-109-Eng-38 and DE FG02-87ER25047, and by National Science Foundation Grant No. DCR-86-02071.